Thesis - Physics.cz

KARLOVA UNIVERZITA V PRAZE
MATEMATICKO-FYZIKÁLNÍ FAKULTA
RELATIVISTICKÉ DISKY
KOLEM KOMPAKTNÍCH OBJEKTŮ
DOKTORSKÁ DISERTAČNÍ PRÁCE
PETR SLANÝ
PRAHA 2005
Relativistic discs around compact objects
Petr Slaný
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava
Bezručovo nám. 13, 746 01 Opava, Czech Republic
Thesis submitted for the degree of Doctor Philosphiae
Faculty of Mathematics and Physics, Charles University in Prague
Supervisor: Prof. RNDr. Zdeněk Stuchlı́k, CSc.
July 2005
Typeset in LATEX 2ε
RODIČŮM, DANCE A KUBOVI
OBSAH
Abstrakt (in English)
Předmluva
1 Vliv repulzivnı́ kosmologické konstanty na akrečnı́ disky
1.1 Výchozı́ představy teorie akrečnı́ch disků . . . . . . .
1.2 Vliv kosmické repulze na tenké disky . . . . . . . . .
1.3 Vliv kosmické repulze na tlusté disky . . . . . . . . .
1.4 Shrnutı́ . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Aschenbachův efekt
2.1 Shrnutı́ . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference
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ABSTRACT
Presented doctoral thesis consists of the commented series of four original papers written
in collaboration with my supervisor Prof. Zdeněk Stuchlı́k and other our colleagues.
[1] Z. Stuchlı́k, P. Slaný and S. Hledı́k: Equilibrium configurations of perfect fluid orbiting
Schwarzschild–de Sitter black holes, Astronomy and Astrophysics 363, 425 (2000).
[2] Z. Stuchlı́k and P. Slaný: Equatorial circular orbits in the Kerr–de Sitter spacetimes,
Physical Review D 69, 064001 (2004).
[3] P. Slaný and Z. Stuchlı́k: Relativistic thick discs in the Kerr–de Sitter backgrounds,
accepted for publication in Classical and Quantum Gravity.
[4] Z. Stuchlı́k, P. Slaný, G. Török and M. A. Abramowicz: Aschenbach effect: Unexpected
topology changes in the motion of particles and fluids orbiting rapidly rotating Kerr
black holes, Physical Review D 71, 024037 (2005).
In Section 1 and papers [1–3], influence of a cosmic repulsion, represented by positive
value of the cosmological constant Λ, on accretion processes in black-hole (nakedsingularity) backgrounds is analysed. The presence of Λ > 0 substantially changes the
asymptotic structure of black-hole spacetimes, as these become asymptotically de Sitter.
In the framework of Schwarzschild–de Sitter (SdS) and Kerr–de Sitter (KdS) spacetimes,
the equatorial circular motion of test particles (Keplerian motion) and the toroidal equilibrium configurations of barotropic test perfect fluid orbiting with uniform distribution
of the specific angular momentum (`(r, θ ) = const), determined by the structure of equipotential surfaces in the fluid, are studied. We show that both the geometrically thin and
thick accretion discs must have an outer edge, enabling outflow of matter from the disc
into the outer space, located close to the static radius where the gravitational attraction
of a black hole is just compensated by the cosmological repulsion. Recall that no such an
outer edge is naturally defined in the case of accretion discs orbiting a single black hole
(naked singularity) in asymptotically flat Schwarzschild or Kerr background. Moreover,
cosmic repulsion enables the existence of a new type of stationary tori called the excretion
disc, in which the outflow through the outer cusp (edge) is only possible and is driven by
the same mechanism as the accretion in the case of thick accretion discs, i.e., by the violation of hydrostatic equilibrium. In addition, there is some indication that jets produced
by thick discs could be significantly collimated after crossing the static radius because
of the strong collimation of open equipotential surfaces along the rotational axis. In the
case of KdS naked singularities with the spacetime parameters sufficiently close to the
extreme black-hole state, the configurations with two separated tori can exist. The inner
disc is always a counterrotating accretion disc, while the outer one can be a corotating or
counterrotating excretion disc, as well as a counterrotating accretion disc.
v
vi
Relativistické disky kolem kompaktnı́ch objektů
In Section 2 and paper [4], the Aschenbach effect residing in non-monotonic behaviour
of the orbital velocity related to locally non-rotating frames (LNRF), as discovered by
Aschenbach in the case of Keplerian motion of test particles around rapidly rotating
Kerr black holes with spin a/M > 0.9953 [Aschenbach, 2004], is discussed. We show
that there is a corresponding behaviour of the orbital velocity relative to the LNRF
also for non-Keplerian motion of test perfect fluid tori (with uniform distribution of
the specific angular momentum) orbiting the Kerr black holes with a/M > 0.99979.
Positive radial gradient of the orbital velocity occurs just above the center of the tori, in
the region where stable equatorial circular geodesics of the Kerr spacetime are allowed.
Aschenbach proposed to relate the maximal positive value of the velocity gradient to the
frequency of possible oscillations of Keplerian discs. It is shown that the characteristic
frequencies related to the static observer at infinity reach for both the Keplerian discs
and fluid tori the maximal value for a certain spin of the Kerr black hole. The global
character of the phenomenon is given in terms of topology changes of the equivelocity
surfaces: in addition to the surface crossing itself under the inner edge of the torus, another
self-crossing surface together with toroidal equivelocity surfaces exist around the circle
corresponding to the minimum of the equatorial LNRF velocity profile. The whole effect
is located in a small region inside the ergosphere of a given spacetime.
Ačkoliv vesmı́r se vůbec nijak nezavázal, že by měl dávat nějaký smysl,
studenti v doktorandském studiu by o to usilovat měli.
Robert P. Kirchschner
Předkládaná doktorská práce sestává z komentovaného souboru čtyř původnı́ch článků odrážejı́cı́ch témata, jimž jsem se během svého doktorského studia věnoval v rámci vlastnı́ho výzkumu
pod vedenı́m Prof. RNDr. Zdeňka Stuchlı́ka, CSc., a na jejichž vzniku se ve dvou přı́padech
podı́leli i dalšı́ naši kolegové. Jedná se o tyto práce:
[1] Z. Stuchlı́k, P. Slaný a S. Hledı́k: Equilibrium configurations of perfect fluid orbiting
Schwarzschild–de Sitter black holes, Astronomy and Astrophysics 363, 425 (2000).
[2] Z. Stuchlı́k a P. Slaný: Equatorial circular orbits in the Kerr–de Sitter spacetimes, Physical
Review D 69, 064001 (2004).
[3] P. Slaný a Z. Stuchlı́k: Relativistic thick discs in the Kerr–de Sitter backgrounds, přijato
k publikovánı́ v Classical and Quantum Gravity.
[4] Z. Stuchlı́k, P. Slaný, G. Török a M. A. Abramowicz: Aschenbach effect: Unexpected
topology changes in the motion of particles and fluids orbiting rapidly rotating Kerr black
holes, Physical Review D 71, 024037 (2005).
Články [1–3] jsou věnovány studiu vlivu kosmické repulze, charakterizované kladnou hodnotou
kosmologické konstanty Λ, na akrečnı́ procesy v poli černých děr, přı́p. nahých singularit. Tato
problematika je blı́že diskutována v kapitole 1. Článek [4] se zabývá Aschenbachovým efektem,
projevujı́cı́m se jak u geodetického, tak i negeodetického pohybu hmoty anomálnı́m chovánı́m
gradientu orbitálnı́ rychlosti vztažené vůči lokálně nerotujı́cı́m systémům v přı́padě velmi rychle
rotujı́cı́ch Kerrových černých děr. Tento efekt je dále diskutován v kapitole 2.
Na tomto mı́stě bych rád poděkoval všem, kdo nemalou měrou přispěli k tomu, že moje
světočára procházı́ událostmi spojenými s těmito řádky. Na prvnı́m mı́stě jsou to rodiče, kteřı́
od útlého mládı́ ve mně probouzeli touhu po vzdělánı́ a hlubšı́m pochopenı́ řádu přı́rody. Za
jejich všestrannou podporu a možnost jı́t svou cestou jim patřı́ můj největšı́ dı́k. Mezi ty, kdo se
zasloužili o to, že mi fyzika nadobro přirostla k srdci, patřı́ zejména mı́ učitelé na všech stupnı́ch
škol, a taky strýc Eduard, jehož fyzikálnı́ myšlenı́ a rozhled pro mě byli tou pravou motivacı́,
proč dělat fyziku. Za to, že mı́sto metrologii, kterou jsem vystudoval na Přı́rodovědecké fakultě
Univerzity Palackého v Olomouci, se věnuji relativitě, vděčı́m Dr. Lukáši Richterkovi, kterému
se mimo jiné podařilo přesvědčit Prof. Jiřı́ho Bičáka, aby na půdě olomoucké Alma Mater
uspořádal seminář o kosmologii. O pár let později se situace opakovala, jen mı́sto Prof. Bičáka
poctil Olomouc svou přı́tomnostı́ Prof. Zdeněk Stuchlı́k, a aniž to kdokoliv z nás mohl tušit, jeho
přednáška o „rehabilitaci kosmologické konstanty“ již tenkrát předznamenala budoucı́ plodnou
vii
viii
Relativistické disky kolem kompaktnı́ch objektů
spolupráci, o nı́ž chci věřit, že potrvá, co nám oběma budou sı́ly stačit. Prof. Stuchlı́k je zdrojem
nepřeberného množstvı́ nápadů a inspiracı́ a bez jeho přispěnı́ by výsledky mé práce jen stěžı́
spatřily světlo světa v podobě publikovaných článků. Za vše, co pro mě doposud udělal a já měl
možnost se od něj naučit jsem mu velmi zavázán a děkuji mu za to. Mé dı́ky si zasloužı́ i všichni
mı́ studenti, s nimiž jsem měl tu možnost podělit se o své skromné znalosti a z nichž někteřı́
se posléze stali mými kolegy na doktorandském studiu a kamarády. Jejich všetečné dotazy
a postřehy pro mě byly živoucı́m impulsem k dalšı́ práci. Za kolegiálnı́ podporu, nesčetné
konzultace a okouzlenı́ Linuxem a LATEXem, speciálně pak za „latexovskou třı́du“ a mnoho
rad spojených s jejı́ úpravou pro účely této disertace, patřı́ velký dı́k Dr. Stanislavu Hledı́kovi.
Nakonec, avšak nikoliv nejmenšı́ měrou, děkuji za všemožnou podporu, pochopenı́ a velkou
mı́ru tolerance své ženě Daniele, která dala našemu životu nový řád – syna Jakuba.
Paskov, 10. července 2005
Petr Slaný
1 VLIV REPULZIVNÍ KOSMOLOGICKÉ KONSTANTY NA AKREČNÍ DISKY
Kosmologická pozorovánı́ uskutečněná v devadesátých letech 20. stoletı́ až po současnost přesvědčivě ukazujı́, že docházı́ ke zrychlujı́cı́ se expanzi současného vesmı́ru. Možné vysvětlenı́
tohoto jevu, silně podporované pestrou škálou nezávislých kosmologických testů (např. analýza anizotropie reliktnı́ho mikrovlnného zářenı́, měřenı́ vzdálenostı́ galaxiı́ s velkými rudými
posuvy pomocı́ supernov, měřenı́ současné hodnoty Hubbleova parametru, studium efektu gravitačnı́ čočky na kvasarech), je možné hledat v rámci inflačnı́ kosmologie bud’ v nenulové
energii vakua, tj. v nenulové, byt’velmi malé, kladné hodnotě kosmologické konstanty Λ, nebo
v působenı́ podobně se chovajı́cı́ nové formy hmoty, tzv. kvintesence. Obě možnosti bývajı́
často zahrnuty pod společný termı́n temná energie. Pozorovánı́ ukazujı́, že geometrie současného vesmı́ru je s velkou přesnostı́ plochá s odpovı́dajı́cı́ současnou hodnotou hustoty energie
vakua [Spergel et al., 2003]
%vac(0) ≈ 0.73%crit(0) ,
(1)
kde současná hodnota kritické hustoty energie %crit(0) je určena1 současnou hodnotou Hubbleova
parametru H0
3H02
, H0 = 100h km s−1 Mpc−1 .
(2)
8π
Vezmeme-li bezrozměrný parametr h ≈ 0.7, obdržı́me hodnotu „reliktnı́ “ repulzivnı́ kosmologické konstanty
%crit(0) =
Λ0 = 8π%vac(0) ≈ 1.3 × 10−56 cm−2 .
(3)
Je všeobecně známo, že repulzivnı́ kosmologická konstanta, Λ > 0, ovlivňuje dynamiku
vesmı́ru jako celku, vedoucı́ nakonec k exponenciálně rostoucı́ expanzi. Tato práce ukazuje, že
Λ > 0 může hrát významnou roli i v přı́padě astrofyzikálnı́ch procesů, jako je disková akrece
na supermasivnı́ černou dı́ru, ke které zřejmě docházı́ v jádrech aktivnı́ch galaxiı́.
1.1 Výchozı́ představy teorie akrečnı́ch disků
Základnı́ vlastnosti geometricky tenkých akrečnı́ch disků, vyznačujı́cı́ch se malými akrečnı́mi
toky a zanedbatelným vlivem tlakových gradientů v disku, jsou popsány geodetickým pohybem
testovacı́ch částic po stabilnı́ch kruhových orbitách v ekvatoriálnı́ rovině černoděrového prostoročasu [Novikov and Thorne, 1973]. Tyto disky se často nazývajı́ „Keplerovské“. V přı́padě
rotujı́cı́ch prostoročasů bylo ukázáno [Bardeen and Petterson, 1975], že jakkoliv skloněný disk
bude v důsledku efektu vlečenı́ inerciálnı́ch soustav nakonec naveden do ekvatoriálnı́ roviny
daného prostoročasu. Hmota v tenkém disku v podstatě spiráluje po stabilnı́ch kvazi-kruhových
orbitách, přitom ztrácı́ svůj moment hybnosti a energii (v důsledku třenı́ mezi sousednı́mi vrstvami diferenciálně rotujı́cı́ho disku) a propadá se hlouběji v gravitačnı́m poli centrálnı́ černé
dı́ry. Vnitřnı́ okraj disku ležı́ na nejvnitřnějšı́ stabilnı́ orbitě, r in ≈ rms , kde rms je poloměr
mezně stabilnı́ kruhové geodetiky, a maximálnı́ účinnost akrece odpovı́dá vazebné energii
1 V této práci jsou použı́vány: signatura metriky (−, +, +, +), geometrická soustava jednotek (c = G = 1) a
standardnı́ prostoročasové souřadnice (t, r, θ, ϕ) odpovı́dajı́cı́ ve statickém prostoročase Schwarzschildovým
souřadnicı́m a v rotujı́cı́m prostoročase Boyerovým–Lindquistovým souřadnicı́m.
1
2
Relativistické disky kolem kompaktnı́ch objektů
částice o jednotkové klidové hmotnosti na mezně stabilnı́ kruhové geodetice, η = 1 − E ms .
V přı́padě Keplerovského disku kolem Schwarzschildovy černé dı́ry η ≈ 0.057, zatı́mco pro
korotujı́cı́ Keplerovský disk obı́hajı́cı́ extrémnı́ Kerrovu černou dı́ru η ≈ 0.42. Připustı́me-li
existenci hypotetických nahých singularit, pak v přı́padě „pomalu rotujı́cı́ “ Kerrovy nahé singularity může účinnost akrece v korotujı́cı́m disku přesáhnout účinnost anihilačnı́ho procesu;
pro meznı́ hodnotu specifického momentu hybnosti Kerrovy nahé singularity (a/M → 1)
η ≈ 1.58 [Stuchlı́k, 1980].
Pokud tlakové gradienty v disku nejsou zanedbatelné, vznikne tzv. geometricky tlustý disk
charakteristický velkými akrečnı́mi toky. V tomto přı́padě již nelze modelovat disk pomocı́ geodetického pohybu hmoty, namı́sto toho základnı́ představy o chovánı́ a vlastnostech tlustých
disků lze zı́skat studiem rovnovážných konfiguracı́ dokonalé kapaliny obı́hajı́cı́ v čistě azimutálnı́m směru centrálnı́ černou dı́ru, přı́p. nahou singularitu. Je známo [Boyer, 1965, Abramowicz,
1974], že hranice jakékoliv stacionárnı́ konfigurace tvořené barotropnı́ dokonalou kapalinou
musı́ být ekvipotenciálnı́ plocha. Analytická plně relativistická teorie, splňujı́cı́ tuto „Boyerovu
podmı́nku“ pro testovacı́ disky v obecném stacionárnı́m a osově symetrickém prostoročase,
byla rozpracována Abramowiczem a jeho spolupracovnı́ky a aplikována na barotropnı́ disky
obı́hajı́cı́ Schwarzschildovu a Kerrovu černou dı́ru [Abramowicz et al., 1978, Kozłowski et al.,
1978,Jaroszyński et al., 1980]. Teorie je stručně rekapitulována v úvodnı́ch částech pracı́ [1, 3].
Jejı́ souřadnicově nezávislá formulace byla publikována v [Abramowicz, 1974].
Zvláštnı́ pozornost je věnována rovnovážným konfiguracı́m s konstantnı́m rozloženı́m specifického momentu hybnosti2 podél disku,
Uϕ (r, θ)
= const,
(4)
Ut (r, θ)
kde Ut a Uϕ jsou časová a azimutálnı́ složka kovariantnı́ 4-rychlosti, a to z několika důvodů.
Předně jsou tyto konfigurace mezně stabilnı́ s ohledem na obecnou podmı́nku stability barotropnı́ho tělesa vůči lokálnı́m osově symetrickým perturbacı́m, ∇` ≥ 0 [Seguin, 1975], dále
odpovı́dajı́ konfiguracı́m s maximálnı́ luminozitou [Abramowicz et al., 1980,Frank et al., 2002]
a v neposlednı́ řadě nejvı́ce odrážejı́ geometrické vlastnosti prostoročasu. Navı́c se dá očekávat, že v nejvnitřnějšı́ části blı́zce Keplerovského disku (r ≈ r ms ) bude ` ≈ const, nebot’ zde
d`/dr ≈ 0 [Kozłowski et al., 1978].
Plochy konstantnı́ho tlaku odpovı́dajı́ v přı́padě konfiguracı́ s `(r, θ) = const ekvipotenciálnı́m plochám potenciálu
`(r, θ) ≡ −
W (r, θ) = ln Ut (r, θ),
(5)
kde časová složka kovariantnı́ 4-rychlosti je obecně dána metrikou prostoročasu a funkcı́ `(r, θ):
Ut2
=
2
gtϕ
− gtt gϕϕ
gtt `2 + 2gtϕ ` + gϕϕ
;
(6)
je-li splněna podmı́nka (4), je funkce Ut (r, θ) zcela určena geometriı́ prostoročasu. Ekvipotenciálnı́ plochy mohou být trojı́ho typu: uzavřené – odpovı́dajı́cı́ toroidálnı́m konfiguracı́m, otevřené
– popisujı́cı́ dynamické procesy jako jsou např. výtrysky hmoty podél osy rotace [Lynden-Bell,
1969, Blandford, 1987] a sebeprotı́najı́cı́ (kritické) – umožňujı́cı́ ve specifických přı́padech od-
2 V raných pracı́ch Abramowicze a spol. se tato veličina nazývá „hustota momentu hybnosti“.
1. Vliv Λ > 0 na akrečnı́ disky
3
tok hmoty z disku skrz mı́sto překřı́ženı́ ekvipotenciály tzv. „Paczy ńského mechanismem“ (viz
dále).
Abramowicz et al. [1978] ukázali, že v černoděrových prostoročasech existujı́ stacionárnı́
toroidálnı́ konfigurace pro konstantnı́ specifický moment hybnosti ` > ` ms , kde `ms odpovı́dá
specifickému momentu hybnosti částice na mezně stabilnı́ kruhové geodetice v ekvatoriálnı́
rovině daného prostoročasu, nicméně ne všechny odpovı́dajı́ akrečnı́m diskům.
Pro
` < `ms
(7)
se v barotropnı́ dokonalé kapalině nevyskytujı́ toroidálnı́ ekvipotenciálnı́ plochy umožňujı́cı́
existenci stacionárnı́ch diskových struktur (Obr. 1a), je-li
` = `ms ,
(8)
dostáváme mezně uzavřenou kritickou ekvipotenciálu s kritickým bodem v ekvatoriálnı́ rovině
na mezně stabilnı́ kruhové geodetice, r crit = rms . Tato konfigurace popisuje disk v podobě
nekonečně tenkého nestabilnı́ho prstence na r = r ms . (Obr. 1b).
Jestliže
`ms < ` < `mb ,
(9)
kde `mb odpovı́dá specifickému momentu hybnosti částice na mezně vázané ekvatoriálnı́ kruhové geodetice, kritická ekvipotenciála je uzavřená a umožňuje akreci hmoty na centrálnı́ černou
dı́ru (Obr. 1c). Paczyńského mechanismus odtoku hmoty z disku vycházı́ z narušenı́ podmı́nky
hydrostatické rovnováhy při překročenı́ kritické ekvipotenciály povrchem disku [Kozłowski
et al., 1978]. Hmota nad kritickou ekvipotenciálou již dále nenı́ v mechanické rovnováze a přetéká přes mı́sto, kde se ekvipotenciála křı́žı́, na černou dı́ru. Situace je podobná té, k nı́ž docházı́
v těsných binárnı́ch systémech, kdy jedna ze složek vyplnı́ svůj „Rocheův lalok“ a hmota z nı́
začne přetékat skrz Lagrangeův bod L 1 na druhou složku. Poznamenejme ale, že na rozdı́l od
binárnı́ho systému, kde uvažujeme dva gravitujı́cı́ objekty, zde máme gravitujı́cı́ objekt pouze
jeden – centrálnı́ černou dı́ru (přı́p. nahou singularitu), disk je brán jako testovacı́. Navı́c, pro
disky kolem izolované Schwarzschildovy nebo Kerrovy černé dı́ry nemáme k dispozici přirozený mechanismus pro odtok hmoty z disku vně soustavu disk–černá dı́ra (analog Lagrangeova
bodu L 2 ), odnášejı́cı́ přebytečný moment hybnosti. Mı́sto protnutı́ kritické ekvipotenciály odpovı́dá vnitřnı́mu okraji disku, r in ≈ rcrit , který se dı́ky existenci tlakových gradientů v disku
posunul (oproti Keplerovským diskům) pod mezně stabilnı́ orbitu, r mb < rin < rms , kde rmb je
poloměr mezně vázané kruhové geodetiky. Uvnitř kritické ekvipotenciály se vyskytujı́ uzavřené
(toroidálnı́) ekvipotenciálnı́ plochy, odpovı́dajı́cı́ možným hranicı́m dalšı́ch toroidálnı́ch struktur, které bychom mohli nazývat „toroidálnı́ hvězdy“ (bez možnosti odtoku hmoty z „hvězdy“
řı́zeného Paczyńského mechanismem).
Když
` = `mb ,
(10)
kritická ekvipotenciála je mezně otevřená s kritickým bodem na mezně vázané orbitě, r crit =
rmb , a přı́slušný akrečnı́ disk dosahuje maximálnı́ tloušt’ky a teoreticky nekonečného rozpětı́
(Obr. 1d). Potenciálový rozdı́l mezi povrchem (daným kritickou ekvipotenciálou) a centrem
disku, ∆W = Wcrit − Wcent , dosahuje taktéž maximálnı́ hodnoty [Abramowicz et al., 1978].
4
Relativistické disky kolem kompaktnı́ch objektů
15
W>0
W=0
15
W<0
W>0
W=0
10
5
5
r cos Θ
r cos Θ
W<0
10
0
-5
-5
-10
-15
0
-10
{=3
(a)
5
10
20
15
r sin Θ
25
-15
!!!!!!!!!!!
{=3 3  2
(b)
30
5
15
10
15
W>0
W>0
W=0
10
20
15
r sin Θ
30
25
W<0
W=0
10
W<0
5
r cos Θ
r cos Θ
5
0
-5
-5
-10
-15
0
-10
{=3.84
(c)
5
10
15
20
15
r sin Θ
25
-15
{=4
(d)
30
5
15
W=0
10
W<0
5
r cos Θ
r cos Θ
30
W=0
10
W<0
5
0
-5
0
-5
-10
-15
25
W>0
W>0
10
20
15
r sin Θ
-10
{=4.5
(e)
5
10
20
15
r sin Θ
25
30
-15
{=5.3
(f)
5
10
20
15
r sin Θ
25
30
Obr. 1. Meridionálnı́ řezy ekvipotenciálnı́mi plochami v dokonalé kapalině s uniformně rozloženým
specifickým momentem hybnosti `(r, θ ) obı́hajı́cı́ Schwarzschildovu černou dı́ru (tmavý půlkruh).
Vystı́novaná oblast odpovı́dá výskytu uzavřených – toroidálnı́ch – ploch, křı́ž označuje centrum diskové
konfigurace. (a) ` < `ms , (b) ` = `ms , (c) `ms < ` < `mb , (d) ` = `mb , (e) `mb < ` < `ph(c) , (f)
` > `ph(c) .
1. Vliv Λ > 0 na akrečnı́ disky
5
V přı́padě Schwarzschildovy černé dı́ry
√ ∆W ≈ 0.043, zatı́mco pro korotujı́cı́ disk kolem
extrémnı́ Kerrovy černé dı́ry ∆W = ln 3 ≈ 0.55.
Je-li
`mb < ` < `ph(c) ,
(11)
kde `ph(c) odpovı́dá impaktnı́mu parametru fotonové kruhové geodetiky v ekvatoriálnı́ rovině,
kritická ekvipotenciála je otevřená a oddělená od uzavřených ekvipotenciál dalšı́mi otevřenými
(cylindrickými) plochami (Obr. 1e). Akrece prostřednictvı́m Paczy ńského mechanismu nenı́
možná a přı́slušné toroidálnı́ struktury odpovı́dajı́ „toroidálnı́m hvězdám“.
Přı́pad
` > `ph(c)
(12)
v podstatě odpovı́dá předchozı́ situaci s tı́m rozdı́lem, že kritická ekvipotenciála neexistuje
(Obr. 1f). Mı́sto, kde bychom očekávali překřı́ženou ekvipotenciálu totiž spadá do oblasti, v nı́ž
by se hmota s daným specifickým momentem hybnosti pohybovala nadsvětelnou rychlostı́,
a která je tudı́ž pro takovouto hmotu zakázaná.3
Obr. 1 znázorňuje meridionálnı́ řezy strukturou ekvipotenciálnı́ch ploch jednotlivých konfiguracı́ dokonalé kapaliny s výše diskutovanými hodnotami konstantnı́ho specifického momentu
hybnosti `, obı́hajı́cı́mi Schwarzschildovu černou dı́ru. Kvalitativně stejné profily ekvipotenciálnı́ch ploch bychom zı́skali i v přı́padě testovacı́ kapaliny obı́hajı́cı́ Kerrovu černou dı́ru.
1.2 Vliv kosmické repulze na tenké disky
Černoděrové prostoročasy s repulzivnı́ kosmologickou konstantou (Λ > 0) již nejsou asymptoticky ploché, nýbrž asymptoticky desitterovské. To mimo jiné způsobuje, že kromě horizontu
událostı́ černé dı́ry obsahujı́ i kosmologický horizont, za nı́mž je prostoročas dynamický, neumožňujı́cı́ existenci stacionárnı́ch pozorovatelů. Je-li charakter centrálnı́ singularity prostoročasu časupodobný, rozlišujeme obecně dva horizonty černé dı́ry – vnitřnı́ a vnějšı́, mezi nimiž
je prostoročas dynamický, nicméně pod vnitřnı́m horizontem je opět stacionárnı́. V dalšı́m se
omezı́me pouze na ty oblasti černoděrových prostoročasů, které ležı́ mezi horizontem událostı́
černé dı́ry (vnějšı́ horizont) a kosmologickým horizontem.
Tenké disky mohou existovat pouze v těch oblastech prostoročasu, v nichž je možný pohyb po
stabilnı́ch kruhových geodetikách. Ukazuje se, že v důsledku kosmické repulze existuje kromě
vnitřnı́ mezně stabilnı́ orbity, r ms(i) , také vnějšı́ mezně stabilnı́ orbita, r ms(o) , ležı́cı́ blı́zko tzv.
statického poloměru, což je mı́sto, kde kosmická repulze právě vyvažuje gravitačnı́ přitažlivost
centra. Statický poloměr je nestabilnı́ (vůči radiálnı́m perturbacı́m) kruhová orbita, na nı́ž má
geodetický pozorovatel nenulovou pouze časovou složku 4-rychlosti U t . Stabilnı́ kruhové orbity
ležı́ v oblasti mezi vnitřnı́ a vnějšı́ mezně stabilnı́ geodetikou, r ms(i) < r < rms(o) . Keplerovské
disky tedy majı́ v prostoročasech s repulzivnı́ kosmologickou konstantou přirozeným způsobem
definovány oba okraje disku: r in ≈ rms(i) , rout ≈ rms(o) . Maximálnı́ účinnost akrece je dána
rozdı́lem specifických energiı́ částice na vnějšı́ a vnitřnı́ mezně stabilnı́ orbitě,
η = E ms(o) − E ms(i) .
(13)
3 Výše uvedené nerovnosti (9)–(12) platı́ v přı́padě rotujı́cı́ (Kerrovy) černé dı́ry jen pro korotujı́cı́ disk, pro
protirotujı́cı́ disk (` < 0) je nutné v přı́slušných relacı́ch obrátit znaménka nerovnosti.
6
1.2.1
Relativistické disky kolem kompaktnı́ch objektů
Schwarzschildův–de Sitterův prostoročas
Vlastnosti Schwarzschildova–de Sitterova (SdS) prostoročasu zı́skané studiem chovánı́ „efektivnı́ch potenciálů“ řı́dı́cı́ch radiálnı́ pohyb testovacı́ch částic a fotonů, a analýzou únikových
světelných kuželů jsou uvedeny v článku [Stuchlı́k and Hledı́k, 1999], kde je souběžně diskutován i přı́pad „atraktivnı́ “ kosmologické konstanty (Λ < 0) popisujı́cı́ Schwarzschildovu–antide Sitterovu geometrii (SadS). Rozdı́ly v charakteru SdS a SadS prostoročasů oproti přı́padu
bez kosmologické konstanty (Schwarzschildův prostoročas), projevujı́cı́ se u SdS prostoročasu
zejména v blı́zkosti statického poloměru a dále až po kosmologický horizont, jsou znázorněny
pomocı́ vnořovacı́ch diagramů řezů konstantnı́ho času centrálnı́ rovinou obyčejné i optické
referenčnı́ geometrie. Práce [1] shrnuje a dále analyzuje ty vlastnosti SdS a SadS prostoročasů,
které majı́ přı́mou souvislost s tenkými a potažmo tlustými disky. Zde si co nejpřı́močařeji
naznačme způsob nalezenı́ vlastnostı́ kruhových orbit testovacı́ch částic v SdS prostoročase.
Geometrie SdS (a SadS) prostoročasu je popsána výrazem pro prostoročasový element
¶
µ
¶−1
µ
Λ
Λ
2M
2M
− r 2 dt 2 + 1 −
− r2
dr 2 + r 2 (dθ 2 + sin2 θdϕ 2 ),
ds 2 = − 1 −
r
3
r
3
(14)
kde M je celková hmotnost v centru. Mı́sto kosmologické konstanty Λ se s výhodou použı́vá
bezrozměrný kosmologický parametr
1
y = ΛM 2 ;
3
(15)
SdS prostoročasu pak odpovı́dá y > 0. Pro jednoduchost dále položı́me M = 1, čı́mž se
všechny ostatnı́ „geometrizované“ veličiny stanou bezrozměrnými. Horizonty událostı́ SdS
prostoročasu jsou dány pseudosingularitami metriky (14), konkrétně podmı́nkou
µ
¶
2
−1
2
grr ≡ 1 − − yr = 0,
(16)
r
vedoucı́ na relaci [Stuchlı́k and Hledı́k, 1999]
y = yh (r ) ≡
r −2
,
r3
(17)
která implicitně pro danou hodnotu kosmologického parametru y určuje polohu černoděrového
(rh ) a kosmologického (r c ) horizontu (Obr. 2). Prostoročas je v celé oblasti mezi horizonty,
rh < r < rc , statický. Analýza průběhu funkce yh (r ) vede ke zjištěnı́, že pro určitou kritickou
hodnotu kosmologického parametru,
yc(SdS) =
1 .
= 0.037,
27
(18)
oba horizonty splývajı́ na r = 3, kde, jak uvidı́me dále, se nacházı́ fotonová kruhová geodetika.
Pro y > yc(SdS) je prostoročas na všech r > 0 dynamický s nahou singularitou v centru.
S existencı́ Killingových vektorových polı́ X(t) = ∂/∂t a X(ϕ) = ∂/∂ϕ jsou spojeny pohybové
konstanty E = −P·X(t) a Φ = P·X(ϕ) , odpovı́dajı́cı́ průmětům 4-hybnosti částice P na přı́slušné
Killingovo vektorové pole. Jelikož SdS prostoročas nenı́ asymptoticky plochý, nelze pohybové
konstanty E a Φ interpretovat způsobem známým ze Schwarzschildova prostoročasu, tedy jako
energii a axiálnı́ složku momentu hybnosti částice v nekonečnu.
1. Vliv Λ > 0 na akrečnı́ disky
7
Pohyb testovacı́ch částic a fotonů probı́há po geodetikách daného prostoročasu, určených
podmı́nkami
∇P P = 0,
(19)
2
P · P = −m ,
(20)
kde m je klidová hmotnost částic (v přı́padě fotonů m = 0). Vzhledem ke sférické symetrii
prostoročasu bude pohyb konkrétnı́ částice probı́hat v některé z centrálnı́ch rovin; s výhodou
volı́me ekvatoriálnı́ rovinu (θ = π/2). V dalšı́m si zaved’me specifickou energii a specifický
moment hybnosti testovacı́ částice a impaktnı́ parametr fotonové trajektorie 4 relacemi
E
Φ
Φ
, L≡ , `≡ .
(21)
m
m
E
Pohyb testovacı́ch částic (m 6= 0) v ekvatoriálnı́ rovině SdS prostoročasu je určen „efektivnı́m
potenciálem“ [Stuchlı́k, 1983]
µ
¶
¶µ
2
L2
2
Vef (r ; y, L) ≡ 1 − − yr
(22)
1+ 2
r
r
E≡
a je povolen pouze v těch oblastech, kde E 2 ≥ Vef (r ; y, L); rovnost odpovı́dá bodům obratu
radiálnı́ho pohybu testovacı́ch částic.
Pohyb fotonů (m = 0) v ekvatoriálnı́ rovině SdS prostoročasu je dán „efektivnı́m potenciálem“ [Stuchlı́k and Hledı́k, 1999]
`2ph (r ; y) ≡
r3
r − 2 − yr 3
(23)
a je omezen pouze na ty oblasti, v nichž impaktnı́ parametr ` fotonové dráhy slňuje relaci:
`2 ≤ `2ph (r ; y); rovnost odpovı́dá bodům obratu radiálnı́ho pohybu fotonů.
Lokálnı́ extrémy efektivnı́ho potenciálu (22), tj. mı́sta, kde ∂ Vef /∂r = 0, určujı́ polohu kruhových orbit testovacı́ch částic; minima (maxima) odpovı́dajı́ stabilnı́m (nestabilnı́m) kruhovým
orbitám. Specifická energie a specifický moment hybnosti částice na kruhové orbitě jsou dány
výrazy [Stuchlı́k, 1983]
r − 2 − yr 3
E(r ; y) = √
,
r (r − 3)
s
1 − yr 3
L(r ; y) = r
,
r −3
(24)
(25)
z nichž je zřejmé, že kruhové orbity testovacı́ch částic mohou existovat pouze v oblasti
3 < r ≤ rs (y) ≡ y −1/3 ,
(26)
kde rs (y) odpovı́dá statickému poloměru SdS prostoročasu. Zdola je přı́pustná oblast omezena
fotonovou kruhovou orbitou, jak je patrno z průběhu efektivnı́ho potenciálu (23) řı́dı́cı́ho pohyb
fotonů – má zde jediný lokálnı́ extrém (minimum), přı́p. z chovánı́ funkcı́ (24), (25), které na
4 Veličina ` může taktéž charakterizovat pohyb testovacı́ch částic. V tom přı́padě ji ovšem trochu nešt’astně taky
nazýváme „specifický moment hybnosti“ s přesvědčenı́m, že nedojde k záměně s veličinou L. V teorii tlustých
disků je to právě `, co charakterizuje rotaci elementů kapaliny.
8
Relativistické disky kolem kompaktnı́ch objektů
0.01
y
0.001
0.0001
0.00001
1
5
10
50 100
r
Obr. 2. Charakteristické poloměry SdS prostoročasů a oblasti stability kruhových orbit v závislosti na
kosmologickém parametru y. Plná křivka určuje polohu černoděrového a kosmologického horizontu,
čerchovaně je vyznačen statický poloměr, tečkovaně kruhová fotonová orbita a čárkovaně polohy mezně
vázaných orbit. Stabilnı́ kruhové orbity existujı́ pouze ve tmavě vystı́nované oblasti (jejı́ hranice je
tvořena mezně stabilnı́mi orbitami), světle vystı́novaná oblast obsahuje pouze nestabilnı́ kruhové orbity.
r = 3 divergujı́. Poznamenejme, že radiálnı́ souřadnice fotonové orbity v SdS prostoročase,
rph(c) = 3, je dána nezávisle na kosmologickém parametru y a odpovı́dá radiálnı́ souřadnici
fotonové orbity ve Schwarzschildově prostoročase.
Funkce (24) a (25) majı́ společné lokálnı́ extrémy, které odpovı́dajı́ polohám mezně stabilnı́ch
orbit. Ty jsou implicitně určeny relacı́ [Stuchlı́k and Hledı́k, 1999]
y = yms (r ) ≡
r −6
.
− 15)
r 3 (4r
(27)
Analýza funkce yms (r ) ukazuje, že existuje jistá meznı́ hodnota kosmologického parametru
připouštějı́cı́ existenci stabilnı́ch kruhových orbit
12 .
= 0.00024;
(28)
154
pro 12/154 < y < 1/27 jsou všechny kruhové orbity nestabilnı́ vůči radiálnı́m perturbacı́m.
Je-li 0 < y < 12/154 , lze oblast kruhových geodetik (26) rozdělit podle jejich stability na tři
na sebe navazujı́cı́ podoblasti, oddělené vnitřnı́ resp. vnějšı́ mezně stabilnı́ orbitou, r ms(i) resp.
rms(o) (Obr. 2):
yc(ms) =
(i) oblast nestabilnı́ch kruhových geodetik: 3 < r < r ms(i) (y),
(ii) oblast stabilnı́ch kruhových geodetik: r ms(i) (y) < r < rms(o) (y),
(iii) oblast nestabilnı́ch kruhových geodetik: r ms(o) (y) < r ≤ rs (y).
Statický poloměr je tedy nestabilnı́ kruhová geodetika, kde navı́c L = 0.
Pro teorii tlustých disků hraje důležitou roli poloha mezně vázané orbity. Stejně jako v přı́padě
mezně stabilnı́ch orbit i v tomto přı́padě existujı́ dvě mezně vázané orbity – vnitřnı́ a vnějšı́,
rmb(i) a rmb(o) , a to pouze v prostoročasech připouštějı́cı́ch stabilnı́ kruhové orbity (y < y c(ms) ).
Polohu statického poloměru a polohy mezně stabilnı́ch a mezně vázaných orbit v závislosti na
kosmologickém parametru y znázorňuje Obr. 2. Meznı́ vázané orbity jsou nestabilnı́ geodetiky,
jejichž poloha je určena rovnostı́ obou lokálnı́ch maxim efektivnı́ho potenciálu (22) pro dané y
Vef (rmb(i) ; y, L mb ) = Vef (rmb(o) ; y, L mb ),
(29)
1. Vliv Λ > 0 na akrečnı́ disky
1.1
y=10-6
1.05
Vef Hr,LL
9
1
0.95
0.9
mbHiL
0.85
mbHoL
msHiL
0
0.5
1
msHoL
1.5
log r
2
2.5
Obr. 3. Průběh „efektivnı́ho potenciálu testovacı́ch částic“ v ekvatoriálnı́ rovině SdS prostoročasu (y =
10−6 ) v závislosti na specifickém momentu hybnosti částice L. Lokálnı́ minima (maxima) odpovı́dajı́
stabilnı́m (nestabilnı́m) kruhovým geodetikám. Čárkovaná (čerchovaná) křivka definuje vnitřnı́ (vnějšı́)
mezně stabilnı́ kruhovou geodetiku, ms(i) a ms(o), plná černá křivka splňujı́cı́ podmı́nku (29) určuje
polohu vnitřnı́ a vnějšı́ mezně vázané kruhové geodetiky, mb(i) a mb(o). Šedé křivky znázorňujı́ průběh
efektivnı́ho potenciálu pro L < L ms(i) , L ms(i) < L < L mb , L mb < L < L ms(o) a L > L ms(o) . Statický
poloměr uvažovaného SdS prostoročasu ležı́ na r = 100.
kde L mb je přı́slušná hodnota specifického momentu hybnosti vedoucı́ na rovnost (29). Charakteristické chovánı́ efektivnı́ho potenciálu (22) v závislosti na specifickém momentu hybnosti
částice je zobrazeno na Obr. 3, kde efektivnı́ potenciály, definujı́cı́ vnitřnı́ a vnějšı́ mezně stabilnı́ a mezně vázané orbity, jsou speciálně vyznačeny. Všimněme si kvalitativně odlišného
chovánı́ efektivnı́ho potenciálu v blı́zkosti statického poloměru (relace (26)) a za nı́m ve srovnánı́ s dobře známým průběhem odpovı́dajı́cı́ho efektivnı́ho potenciálu v asymptoticky plochém
Schwarzschildově prostoročase (y = 0).
Energie částic na vnitřnı́ a vnějšı́ mezně stabilnı́ orbitě, E ms(i) a E ms(o) , je určena implicitně
relacemi (24) a (27). Jejich průběh v závislosti na kosmologickém parametru y ukazuje Obr. 4a.
Obr. 4b graficky znázorňuje závislost účinnosti akrece η, definované relacı́ (13), na kosmologickém parametru. Vidı́me, že vliv kosmické repulze se projevuje ve snı́ženı́ maximálnı́ účinnosti
akrece, které je, nicméně, patrné až pro většı́ hodnoty kosmologického parametru y.
1.2.2
Kerrův–de Sitterův prostoročas
Studium pohybu testovacı́ch částic a fotonů v SdS prostoročase nám pomohlo odhalit projevy
kosmické repulze v nejjednoduššı́m černoděrovém prostoročase, nicméně pro astrofyzikálnı́
aplikace je důležité analyzovat vliv kosmické repulze na pohyb testovacı́ch částic a fotonů
v rotujı́cı́m (Kerrově) černoděrovém prostoročase. Práce [2] podrobným způsobem rozebı́rá
pohyb testovacı́ch částic v ekvatoriálnı́ rovině Kerrova–de Sitterova prostoročasu, byt’částečně
se věnuje i pohybu fotonů. Detailnı́ analýzu pohybu fotonů v ekvatoriálnı́ rovině obecnějšı́ho
Kerrova–Newmanova–de Sitterova prostoročasu je možno najı́t v článku [Stuchlı́k and Hledı́k,
2000], kde je pro úplnost uvažován i anti-desitterovský přı́pad „atraktivnı́“ kosmologické
konstanty (Λ < 0). Na tomto mı́stě si shrneme a okomentujeme nejdůležitějšı́ výsledky plynoucı́
z práce [2].
Geometrie Kerrova–de Sitterova (KdS) prostoročasu v Boyerových – Lindquistových souřadnicı́ch je popsána výrazem
10
Relativistické disky kolem kompaktnı́ch objektů
0.99
0.05
0.98
0.04
0.97
0.03
Η
Ems
1
0.96
0.02
0.95
0.01
0.94
0.93
-12
-10
-8
log y
(a)
-6
-4
0
-12
-10
-8
log y
-6
-4
(b)
Obr. 4. (a) Specifická energie částice na vnitřnı́ (dolnı́ křivka) resp. vnějšı́ (hornı́ křivka) mezně stabilnı́
√
orbitě kolem SdS černé dı́ry v závislosti na kosmologickém parametru
y. Pro y → 0, E ms(i) → 2 2/3 a
√
√
E ms(o) → 1, zatı́mco pro y → 12/154 , E ms(o) → E ms(i) → 6 3/5 5. (b) Maximálnı́ účinnost akrece
daná rozdı́lem specifických energiı́ částice na vnějšı́ a vnitřnı́ mezně stabilnı́ orbitě kolem SdS černé dı́ry
v závislosti na kosmologickém parametru y. Pro y → 0, η → 0.057, kdežto pro y → 12/15 4 , η → 0.
ds 2 = −
kde
¡ 2
¢ ¤2 ρ 2 2 ρ 2 2
∆θ sin2 θ £
∆r
2
2
2
(dt
−
a
sin
θdϕ)
+
dr +
dθ , (30)
adt
−
r
+
a
dϕ +
I 2ρ 2
I 2ρ 2
∆r
∆θ
¡
¢
1
∆r = − Λr 2 r 2 + a 2 + r 2 − 2Mr + a 2 ,
(31)
3
1
1
(32)
∆θ = 1 + Λa 2 cos2 θ, I = 1 + Λa 2 , ρ 2 = r 2 + a 2 cos2 θ;
3
3
hmotnostnı́ parametr M a rotačnı́ parametr a odpovı́dajı́ celkové hmotnosti a specifickému
momentu hybnosti centra. Opět je výhodné zavést kosmologický parametr y vztahem (15) a
položit M = 1.
Horizonty KdS prostoročasu jsou implicitně dány podmı́nkou ∆r = 0, vedoucı́ na relaci
y = yh (r ; a) ≡
r 2 − 2r + a 2
.
r 2 (r 2 + a 2 )
(33)
Analýza chovánı́ funkce yh (r ; a) v závislosti na hodnotě rotačnı́ho parametru a vede ke zjištěnı́,
že existuje jistá kritická hodnota
q
√
3(3 + 2 3) .
acrit =
= 1.101
(34)
4
taková, že pro a > acrit funkce yh (r ; a) nemá lokálnı́ extrém, je pro všechna r > 0 kladná
a monotonně klesajı́cı́, což pro všechna y > 0 indikuje existenci pouze jediného horizontu –
kosmologického. Poloha inflexnı́ho bodu funkce yh (r ; a) (odpovı́dajı́cı́ a = acrit ) udává meznı́
(kritickou) hodnotu kosmologického parametru připouštějı́cı́ existenci černých děr, 5
yc(KdS) =
16
.
= 0.059.
√
3
(3 + 2 3)
5 Ve stejném duchu je a
crit meznı́ hodnotou rotačnı́ho parametru připouštějı́cı́ existenci černých děr.
(35)
1. Vliv Λ > 0 na akrečnı́ disky
0.1
KdS
0
0.98
1
1.03
acrit U 1.101
1.2
0.075
yh Hr; aL
11
0.05
0.025
0
-0.025
-0.05
KadS
0
2
4
6
8
10
r
Obr. 5. Funkce horizontů KdS prostoročasů. Hodnoty rotačnı́ho parametru a pro jednotlivé křivky udává
legenda. Záporné hodnoty funkce yh (r ; a) určujı́ horizonty Kerrových–anti-de Sitterových prostoročasů
(KadS).
Pro a > acrit (nebo y > yc(KdS) ) popisuje (30) pouze nahé singularity. Pro a < acrit má funkce
yh (r ; a) dva lokálnı́ extrémy, ymin (a) a ymax (a), umožňujı́cı́ existenci až třı́ horizontů – vnitřnı́
a vnějšı́ černoděrový a kosmologický. Černé dı́ry existujı́ pro kosmologické parametry splňujı́cı́
relaci ymin (a) < y < ymax (a). Poznamenejme, že pro 0 < a < 1 je ymin (a) < 0. Stručnou
diskusi extrémnı́ch přı́padů, kdy některé dva, přı́p. všechny tři horizonty splývajı́, lze nalézt
v pracı́ch [2, 3]. Průběh funkce yh (r ; a) pro některé hodnoty rotačnı́ho parametru a ukazuje
Obr. 5. Stacionárnı́ oblasti prostoročasu jsou dány podmı́nkou y < y h (r ; a) a na rozdı́l od
asymptoticky ploché oblasti Kerrova prostoročasu je vliv rotace KdS prostoročasu (projevujı́cı́
se vlečenı́m lokálnı́ch inerciálnı́ch systémů) významný i v blı́zkosti kosmologického horizontu,
tedy v celé stacionárnı́ oblasti prostoročasu.
Rozbor rovnic geodetiky (19), (20) v Carterově tvaru [Carter, 1973] vede k nalezenı́ „efektivnı́ho potenciálu“ řı́dı́cı́ho pohyb testovacı́ch částic v ekvatoriálnı́ rovině (θ = π/2) KdS
prostoročasu. Analogicky jako v přı́padě SdS prostoročasu souvisı́ se stacionaritou a osovou
symetriı́ KdS prostoročasu existence přı́slušných pohybových konstant E a Φ, které ze stejného
důvodu jako v SdS prostoročase nelze ovšem interpretovat jako energii a axiálnı́ složku momentu hybnosti částice v nekonečnu. Definujeme-li specifickou energii a specifický moment
hybnosti částice relacemi
IE
IΦ
, L≡
(36)
m
m
a zavedeme-li nový axiálnı́ parametr (odpovı́dajı́cı́ předefinovanému specifickému momentu
hybnosti)
E≡
X ≡ L − a E,
(37)
lze efektivnı́ potenciál psát ve velmi jednoduchém tvaru
·
¸
q ¡
¢
1
2
2
E (+) (r ; a, y, X ) ≡ 2 a X + ∆r r + X
.
r
(38)
Pohyb je povolen pouze v těch oblastech, kde E ≥ E (+) (r ; a, y, X ); rovnost odpovı́dá bodům
obratu radiálnı́ho pohybu testovacı́ch částic.
Lokálnı́ extrémy efektivnı́ho potenciálu, tj. mı́sta, kde ∂ E (+) /∂r = 0, určujı́ polohu ekvatoriálnı́ch kruhových orbit testovacı́ch částic, a opět minima odpovı́dajı́ stabilnı́m a maxima
12
Relativistické disky kolem kompaktnı́ch objektů
nestabilnı́m kruhovým geodetikám. Specifická energie a specifický moment hybnosti částice
na kruhové orbitě jsou dány výrazy
¡
¢
¡
¢1/2
r 3/2 − 2r 1/2 − r 3/2 r 2 + a 2 y ± a 1 − yr 3
(39)
E ± (r ; a, y) =
h
¡
¢
¡
¢1/2 i1/2 ,
3/2
2
1/2
3
3/4
r
1 − a y − 3r ± 2a 1 − yr
r
¡ 2
¢
¡
¢1/2 £
¡
¢ ¤
r + a 2 1 − yr 3
∓ 2ar 1/2 + ar 3/2 r 2 + a 2 y
L ± (r ; a, y) = ±
(40)
h
¡
¢
¡
¢1/2 i1/2 ,
3/4
3/2
2
1/2
3
r
r
1 − a y − 3r ± 2a 1 − yr
které majı́ tu výhodu, že v limitě y → 0 přecházejı́ přı́mo na výrazy pro energii a moment
hybnosti částice na kruhové orbitě v Kerrově prostoročase publikované v článku [Bardeen et al.,
1972] (provedeme-li v nich limitu M → 1). V práci [2] je použito jejich alternativnı́ vyjádřenı́,
které v limitě y → 0 přecházı́ na odpovı́dajı́cı́ kerrovské výrazy uvedené v [Chandrasekhar,
1983]. Podobně jako v Kerrově prostoročase i v KdS prostoročase existujı́ dvě třı́dy orbit
označené plus a minus podle znamének ± v relacı́ch (39) a (40).
Podmı́nky existence kruhových orbit obou třı́d jsou podrobně diskutovány v článku [2].
„Minusová“ třı́da orbit existuje jen v KdS prostoročasech, jejichž kosmologický parametr
vyhovuje podmı́nce
1 .
= 0.037
27
a jejichž rotačnı́ parametr je omezen relacı́
y < yc(SdS) =
2
a 2 ≤ ac(s)
(y) ≡
1 − 3y 1/3
.
y
(41)
(42)
Oblast výskytu „minusových“ orbit je omezena zdola protirotujı́cı́ fotonovou kruhovou geodetikou, jejı́ž poloha je nynı́ funkcı́ obou prostoročasových parametrů a, y, a shora statickým
poloměrem rs (y) ≡ y −1/3 , jehož poloha je určena stejným výrazem jako v SdS geometrii
(relace (26)), nezávislým na rotačnı́m parametru KdS prostoročasu.
Orbity „plusové“ třı́dy existujı́ ve všech KdS prostoročasech. Speciálně v prostoročasech,
jejichž parametry splňujı́ relace (41) a (42), jsou „plusové“ orbity kolem černých děr 6 omezeny
zdola korotujı́cı́ fotonovou kruhovou geodetikou a shora statickým poloměrem, v přı́padě
nahých singularit je přı́pustná oblast jejich výskytu omezena pouze shora, a to statickým
poloměrem. V prostoročasech nepřipouštějı́cı́ch „minusové“ orbity (což jsou bud’prostoročasy,
jejichž kosmologický parametr sice splňuje relaci (41), ale jejich rotačnı́ parametr je většı́ než
kritická hodnota ac(s) (y), nebo prostoročasy s kosmologickým parametrem většı́m než y c(SdS) )
je hornı́ mez výskytu „plusových“ orbit dána nikoliv statickým poloměrem, nýbrž protirotujı́cı́
fotonovou kruhovou geodetikou; dolnı́ mez pak opět tvořı́ v přı́padě černých děr (ty mohou
existovat pouze v KdS prostoročasech s y < yc(KdS) ) korotujı́cı́ fotonová kruhová geodetika,
v přı́padě nahých singularit je omezenı́ dáno prstencovou singularitou (r = 0, θ = π/2).
V prostoročasech připouštějı́cı́ch „minusové“ orbity obě třı́dy splývajı́ na statickém poloměru,
a jen v těchto prostoročasech má statický poloměr fyzikálnı́ význam.
Analýza stability kruhových orbit vůči radiálnı́m perturbacı́m vede k závěru, že pro každou
třı́du orbit existuje jistá maximálnı́ hodnota kosmologického parametru, pro niž již neexistujı́
6 Při diskuzi černoděrových prostoročasů se omezı́me pouze na orbity ve stacionárnı́ oblasti mezi vnějšı́m horizon-
tem černé dı́ry a kosmologickým horizontem.
1. Vliv Λ > 0 na akrečnı́ disky
NS(0)
-1
log y
-2
ČD(0)
E+
-3
NS(+)
<
ČD(+)
0
-4
-5
13
ČD(±)
0.5
NS(±)
1
1.5
a2
2
2.5
Obr. 6. Rozdělenı́ KdS prostoročasů podle výskytu stabilnı́ch ekvatoriálnı́ch kruhových orbit testovacı́ch
částic. Čárkovaná křivka odděluje černoděrové prostoročasy (ČD) od prostoročasů s nahou singularitou
(NS). Výskyt stabilnı́ch kruhových orbit dané třı́dy je charakterizován odpovı́dajı́cı́m znaménkem (±);
(0) označuje prostoročasy bez stabilnı́ch kruhových orbit. Vystı́novaná oblast odpovı́dá NS prostoročasům, v nichž mohou být „plusové“ stabilnı́ orbity protirotujı́cı́, čerchovaně je vyznačena podoblast
prostoročasů připouštějı́cı́ch protirotujı́cı́ stabilnı́ „plusové“ orbity se zápornou energiı́.
stabilnı́ orbity dané třı́dy. Tyto maximálnı́ hodnoty pro „minusovou“ resp. „plusovou“ třı́du
jsou:
yc(ms−) =
12 .
= 0.00024
154
resp.
yc(ms+) =
100
.
= 0.069.
√
3
(5 + 2 10)
(43)
Obr. 6 názorně ukazuje v KdS parametrickém prostoru (y, a 2 ) výskyt stabilnı́ch ekvatoriálnı́ch
kruhových orbit jednotlivých třı́d. Podobně jako v SdS prostoročase i zde existujı́ u každé třı́dy
dvě mezně stabilnı́ orbity – vnitřnı́ a vnějšı́, r ms(i) a rms(o) , vymezujı́cı́ oblast stabilnı́ch orbit
dané třı́dy.
V KdS prostoročase připouštějı́cı́m stabilnı́ kruhové geodetiky dané třı́dy můžeme pro tuto
třı́du orbit dále definovat vnitřnı́ a vnějšı́ mezně vázanou orbitu, r mb(i) a rmb(o) , pomocı́ rovnosti
lokálnı́ch maxim efektivnı́ho potenciálu (38)
E (+) (rmb(i) ; a, y, X mb ) = E (+) (rmb(o) ; a, y, X mb ),
(44)
kde X mb je přı́slušná hodnota axiálnı́ho parametru X vedoucı́ pro danou třı́du orbit na rovnost
(44). Mezně vázané orbity jsou nestabilnı́ kruhové geodetiky.
Poloha jednotlivých astrofyzikálně důležitých orbit v KdS prostoročasech, jejichž kosmologický parametr vyhovuje relaci y < yc(ms−) = 12/154 , je kvalitativně shodná se situacı́ zobrazenou na Obr. 7. Z obrázku je patrno, že při přechodu od černých děr k nahým singularitám
docházı́ u vnitřnı́ i vnějšı́ mezně vázané orbity „plusové“ třı́dy ke skoku v jejı́ poloze. V přı́padě
KdS prostoročasů s kosmologickým parametrem 12/15 4 ≤ y < yc(ms+) (tyto prostoročasy
nepřipouštějı́ stabilnı́, ani vázané ekvatoriálnı́ kruhové orbity „minusové“ třı́dy) je odpovı́dajı́cı́
vzájemná poloha zbylých orbit analogická té, znázorněné na Obr. 7, s tı́m, že mezně stabilnı́
a mezně vázané „plusové“ orbity existujı́ až od jisté minimálnı́ hodnoty rotačnı́ho parametru
a > 0, viz oblasti ČD(+) a NS(+) na Obr. 6.
Z rozboru vlastnostı́ ekvatoriálnı́ch kruhových geodetik kolem Kerrovy černé dı́ry je známo
[Bardeen et al., 1972], že „plusová“ třı́da odpovı́dá korotujı́cı́m, zatı́mco „minusová“ třı́da
protirotujı́cı́m orbitám z pohledu stacionárnı́ho pozorovatele v nekonečnu. Poznamenejme,
14
Relativistické disky kolem kompaktnı́ch objektů
40 ČD
35
NS
statický poloměr
mb(o)+
ČD
30
y = 2 × 10−5
mb(o)−
30
25
ms(o)+
ms(o)−
r
r
20
20
15
10
10
ms(i)−
mb(i)−
5
0
0
2
4
6
8
10
12
0
a
0.5
a
1
ph−
ms(i)+
mb(i)+
ph+
bh+
Obr. 7. Vzájemná poloha astrofyzikálně významných orbit v ekvatoriálnı́ rovině KdS prostoročasů
v závislosti na rotačnı́m parametru a pro typicky malé hodnoty kosmologického parametru y < 12/15 4 .
Levý obrázek pokrývá prostoročasy černých děr (ČD) i nahých singularit (NS), pravý obrázek přibližuje
situaci v černoděrových prostoročasech odděleně. Plná křivka znázorňuje polohu vnějšı́ho horizontu
černé dı́ry (bh+), tečkovaně je znázorněna poloha fotonových (ph) orbit, čárkovaně poloha mezně
vázaných (mb) orbit a čerchovaně poloha mezně stabilnı́ch (ms) orbit. Orbity „minusové“ třı́dy jsou
znázorněny tučně.
že ke stejnému závěru dojdeme, určı́me-li směr orbit z pohledu lokálně nerotujı́cı́ch soustav
(LNRF). Stuchlı́k [1980] ale ukázal, že podobná klasifikace obecně neplatı́ v přı́padě Kerrových
√
nahých singularit, kde pro dostatečně malé hodnoty rotačnı́ho parametru, 1 < a < 3 3/4,
existujı́ v blı́zkosti prstencové singularity „plusové“ orbity, které jsou protirotujı́cı́.
V KdS prostoročase, jelikož nenı́ asymptoticky plochý, můžeme směr orbit určit pouze
s pomocı́ LNRF. V práci [2] je ukázáno, že znaménko lokálně měřené azimutálnı́ složky 4hybnosti (v LNRF) je dáno znaménkem pohybové konstanty L, jak bychom ostatně intuitivně
předpokládali. Orbitu nazýváme korotujı́cı́ resp. protirotujı́cı́, je-li L > 0 resp. L < 0 pro danou
orbitu. Zde si uvedeme ekvivalentnı́ kriterium, které využı́vá porovnánı́ úhlových rychlostı́,
Ω = dϕ/dt, částice na kruhové orbitě,
ΩK± = ±
r 3/2 /(1
1
,
− yr 3 )1/2 ± a
(45)
a lokálně nerotujı́cı́ho pozorovatele v ekvatoriálnı́ rovině (θ = π/2)
a(r 2 + a 2 − ∆r )
.
(46)
(r 2 + a 2 )2 − a 2 ∆r
Přı́slušná orbita je korotujı́cı́ resp. protirotujı́cı́, je-li ΩK > ΩLNRF resp. ΩK < ΩLNRF . Ukazuje
se, že v KdS prostoročase ani v přı́padě černých děr nelze konkrétnı́ třı́du jednoznačně spojit
s určitým směrem oběhu. Podobně jako v Kerrově prostoročase i v KdS prostoročase jsou orbity
„minusové“ třı́dy pouze protirotujı́cı́. Orbity „plusové“ třı́dy jsou většinou korotujı́cı́, nicméně
opět existujı́ i protirotujı́cı́ „plusové“ orbity. Ve všech KdS prostoročasech (černoděrových
ΩLNRF =
1. Vliv Λ > 0 na akrečnı́ disky
15
i s nahou singularitou) k nim patřı́ orbity blı́zko hornı́ hranice oblasti výskytu „plusových“ orbit
(tj. blı́zko statického poloměru, či protirotujı́cı́ fotonové orbity); tyto orbity jsou nestabilnı́.
Navı́c v přı́padě nahých singularit s dostatečně malými hodnotami rotačnı́ho parametru,
amin(ns) (y) < a < amax(L=0) (y),
(47)
kde v závislosti na kosmologickém parametru y < yc(ms+) nabývá spodnı́ resp. hornı́ mez
intervalu hodnot
√
√
√
.
.
(48)
1 ≤ amin(ns) (y) . 1.41 = 1.19 resp. 3 3/4 ≤ amax(L=0) (y) . 2.44 = 1.56,
existujı́ taktéž v blı́zkosti prstencové singularity protirotujı́cı́ „plusové“ orbity; tyto orbity jsou
jak nestabilnı́ (ležı́cı́ pod vnitřnı́ mezně stabilnı́ orbitou), tak stabilnı́ (Obr. 6). Pro dostatečně
velké hodnoty kosmologického parametru, y ∼ 0.06, jsou protirotujı́cı́ všechny stabilnı́ orbity a
obě dřı́ve oddělené oblasti „plusových“ protirotujı́cı́ch orbit se spojı́. Poznamenejme, že v KdS
prostoročasech, připouštějı́cı́ch existenci obou třı́d kruhových orbit, odpovı́dá orbita, ležı́cı́ na
statickém poloměru, nestabilnı́ protirotujı́cı́ kruhové geodetice.
Rozbor vlastnostı́ specifické energie a specifického momentu hybnosti na kruhových orbitách
vede ke zjištěnı́, že v KdS prostoročase existujı́ „plusové“ protirotujı́cı́ kruhové orbity se zápornou energiı́. Na rozdı́l od Kerrova prostoročasu [Stuchlı́k, 1980] se ovšem mohou vyskytovat
i v černoděrovém přı́padě (vně černé dı́ry), a to pro dostatečně velké hodnoty kosmologického
parametru, y ∼ 0.04, a libovolný rotačnı́ parametr a > 0. Tyto orbity, ležı́cı́ blı́zko protirotujı́cı́
kruhové fotonové orbity, která zde tvořı́ hornı́ mez výskytu orbit „plusové“ třı́dy, jsou však
nestabilnı́. Situace do jisté mı́ry analogická kerrovskému přı́padu nastává u nahých singularit
s dostatečně malými hodnotami rotačnı́ho parametru,
amin(ns) (y) < a < amax(E=0) (y),
(49)
kde v závislosti na kosmologickém parametru y < yc(ms+) nabývá hornı́ mez intervalu hodnot
r
√
4 2
.
≤ amax(E=0) (y) . 1.47 = 1.21
(50)
3 3
(spodnı́ mez již byla diskutována výše – relace (48)), u nichž se v blı́zkosti prstencové singularity
vyskytujı́ stabilnı́ kruhové orbity se zápornou energiı́ (zahrnujı́cı́ vnitřnı́ mezně stabilnı́ orbitu),
viz Obr. 6. Navı́c pro dostatečně velké hodnoty kosmologického parametru, y ∼ 0.06, jsou
všechny stabilnı́ „plusové“ orbity protirotujı́cı́ se zápornou energiı́.
Z vlastnostı́ tenkých disků v Kerrově prostoročase vı́me, že největšı́ účinnosti přeměny
klidové hmotnosti částic v jiné formy energie dosahujı́ při akreci na extrémnı́ černou dı́ru přı́p.
meznı́ nahou singularitu.7 V přı́padě KdS prostoročasů je situace analogická. Průběh specifické
energie částice na vnitřnı́ a vnějšı́ mezně stabilnı́ „plusové“ orbitě kolem KdS extrémnı́ černé
dı́ry a nahé singularity, E ms(i) a E ms(o) , v závislosti na kosmologickém parametru y je znázorněn
na Obr. 8a, b, z nichž je patrná výrazná nespojitost energie na vnitřnı́ mezně stabilnı́ orbitě
při přechodu od černých děr k nahým singularitám. Energie na vnějšı́ mezně stabilnı́ orbitě
podobnou nespojitost nevykazuje. Závislost účinnosti akrece na extrémnı́ černou dı́ru a meznı́
nahou singularitu, definované obecně relacı́ (13), na kosmologickém parametru je znázorněna
na Obr. 8c. Z Obr. 8 vidı́me, že vliv repulzivnı́ kosmologické konstanty se projevuje ve snı́ženı́
velikosti nespojitosti energie na vnitřnı́ mezně stabilnı́ orbitě při přechodu od černých děr
7 Meznı́ nahou singularitou nazýváme nahou singularitu, která je infinitezimálně blı́zko stavu extrémnı́ černé dı́ry.
-6 -5 -4 -3 -2 -1
log y
(a)
1
0.8
0.6
0.4
0.2
0
Η
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
Relativistické disky kolem kompaktnı́ch objektů
Ems HoL
Ems HiL
16
-10 -8
-6
log y
(b)
-4
-2
1.5
1.25
1
0.75
0.5
0.25
0
-6 -5 -4 -3 -2 -1
log y
(c)
Obr. 8. Specifická energie částice na vnitřnı́ (a) a vnějšı́ (b) mezně stabilnı́ „plusové“ orbitě kolem
KdS extrémnı́ černé dı́ry
parametru y. Pro
√ a meznı́ nahé singularity v závislosti na kosmologickém
√
y → 0, E ms(i) → 1/ 3 v přı́padě černých děr, resp. E ms(i) → −1/ 3 v přı́padě nahých singularit,
kdežto E ms(o) → 1 pro oba přı́pady, zatı́mco pro y → yc(KdS) , E ms(o) → E ms(i) → 0. (c) Maximálnı́
účinnost akrece daná rozdı́lem specifických energiı́ částice na vnějšı́ a vnitřnı́ mezně stabilnı́ orbitě
kolem KdS extrémnı́ černé dı́ry (plná křivka) a meznı́ nahé singularity (čárkovaná křivka) v závislosti
na kosmologickém parametru y. Pro y → 0, η → 0.42 v přı́padě černých děr, resp. η → 1.58 v přı́padě
nahých singularit, kdežto pro y → yc(KdS) , η → 0 pro oba přı́pady.
k nahým singularitám a v poklesu maximálnı́ účinnosti akrece. Oba projevy jsou nicméně
patrné až pro velké hodnoty kosmologického parametru y.
S akrecı́ hmoty v „plusovém“ disku na „pomalu rotujı́cı́“ nahou singularitu je (stejně jako
v kerrovském přı́padě [Stuchlı́k, 1981]) spojena možnost přeměny nahé singularity v černou
dı́ru, která by mohla mı́t observačně významné důsledky v náhlém poklesu luminozity vnitřnı́
části disku. Důvod je v tom, že hmota v oblasti mezi vnitřnı́m okrajem disku kolem nahé
singularity a mı́stem, kde se ustavı́ vnitřnı́ okraj disku kolem budoucı́ černé dı́ry (souřadnicová
vzdálenost těchto mı́st je nepatrná, nicméně jejich vlastnı́ vzdálenost je obrovská), se octne na
nestabilnı́ch orbitách a bude prakticky okamžitě pohlcena vznikajı́cı́ černou dı́rou.
1.3 Vliv kosmické repulze na tlusté disky
Tlusté disky podobně jako tenké mohou existovat jen v těch oblastech prostoročasu, kde
je možný pohyb po stabilnı́ch kruhových geodetikách, nebot’ hmota v centru disku (tlakové
gradienty zde nabývajı́ extrémnı́ch hodnot) sleduje stabilnı́ kruhovou geodetiku. Vliv repulzivnı́
kosmologické konstanty na chovánı́ ekvipotenciálnı́ch ploch v rotujı́cı́ kapalině SdS (práce [1])
a KdS (práce [3]) prostoročasu byl studován pouze u konfiguracı́ s konstantnı́m rozloženı́m
specifického momentu hybnosti (4), nicméně se zdá, že tyto konfigurace odrážejı́ všechny
topologické vlastnosti ekvipotenciálnı́ch ploch pro libovolné stabilnı́ rozloženı́ specifického
momentu hybnosti `(r, θ) v disku. Kvalitativně nové rysy ve vlastnostech ekvipotenciálnı́ch
ploch při srovnánı́ s asymptoticky plochými geometriemi se objevujı́ v blı́zkosti statického
poloměru daného desitterovského prostoročasu a spočı́vajı́ ve dvou efektech: (i) výskyt dalšı́
sebeprotı́najı́cı́ se (kritické) plochy s kritickým bodem (mı́stem sebekřı́ženı́) těsně pod statickým
poloměrem, (ii) výrazná kolimace otevřených ekvipotenciál jdoucı́ch podél osy rotace disku,
počı́najı́cı́ na statickém poloměru a pokračujı́cı́ dál za nı́m.
Ve většině konfiguracı́ tak existujı́ dvě kritické ekvipotenciálnı́ plochy, 8 jedna protı́najı́cı́
se v blı́zkosti centrálnı́ černé dı́ry, či nahé singularity (vnitřnı́ kritický bod), druhá s mı́stem
sebeprotnutı́ v blı́zkosti statického poloměru (vnějšı́ kritický bod). Obě kritické ekvipotenciály
8 Speciálnı́ přı́pady, kdy existujı́ 1 nebo 3 kritické ekvipotenciálnı́ plochy jsou diskutovány nı́že.
1. Vliv Λ > 0 na akrečnı́ disky
17
3
Wcrit(i) = Wcrit(o)
2
1
1
0
-1
Hlog rL cos Θ
2
Hlog rL cos Θ
Hlog rL cos Θ
Wcrit(i) < Wcrit(o)
0
-1
-2
-2
0.5
1
1.5
Hlog rL sin Θ
(a)
2
1
0
-1
-2
` = `mb
`ms(i) < ` < `mb
0
Wcrit(i) > Wcrit(o)
2
0
0.5
1
1.5
Hlog rL sin Θ
(b)
2
`mb < ` < `ms(o)
-3
0
0.5
1
1.5
Hlog rL sin Θ
2
(c)
Obr. 9. Meridionálnı́ řezy strukturou ekvipotenciálnı́ch ploch popisujı́cı́ tři základnı́ diskové konfigurace
barotropnı́ kapaliny (vystı́nované oblasti) v SdS a KdS prostoročasech: (a) akrečnı́ disk, (b) mezně vázaný
akrečnı́ disk, (c) exkrečnı́ disk.
mohou být jak otevřené, tak mezně uzavřené – v tom přı́padě umožňujı́ odtok hmoty z jimi
uzavřené toroidálnı́ oblasti Paczyńského mechanismem. Důsledkem výskytu „vnějšı́“ kritické
ekvipotenciály je skutečnost, že ve studovaných asymptoticky desitterovských prostoročasech
neexistujı́ toroidálnı́ konfigurace kapaliny, z nichž by při porušenı́ podmı́nky hydrostatické
rovnováhy nebyl nakonec možný řı́zený odtok hmoty. V podstatě rozlišujeme tři základnı́
diskové konfigurace (Obr. 9):
akrečnı́ disk – ekvipotenciála s vnitřnı́m kritickým bodem je mezně uzavřená, ekvipotenciála
s vnějšı́m kritickým bodem je otevřená. Odtok hmoty z disku probı́há přes vnitřnı́ kritický
bod. Jestliže hmota vyplnı́ i druhou kritickou plochu (při malém rozdı́lu hodnot kritických
potenciálů a výrazném přetečenı́ vnitřnı́ kritické plochy je to v principu možné), bude akrečnı́
tok na černou dı́ru/nahou singularitu doprovázen exkrečnı́m tokem přes vnějšı́ kritický bod do
vnějšı́ho prostoru.
mezně vázaný akrečnı́ disk – v tomto přı́padě dostáváme jedinou kritickou ekvipotenciálnı́
plochu protı́najı́cı́ se ve dvou kritických bodech. Při jejı́m vyplněnı́ je odtok hmoty z disku
realizován stejně účinně přes oba kritické body; akrece a exkrece probı́hajı́ současně.
exkrečnı́ disk – ekvipotenciálnı́ plocha s vnitřnı́m kritickým bodem je otevřená, zatı́mco plocha s vnějšı́m kritickým bodem je mezně uzavřená umožňujı́cı́ odtok hmoty z disku vně soustavu disk–černá dı́ra/nahá singularita. Obě kritické plochy jsou odděleny dalšı́mi otevřenými
(cylindrickými) plochami, které v podstatě znemožňujı́ akreci na centrálnı́ černou dı́ru/nahou
singularitu.
Sbı́havé chovánı́ otevřených ekvipotenciál podél osy rotace disku, způsobené repulzivnı́
kosmologickou konstantou, naznačuje možnost kolimace výtrysků hmoty – jetů, které jsou
pozorovány u některých aktivnı́ch galaktických jader, daleko za hranicemi diskové struktury
(jejı́ maximálnı́ rozměr je srovnatelný s velikostı́ statického poloměru).
Společný vliv rotace prostoročasu a kosmické repulze na profil povrchu disku, daný tvarem
mezně uzavřené kritické ekvipotenciálnı́ plochy, je znázorněn na Obr. 10 vpravo, kde jsou
porovnávány mezně vázané akrečnı́ disky obı́hajı́cı́ KdS a SdS černou dı́ru se stejnou hodnotou
kosmologického parametru y. Vidı́me, že korotujı́cı́ disk v KdS prostoročase je tlustšı́ a většı́,
než disk obı́hajı́cı́ v SdS prostoročase, který je zase tlustšı́ a většı́ ve srovnánı́ s protirotujı́cı́m
diskem v KdS prostoročase. Navı́c korotujı́cı́ disk v KdS prostoročase má nejužšı́ „komı́n“ –
18
Relativistické disky kolem kompaktnı́ch objektů
kolimovaný
výtrysk
výtrysk
korotující disk
kolem KdS černé díry
korotující disk
kolem Kerrovy č. d.
disk obíhající
Schw. č. d.
J
J
vnější
kritický
bod
protirotující disk
kolem Kerrovy č. d.
disk obíhající
SdS černou díru
-
statický *
poloměr
A
A
A
A
protirotující disk
kolem KdS černé díry
Obr. 10. Vliv rotace prostoročasu a kosmické repulze na tvar akrečnı́ho disku a možná kolimace
výtrysků podporovaná kosmickou repulzı́. Vlevo jsou znázorněny profily kritických ekvipotenciálnı́ch
ploch v barotropnı́ kapalině, obı́hajı́cı́ Kerrovu (y = 0, a 2 = 0.99, ` ≈ `mb ) a Schwarzschildovu
(y = 0, a = 0, ` ≈ `mb ) černou dı́ru, vpravo je znázorněna analogická situace pro přı́pad kapaliny
v poli KdS (y = 10−6 , a 2 = 0.99, ` = `mb ) a SdS (y = 10−6 , a = 0, ` = `mb ) černé dı́ry. Možný
tvar výtrysků je indikován průběhem otevřených ekvipotenciál podél rotačnı́ osy. (Převzato z [Stuchlı́k,
2005] a upraveno.)
oblast, kde jsou podél rotačnı́ osy vystřelovány částice hmoty ve formě výtrysků. Jejich možná
kolimace, způsobená kosmickou repulzı́, je taktéž vyznačena. Pro srovnánı́ je na Obr. 10 vlevo
znázorněna analogická situace bez kosmologické konstanty.
Abychom zı́skali představu, na jakých škálách, v mezı́ch uvažovaného modelu, mohou
diskutované vlivy kosmické repulze nastávat, uved’me si v kiloparsecı́ch rozměry statického
poloměru pro současnou hodnotu „reliktnı́ “ kosmologické konstanty Λ 0 (relace (3)) a dvě
typické hmotnosti supermasivnı́ch černých děr, které se zřejmě nacházejı́ v jádrech aktivnı́ch
galaxiı́.9 Je-li hmotnost centrálnı́ černé dı́ry 106 M¯ , dostáváme pro statický poloměr vzdálenost ∼ 10 kpc od centra, v přı́padě černé dı́ry o hmotnosti 10 9 M¯ ležı́ statický poloměr ve
vzdálenosti ∼ 110 kpc od centra. Uvedené hodnoty naznačujı́, že projevy kosmické repulze
9 Velikost statického poloměru pro dalšı́ typické hmotnosti černých děr od hvězdných až po supermasivnı́ lze najı́t
v práci [3]. Navı́c v práci [1] jsou uvedeny rozměry vnějšı́ mezně vázané orbity v SdS prostoročase jak pro současnou
hodnotu „reliktnı́“ kosmologické konstanty Λ0 , tak pro odpovı́dajı́cı́ hodnotu „efektivnı́“ kosmologické konstanty
Λeff spojené s energiı́ vakua při předpokládaných fázových přechodech ve velmi raném vesmı́ru na energetických
škálách 100 GeV (elektroslabý přechod) a 1 GeV (uvězněnı́ kvarků), relevantnı́ pro přı́padnou diskovou akreci na
primordiálnı́ černou dı́ru.
1. Vliv Λ > 0 na akrečnı́ disky
19
můžeme očekávat už na vzdálenostech srovnatelných, či dokonce menšı́ch, než jsou rozměry
velkých galaxiı́ a dále, že temná energie by mimo jiné měla hrát důležitou roli při vzniku a
vývoji galaktických diskových struktur tı́m, že klade omezenı́ na jejich maximálnı́ velikost. Je
pochopitelné, že u takto rozsáhlých diskových konfiguracı́, v nichž hmotnost disku o několik
řádů převyšuje hmotnost centrálnı́ černé dı́ry, bude hrát významnou roli vlastnı́ gravitace disku,
která může zı́skané výsledky podstatně změnit.
Základnı́ charakteristiky rovnovážných toroidálnı́ch konfiguracı́ dokonalé kapaliny zı́skané
rozborem chovánı́ potenciálu (5) v SdS a KdS prostoročase si okomentujme pro černé dı́ry a
nahé singularity odděleně.
1.3.1
Tlusté disky kolem černých děr
V obecnějšı́m KdS prostoročase jsou ekvipotenciálnı́ plochy v dokonalé kapalině dány výrazem
pro potenciál ve tvaru
½ 2
¾1/2
ρ
∆r ∆θ sin2 θ
W (r, θ) = ln 2
,
(51)
I ∆θ (r 2 + a 2 − a`)2 sin2 θ − ∆r (` − a sin2 θ)2
kde ∆r , ∆θ , I, ρ 2 jsou dány relacemi (31) a (32). V limitě pro rotačnı́ parametr a → 0 přecházı́
výraz (51) ve vyjádřenı́ potenciálu v SdS prostoročase. Ukazuje se, že uzavřené (toroidálnı́)
ekvipotenciálnı́ plochy, nezbytné pro existenci rovnovážných diskových struktur, existujı́ pouze
pro hodnoty konstantnı́ho specifického momentu hybnosti ` v rozmezı́
`ms(i) < ` < `ms(o) ,
(52)
kde `ms(i) (`ms(o) ) odpovı́dá specifickému momentu hybnosti na vnitřnı́ (vnějšı́) mezně stabilnı́ orbitě v ekvatoriálnı́ rovině daného (KdS či SdS) černoděrového prostoročasu. Je-li
`ms(i) < ` < `mb , kde `mb odpovı́dá specifickému momentu hybnosti na mezně vázaných ekvatoriálnı́ch orbitách, popisuje daná konfigurace ekvipotenciálnı́ch ploch akrečnı́ disk (Obr. 9a),
je-li `mb < ` < `ms(o) , dostáváme exkrečnı́ disk (Obr. 9c); speciálnı́ přı́pad ` = ` mb popisuje
mezně vázaný akrečnı́ disk (Obr. 9b). V prostoročasech s typicky malou hodnotou kosmologic.
kého parametru y (v SdS prostoročase pro y < 1/118125 = 0.0000085) splňuje impaktnı́ parametr fotonové kruhové geodetiky podmı́nku `mb < `ph(c) < `ms(o) , a pokud `ph(c) < ` < `ms(o) ,
je exkrečnı́ disk oddělen od centrálnı́ černé dı́ry „zakázanou oblastı́ “ pro výskyt hmoty s daným
specifickým momentem hybnosti `; v tomto přı́padě „vnitřnı́“ (otevřená) kritická ekvipotenciálnı́ plocha chybı́ stejně jako otevřené plochy neprotı́najı́cı́ ekvatoriálnı́ rovinu a nabalujı́cı́ se
v blı́zkosti rotačnı́ osy na černou dı́ru. V přı́padě KdS černé dı́ry se výše uvedené nerovnosti
týkajı́ korotujı́cı́ch disků; pro protirotujı́cı́ disky (` < 0) jsou závěry stejné, jen je potřeba
v přı́slušných relacı́ch obrátit znaky nerovnostı́.
Existence „vnějšı́“ kritické ekvipotenciály způsobuje, že podobně jako v přı́padě akrečnı́ch
disků, kdy je jejich vnitřnı́ okraj dán polohou vnitřnı́ho kritického bodu, ležı́cı́ho mezi vnitřnı́
mezně vázanou a vnitřnı́ mezně stabilnı́ orbitou, r mb(i) < rin < rms(i) , je v přı́padě exkrečnı́ch
disků jejich vnějšı́ okraj dán polohou vnějšı́ho kritického bodu, ležı́cı́ho mezi vnějšı́ mezně
stabilnı́ a vnějšı́ mezně vázanou orbitou, r ms(o) < rout < rmb(o) . Mezně vázaný akrečnı́ disk
má pak přirozeným způsobem definovány oba okraje, r in ≈ rmb(i) , rout ≈ rmb(o) , a odpovı́dá
největšı́ možné stacionárnı́ diskové struktuře v daném desitterovském prostoročase. Dodejme,
že vnějšı́ kritický bod v podstatě odpovı́dá nejvzdálenějšı́mu okraji libovolného akrečnı́ho
disku, nebot’v přı́padě výrazného přeplněnı́ mezně uzavřené ekvipotenciálnı́ plochy s vnitřnı́m
20
Relativistické disky kolem kompaktnı́ch objektů
kritickým bodem může být následně vyplněna i ekvipotenciála s vnějšı́m kritickým bodem,
a akrece přes vnitřnı́ kritický bod bude doprovázena exkrecı́ přes vnějšı́ kritický bod. Na
druhou stranu, v přı́padě exkrečnı́ch disků se zdá, že jakkoliv velké přetečenı́ mezně uzavřené
ekvipotenciály (vedoucı́ k exkreci) zřejmě nezpůsobı́ následné přetečenı́ ekvipotenciálnı́ plochy
s vnitřnı́m kritickým bodem (oblast mezi oběma kritickými plochami je vyplněna cylindrickými
plochami), a exkrečnı́ odtok z disku nebude doprovázen akrecı́ na centrálnı́ černou dı́ru.
Důležitý je vliv vnějšı́ho kritického bodu na stabilitu akrečnı́ch (a mezně akrečnı́ch) disků.
V článku [Rezzolla et al., 2003] bylo ukázáno, že exkrečnı́ odtok přes vnějšı́ kritický bod
u akrečnı́ch disků kolem SdS černých děr může vést k potlačenı́ tzv. „úprkové“ nestability
(runaway instability) [Abramowicz et al., 1983], problému, který souvisı́ s obecně nesynchronizovanou změnou polohy vnitřnı́ho kritického bodu a skutečného vnitřnı́ho okraje disku,
vynucenou rostoucı́ hmotnostı́ černé dı́ry v důsledku akrece materiálu z disku. Jestliže se kritický bod posouvá stále hlouběji do nitra disku, stále vı́ce hmoty na vnitřnı́ straně disku porušı́
podmı́nku hydrostatické rovnováhy a bude pohlceno černou dı́rou. Tento proces má lavinovitý
charakter vedoucı́ nakonec k pohlcenı́ celého disku. Rezzolla et al. [2003] provedli hydrodynamickou osově symetrickou simulaci adiabatického toku a na základě časové posloupnosti
rovnovážných konfiguracı́ dokonalé kapaliny obı́hajı́cı́ stále hmotnějšı́ SdS černou dı́ru došli
k závěru, že pro širokou škálu počátečnı́ch podmı́nek docházı́ k účinnému potlačenı́ „úprkové“
nestability právě dı́ky exkrečnı́mu toku z disku.
1.3.2
Tlusté disky kolem nahých singularit
Nahé singularity, bez ohledu na Penroseovu hypotézu kosmické cenzury [Penrose, 1969], jsou
stále v popředı́ zájmu teoretiků, např. [de Felice and Yunqiang, 2001, Joshi et al., 2002, Joshi
et al., 2004], a má tedy smysl jim věnovat určitou pozornost. Důležité jsou zejména ty jejich
projevy, které je jednoznačně odlišujı́ od černých děr a na ty se zaměřı́me i v tomto komentáři.
V práci [3] jsou diskutovány vlastnosti rovnovážných toroidálnı́ch konfiguracı́ dokonalé
kapaliny obı́hajı́cı́ KdS nahou singularitu. Ekvipotenciálnı́ plochy jsou opět dány hladinami
potenciálu (51), nynı́ ovšem pro hodnoty prostoročasových parametrů (a, y) odpovı́dajı́cı́ch
prostoročasům s nahou singularitou, v nichž je možný pohyb po stabilnı́ch ekvatoriálnı́ch
kruhových geodetikách (dané třı́dy). Ukazuje se, že kvalitativnı́ rozdı́ly oproti černoděrovým
prostoročasům nastávajı́ v prostoročasech, v nichž existujı́ stabilnı́ protirotujı́cı́ (vůči LNRF)
„plusové“ orbity, a jsou tedy spojeny s disky „plusové“ třı́dy. Disky „minusové“ třı́dy vykazujı́
podobné vlastnosti jako při oběhu kolem černých děr, nicméně v některých přı́padech mohou
vytvářet dvojici s diskem „plusové“ třı́dy.
Uvažujeme-li disky kolem nahých singularit, které jsou dostatečně blı́zko stavu extrémnı́
černé dı́ry (v těchto prostoročasech se vyskytujı́ protirotujı́cı́ „plusové“ ekvatoriálnı́ kruhové
geodetiky se zápornou specifickou energiı́ částice), existujı́ stacionárnı́ toroidálnı́ konfigurace
dokonce pro libovolné konstantnı́ rozloženı́ specifického momentu hybnosti `(r, θ). V závislosti
na hodnotě ` pak můžeme mı́t 1–3 kritické ekvipotenciálnı́ plochy v konfiguraci.
Je-li kritická plocha jediná, pak v přı́padě toroidálnı́ch konfiguracı́ je nutně mezně uzavřená.
K možnostem, které mohou nastat i v černoděrových prostoročasech (přı́slušné konfigurace
popisujı́ mezně vázaný akrečnı́ disk a exkrečnı́ disk oddělený od černé dı́ry „zakázanou oblastı́“)
zde přibývá přı́pad s toliko vnitřnı́m kritickým bodem (na straně prstencové singularity). Tato
konfigurace popisuje akrečnı́ disk, který má dalšı́ specifické vlastnosti: (i) elementy kapaliny
v centru a na vnitřnı́m okraji disku (pohybujı́cı́ se po geodetice) jsou protirotujı́cı́ se zápornou
1. Vliv Λ > 0 na akrečnı́ disky
21
0.6
0.75
6
0.4
0.25
0
-0.25
sinh-1 Hr cos ΘL
4
sinh-1 Hr cos ΘL
sinh-1 Hr cos ΘL
0.5
2
0
-2
-0.5
-4
-0.75
-6
0.6 0.7 0.8 0.9
1
1.1 1.2
!!!!!!!!!!!!!!!!!
sinh-1 H r2 + a2 sin ΘL
0
-0.2
-0.4
-0.6
2
3
4
5
!!!!!!!!!!!!!!!!!
sinh-1 H r2 + a2 sin ΘL
1
6
4
4
0
-2
sinh-1 Hr cos ΘL
6
4
sinh-1 Hr cos ΘL
6
2
2
0
-2
2
0
-2
-4
-4
-4
-6
-6
-6
1
2
3
4
5
!!!!!!!!!!!!!!!!!
sinh-1 H r2 + a2 sin ΘL
(c)
0.9 0.95 1 1.05 1.1 1.15
!!!!!!!!!!!!!!!!!
sinh-1 H r2 + a2 sin ΘL
(b)
(a)
sinh-1 Hr cos ΘL
0.2
1
2
3
4
5
!!!!!!!!!!!!!!!!!
sinh-1 H r2 + a2 sin ΘL
(d)
1
2
3
4
5
!!!!!!!!!!!!!!!!!
sinh-1 H r2 + a2 sin ΘL
(e)
Obr. 11. Meridionálnı́ řezy strukturou ekvipotenciálnı́ch ploch v barotropnı́ kapalině obı́hajı́cı́ KdS nahou
singularitu (vybrané přı́pady). (a) protirotujı́cı́ akrečnı́ disk, (b) protirotujı́cı́ akrečnı́ disk (vnitřnı́ disk;
zvětšeno na obrázku vpravo) a korotujı́cı́ exkrečnı́ disk, (c) dva protirotujı́cı́ akrečnı́ disky, (d) protirotujı́cı́
akrečnı́ disk (vnitřnı́ disk) a protirotujı́cı́ mezně vázaný akrečnı́ disk, (e) protirotujı́cı́ akrečnı́ disk
(vnitřnı́ disk) a protirotujı́cı́ exkrečnı́ disk. Při kresbě řezů byly použity Kerrovy–Schildovy souřadnice;
škálovánı́ os pomocı́ funkce argsinh umožňuje znázornit jak oblast poblı́ž prstencové singularity (šedý
bod v ekvatoriálnı́ rovině), tak v blı́zkosti statického poloměru.
energiı́ a my můžeme očekávat, že celý disk má tyto vlastnosti, (ii) disk je společně s prstencovou
singularitou odřı́znut od vnějšı́ho okolı́ oblastı́, kde se hmota s daným specifickým momentem
hybnosti ` nemůže vyskytovat a (iii) neexistujı́ otevřené ekvipotenciály, jdoucı́ podél rotačnı́
osy disku (Obr. 11a). Poznamenejme, že existuje i varianta s kladnou energiı́ elementů kapaliny
v disku, který ale zůstává protirotujı́cı́.
V přı́padě dvou kritických ploch existuje, na rozdı́l od černých děr, i možnost, že obě kritické
plochy jsou mezně uzavřené, což vede ke dvěma odděleným toroidálnı́m konfiguracı́m pro dané
rozloženı́ specifického momentu hybnosti `(r, θ). Vnitřnı́ disk je vždy protirotujı́cı́ akrečnı́ disk.
Vnějšı́ disk, pokud je oddělen „zakázanou oblastı́“ pro hmotu s daným specifickým momentem
hybnosti `, je korotujı́cı́ nebo protirotujı́cı́ exkrečnı́ disk (Obr. 11b). Ve speciálnı́m přı́padě však
může vnějšı́ disk odpovı́dat protirotujı́cı́mu mezně vázanému akrečnı́mu disku, pak „zakázaná
oblast“ chybı́ (Obr. 11d).
Jsou-li kritické plochy tři, dvě z nich jsou vždy mezně uzavřené a opět dostáváme dvojici
diskových struktur pro dané rozloženı́ specifického momentu hybnosti. Stejně jako v předešlém
přı́padě i zde odpovı́dá vnitřnı́ struktura protirotujı́cı́mu akrečnı́mu disku. Vnějšı́ struktura pak
může odpovı́dat jak akrečnı́mu (Obr. 11c), tak exkrečnı́mu disku (Obr. 11e), oba však mohou
být pouze protirotujı́cı́.
22
Relativistické disky kolem kompaktnı́ch objektů
-1
vnitřnı́
kritický bod
-2
-3
centrum
disku
0.1
0
y = 10−6
a = 1.001aex
.
` = 0.825
H
j
H
0.5 1
5 10
50 100
WHr, Θ=А2L
WHr, Θ=А2L
0
vnitřnı́
kritický bod
-0.1
-0.2
-0.3
Y
H
H
-0.4
1
2
5
r
(a)
y = 10−6
a = 0.999aex
.
` = 2.063
centrum
disku
10
r
20
50
100
(b)
Obr. 12. Průběh potenciálu v ekvatoriálnı́ rovině KdS prostoročasu popisujı́cı́ho (a) nahou singularitu,
(b) černou dı́ru v blı́zkosti stavu extrémnı́ černé dı́ry reprezentované rotačnı́m parametrem a ex . Potenciál
odpovı́dá konfiguracı́m se specifickým momentem hybnosti ` = `mb+ , které popisujı́ mezně vázané
akrečnı́ disky.
Nynı́ se nabı́zı́ otázka, nakolik jsou obě diskové struktury na sobě nezávislé. V přı́padě
přı́tomnosti vysoce účinných disipativnı́ch procesů (blı́žı́cı́ch se téměř účinnosti anihilace)
v akrečnı́m toku z vnějšı́ho disku by mohla akreujı́cı́ hmota mı́sto přı́mého pádu na singularitu
nejdřı́ve vyplnit vnitřnı́ disk a teprve po přetečenı́ jeho kritické ekvipotenciálnı́ plochy skončit
v prstencové singularitě. Na druhou stranu se zdá, že pokud disipace nenı́ dostatečně účinná a
vnitřnı́ disk již z jakéhokoliv důvodu existuje, akreujı́cı́ hmota z vnějšı́ho disku jej zřejmě obteče
a skončı́ přı́mo v singularitě. Lepšı́ vhled do problému dá ovšem až dynamické modelovánı́
akrečnı́ho toku.
S tlustými disky kolem nahých singularit, jejichž rotačnı́ parametr se v limitě blı́žı́ rotačnı́mu
parametru extrémnı́ černé dı́ry, je spojena ještě jedna pozoruhodná skutečnost, která nicméně
nemá přı́mou souvislost s kosmickou repulzı́ a projevuje se i v přı́padě Kerrovy nahé singularity
pro a/M → 1. Jedná se o to, že potenciálový rozdı́l mezi okrajem a centrem mezně vázaného
akrečnı́ho disku, ∆W = Wcrit − Wcent , může v „plusových“ discı́ch dosahovat libovolně velké
hodnoty a v uvažované limitě diverguje, ∆W → ∞, na rozdı́l od „plusových“ disků kolem černých děr, u nichž je hornı́ mez, daná opět limitou extrémnı́ černé dı́ry, konečná. Jejı́ maximálnı́
hodnota odpovı́dá mezně vázanému korotujı́cı́mu akrečnı́mu disku kolem Kerrovy extrémnı́
černé dı́ry, ∆Wmax ≈ 0.55 [Abramowicz et al., 1978] a s rostoucı́ hodnotou kosmologického
parametru 0 < y < yc(ms+) klesá k nule. Přı́čina singulárnı́ho chovánı́ potenciálového rozdı́lu
∆W u nahých singularit spočı́vá v nespojitosti polohy vnitřnı́ mezně vázané orbity při přechodu od nahých singularit k černým dı́rám (viz např. Obr. 7) a s tı́m souvisejı́cı́ nespojitostı́
specifického momentu hybnosti `mb+ , odpovı́dajı́cı́ho „plusovým“ mezně vázaným orbitám.
Podrobněji si situaci ozřejmı́me pro Kerrův prostoročas, tzn. volı́me y = 0. U mezně vázaného
akrečnı́ho disku splývá poloha vnitřnı́ho kritického bodu s mezně vázanou kruhovou geodetikou, rcrit = rmb , a potenciál zde nabývá nulové hodnoty, W crit = 0. V limitě
√ extrémnı́ černé
dı́ry `mb+ → `ms+ = 2, rcent → rms+ = 1 a Ut (rcent , θ = π/2) → 1/ 3. Poznamenejme,
že ačkoliv i rmb+ → 1, ve skutečnosti jsou obě orbity (mezně stabilnı́ a mezně vázaná) odděleny [Bardeen et al., 1972].
√ . Pro potenciál v centru dostáváme s využitı́m relace (5) konečnou
hodnotu Wcent = − ln 3 = −0.55. V přı́padě limitnı́ nahé singularity
√ . spočı́vá poloha mezně
vázané orbity a tedy i vnitřnı́ho kritického bodu na √
r mb+ = 3 − 2 2 = 0.17, čemuž odpovı́dá
.
(ns)
hodnota specifického momentu hybnosti `mb+
= 2( 2 − 1) = 0.83, zatı́mco r cent → 1. Jelikož
(ns)
pro ` = `mb+ a rcent → 1, Ut (rcent , θ = π/2) → 0, dostáváme pro limitnı́ nahou singula-
1. Vliv Λ > 0 na akrečnı́ disky
23
ritu mezně vázaný akrečnı́ disk s nekonečně hlubokou potenciálovou jámou. Pro názornost
je na Obr. 12 ukázán průběh potenciálu v ekvatoriálnı́ rovině KdS prostoročasu obsahujı́cı́ho
jak černou dı́ru, tak nahou singularitu blı́zko stavu extrémnı́ černé dı́ry; ten je reprezentován
odpovı́dajı́cı́ hodnotou rotačnı́ho parametru a ex (y). Hodnoty specifického momentu hybnosti
konfigurace odpovı́dajı́ specifickým momentům hybnosti mezně vázaných orbit.
1.4 Shrnutı́
Vliv repulzivnı́ kosmologické konstanty na akrečnı́ disky má tyto stěžejnı́ projevy:
(i) polohu vnějšı́ho okraje disku omezuje na oblast pod statickým poloměrem daného prostoročasu, čı́mž dává omezenı́ na velikost diskových struktur, které se mohou v daném prostoročase
vyskytovat,
(ii) umožňuje existenci nového typu toroidálnı́ struktury nazvané exkrečnı́ disk, u nı́ž docházı́
při porušenı́ podmı́nky hydrostatické rovnováhy k odtoku hmoty přes vnějšı́ kritický bod ležı́cı́
v blı́zkosti statického poloměru,
(iii) ve speciálnı́m přı́padě mezně vázaného akrečnı́ho disku docházı́ při porušenı́ podmı́nky
hydrostatické rovnováhy ke stejně účinnému odtoku hmoty přes oba (vnitřnı́ a vnějšı́) kritické
body,
(iv) odtok hmoty přes vnějšı́ kritický bod může výrazným způsobem stabilizovat disk,
(v) kolimace otevřených ekvipotenciál v blı́zkosti statického poloměru a dál za nı́m naznačuje možnost kolimace výtrysků hmoty podél rotačnı́ osy daleko za hranicı́ mateřské toroidálnı́
struktury,
(vi) v přı́padě „pomalu rotujı́cı́ “ KdS nahé singularity umožňuje existenci dvou oddělených diskových struktur (pro vhodně zvolenou hodnotu uniformně rozloženého specifického
momentu hybnosti), z nichž vnitřnı́ odpovı́dá akrečnı́mu a vnějšı́ exkrečnı́mu disku.
2 ASCHENBACHŮV EFEKT
V obecném stacionárnı́m a osově symetrickém prostoročase nejsou odpovı́dajı́cı́ Killingovy
vektory X(t) a X(ϕ) vzájemně ortogonálnı́, což mimo jiné vede ke „strhávánı́“ prostoročasu
v okolı́ rotujı́cı́ch těles, projevujı́cı́mu se efekty vlečenı́ lokálnı́ch inerciálnı́ch soustav, přı́p.
Lenseovou–Thirringovou precesı́ setrvačnı́ků. Při studiu lokálnı́ch vlastnostı́ disků obı́hajı́cı́ch
v rotujı́cı́m prostoročase pak hrajı́ důležitou roli soustavy, v nichž je lokálně vliv strhávánı́
odfiltrován. V dalšı́m se zaměřme na akrečnı́ disky obı́hajı́cı́ Kerrovu černou dı́ru.
Lokálně nerotujı́cı́ soustavy (LNRF) jsou v Kerrově prostoročase popsány tetrádou bázových
1-forem, jejı́ž explicitnı́ tvar je uveden v práci [4], přı́p. tetrádou bázových vektorů [Bardeen
et al., 1972]
µ
¶ µ
¶
A 1/2 ∂
∂
+ ΩLNRF
e(t) =
,
(53)
Σ∆
∂t
∂ϕ
µ ¶1/2
∂
∆
,
(54)
e(r ) =
Σ
∂r
1 ∂
(55)
e(θ) = 1/2 ,
Σ ∂θ
µ
¶1/2
Σ
∂
e(ϕ) =
,
(56)
2
A sin θ
∂ϕ
kde ve standardnı́ch jednotkách c = G = M = 1
∆ = r 2 − 2r + a 2 ,
2
2
(57)
2
Σ = r + a cos θ,
2
2 2
2
(58)
2
A = (r + a ) − a ∆ sin θ.
(59)
Úhlová rychlost lokálně nerotujı́cı́ soustavy (vzhledem ke klidovému pozorovateli v nekonečnu)
je dána výrazem
2ar
.
(60)
A
V dalšı́m se budeme zabývat chovánı́m orbitálnı́ rychlosti z pohledu LNRF; jejı́ vyjádřenı́ je
dáno relacı́
U · e(ϕ)
A sin θ
V (ϕ) =
=
(Ω − ΩLNRF ),
(61)
U · e(t)
Σ∆1/2
ΩLNRF =
kde U označuje vektor 4-rychlosti a Ω = dϕ/dt úhlovou rychlost částice, přı́p. elementu
kapaliny. Jestliže se částice pohybuje po kruhové geodetice v ekvatoriálnı́ rovině, je jejı́ pohyb
Keplerovský s odpovı́dajı́cı́ úhlovou rychlostı́ (kruhovou frekvencı́)
1
,
(62)
r 3/2 ± a
kde + označuje korotujı́cı́ a − protirotujı́cı́ orbity. V dalšı́m se soustředı́me pouze na korotujı́cı́
orbity, takže označenı́ ± nebudeme využı́vat.
Aschenbach [2004] si všiml, že v přı́padě vysokého spinu Kerrovy černé dı́ry vykazuje
Keplerovská orbitálnı́ rychlost VK(ϕ) , měřená lokálně nerotujı́cı́m pozorovatelem, anomálnı́
chovánı́ v blı́zkosti vnitřnı́ho okraje Keplerovského disku, projevujı́cı́ se ve změně znaménka
jejı́ho radiálnı́ho gradientu. Přesněji, uvedená anomálie nastává pro spin černé dı́ry
ΩK± = ±
24
2. Aschenbachův efekt
1.2
25
0.578
a = 0.2
a = 0.99616
0.577
V (ϕ)
K
V (ϕ)
K
1
0.8
0.576
0.575
0.6
0.574
rph
2.5
3
rmb
3.5
rms
4
4.5
5
rph
5.5
6
0.573
1
rmb
1.2
rms
1.4
r
(a)
r3:1
r
1.6
1.8
2
(b)
Obr. 13. Průběh Keplerovské orbitálnı́ rychlosti z pohledu LNRF pro nı́zkou a vysokou hodnotu spinu
Kerrovy černé dı́ry. (a) Pro nı́zkou hodnotu spinu orbitálnı́ rychlost monotónně roste s klesajı́cı́ radiálnı́
souřadnicı́ r . (b) Pro spin černé dı́ry a > 0.9953 existuje úzká oblast nad mezně stabilnı́ orbitou (r ms ),
v nı́ž je radiálnı́ gradient orbitálnı́ rychlosti kladný. Do této oblasti spadá i poloha orbity, na nı́ž docházı́
k parametrické rezonanci vertikálnı́ch a radiálnı́ch epicyklických oscilacı́ v poměru 3:1 (r 3:1 ). V grafech
jsou dále vyznačeny polohy korotujı́cı́ fotonové (r ph ) a mezně vázané (rmb ) orbity.
.
a > ac(K) = 0.9953.
(63)
Poznamenejme, že Aschenbach svůj objev učinil v souvislosti s problematikou tzv. kvaziperiodických zdrojů (QPOs), ležı́cı́ch v našı́ Galaxii,10 v jejichž spektru byly pozorovány alespoň
dvě výrazné frekvence v relaci 3:2 a/nebo 3:1, kdy se snažil najı́t fyzikálnı́ opodstatněnı́ pro
svůj heuristický předpoklad o celočı́selné násobnosti Keplerovských kruhových frekvencı́ částice na orbitách vykazujı́cı́ch parametrickou rezonanci mezi vertikálnı́ a radiálnı́ epicyklickou
oscilacı́ [Nowak and Lehr, 1998] právě v poměrech 3:1 a 3:2,
ΩK (r3:1 , a) = n × ΩK (r3:2 , a).
(64)
Dodejme, že při započtenı́ uvedeného ad hoc předpokladu (64), kde jediný možný celočı́selný
násobek odpovı́dá n = 3, můžeme jednoznačně stanovit hodnotu spinu centrálnı́ černé dı́ry.
.
Výsledkem je hodnota a = af = 0.99616, kterou Aschenbach prohlásil za „novou“ hornı́
limitu spinu Kerrovy černé dı́ry indukovanou akrečnı́mi procesy, a která je nepatrně nižšı́ než
„Thorneova“ limita 0.998 [Thorne, 1974]. Situaci ilustruje Obr. 13.
Změnu znaménka radiálnı́ho gradientu orbitálnı́ rychlosti V (ϕ) měřené v LNRF (relace (61)),
projevujı́cı́ se v blı́zkosti velmi rychle rotujı́cı́ch černých děr, nazýváme Aschenbachovým
efektem. V práci [4] je ukázáno, že existence tohoto jevu nenı́ vázána pouze na geodetický
(Keplerovský) pohyb testovacı́ch částic, ale vyskytuje se i v přı́padě negeodetického pohybu
kapaliny s rovnoměrně rozloženým specifickým momentem hybnosti, `(r, θ) = const, nicméně
obı́hajı́cı́ kolem podstatně rychleji rotujı́cı́ Kerrovy černé dı́ry se spinem
.
a > ac(bh) = 0.99964.
(65)
Pokud má efekt nastávat ve stacionárnı́ch kapalinových toroidech, musı́ černá dı́ra rotovat ještě
rychleji, a to se spinem
.
(66)
a > ac(tori) = 0.99979.
10 Jedná se o tři mikrokvazary GRO J1655−40, XTE J1550−564 a GRS 1915+105 a zdroj v centru našı́ Galaxie
SGR A? , viz např. [Török et al., 2005].
26
Relativistické disky kolem kompaktnı́ch objektů
Tabulka 1. Charakteristické frekvence možných oscilacı́ disku souvisejı́cı́ch s Aschenbachovým efektem, odpovı́dajı́cı́ kruhovým frekvencı́m Ω = 2π f definovaným v textu, pro vybrané hodnoty spinu
Kerrovy černé dı́ry a. Frekvence jsou vyčı́sleny v jednotkách (M/M ¯ )−1 Hz, kde M je hmotnost
centrálnı́ černé dı́ry.
1−a
4.5 × 10−3
4 × 10−3
1 × 10−3
5 × 10−4
2 × 10−4
1 × 10−4
1 × 10−5
1 × 10−6
1 × 10−9
Keplerovské disky
f
f∞
f˜
f˜∞
57
207
1938
2727
3658
4274
5824
6773
7838
14
48
294
338
351
338
249
159
32
19
69
487
603
693
729
766
771
771
5
16
85
94
97
96
94
94
94
Kapalinové disky
f
f∞
f˜
f˜∞
nedefinováno
47
503
1741
2590
3705
6
50
105
94
29
9
88
230
274
288
1
10
21
23
24
Samotná oblast výskytu orbit s kladnou hodnotou radiálnı́ho gradientu orbitálnı́ rychlosti se
nacházı́ nad centrem toroidálnı́ konfigurace, v oblasti existence stabilnı́ch kruhových geodetik
daného prostoročasu.
Studujeme-li chovánı́ orbitálnı́ rychlosti z pohledu LNRF mimo ekvatoriálnı́ rovinu θ = π/2,
ukazuje se, že v přı́padě stacionárnı́ch toroidů s konstantnı́m specifickým momentem hybnosti
` souvisı́ Aschenbachův efekt s výskytem uzavřených – toroidálnı́ch – ekvirychlostnı́ch ploch,
jejichž centrum ležı́ v ekvatoriálnı́ rovině v mı́stě, kde má orbitálnı́ rychlost vůči LNRF lokálnı́
minimum.
Existence toroidálnı́ch ekvirychlostnı́ch ploch v kapalině implikuje úvahy o vzniku možných
vertikálnı́ch nestabilit vı́rové či turbulentnı́ povahy, doprovázených přı́p. lokálnı́mi oscilacemi
disku. Ve shodě s Aschenbachem můžeme definovat frekvenci těchto oscilacı́ pomocı́ maximálnı́
kladné hodnoty radiálnı́ho gradientu orbitálnı́ rychlosti z pohledu LNRF,
∂V (ϕ)
|max ,
(67)
∂r
nicméně pro odhad charakteristické frekvence spojené s lokálnı́mi procesy v disku se zdá být
relevantnı́ veličina, vycházejı́cı́ z maximálnı́ hodnoty gradientu orbitálnı́ rychlosti podle vlastnı́
radiálnı́ souřadnice [Stuchlı́k et al., 2004],
Ωosc = 2π
∂V (ϕ)
|max ,
∂ r̃
kde souřadnicově nezávislá vlastnı́ radiálnı́ vzdálenost je definována relacı́
Ω̃osc = 2π
(68)
Σ 1/2
) dr.
(69)
∆
Hodnoty charakteristických frekvencı́ v přı́padě Keplerovských disků i kapalinových toroidů
udává pro vybrané hodnoty spinu centrálnı́ černé dı́ry Tabulka 1, kde jsou kromě frekvencı́
definovaných relacemi (67) a (68), vztažených k lokálně nerotujı́cı́m soustavám, uvedeny
i jejich odpovı́dajı́cı́ hodnoty pro statického pozorovatele v asymptoticky plochém nekonečnu
dr̃ = (grr )1/2 dr = (
2. Aschenbachův efekt
Ω∞ =
resp.
Ω̃∞ =
27
q
2
−(gtt + 2ΩLNRF gtϕ + ΩLNRF
gϕϕ )Ωosc ,
(70)
q
2
−(gtt + 2ΩLNRF gtϕ + ΩLNRF
gϕϕ )Ω̃osc ,
(71)
kde pro jednotlivé složky metrického tenzoru platı́:
∆ − a 2 sin2 θ
,
(72)
Σ
2ar sin2 θ
,
(73)
gtϕ = −
Σ
A sin2 θ
(74)
gϕϕ =
Σ
a výraz na pravé straně relacı́ (70) a (71) je vyčı́slen v mı́stě, kde má přı́slušný gradient
orbitálnı́ rychlosti maximum. Vyjádřenı́ jednotlivých složek metrického tenzoru je uvedeno
v práci [4] – relace (8)–(13). Zatı́mco frekvence (67) a (68) monotónně rostou s rostoucı́m
spinem černé dı́ry, po započtenı́ pohybu LNRF po kruhové orbitě v gravitačnı́m poli – relace
(70) a (71) – se ukazuje, že existuje jistá maximálnı́ hodnota frekvence pozorované statickým
pozorovatelem v asymptoticky plochém nekonečnu; jejı́ přibližná hodnota je v Tabulce 1
vyznačena tučně. Samotný mechanismus přı́padného vzniku uvažovaných oscilacı́ nenı́ ovšem
přı́liš jasný, a pochopitelně nenı́ ani jisté, že oscilace pozorované u mikrokvazarů souvisı́
s anomálnı́m chovánı́m gradientu orbitálnı́ rychlosti tak, jak navrhuje Aschenbach. Problematika
nutně vyžaduje podrobnějšı́ zkoumánı́ využı́vajı́cı́ v přı́padě kapalinových toroidů numerické
hydrodynamické modelovánı́ akrečnı́ho disku kolem velmi rychle rotujı́cı́ Kerrovy černé dı́ry,
jejı́ž spin splňuje relaci (66).
Strukturu ekvirychlostnı́ch ploch v kapalině s uniformnı́m rozloženı́m specifického momentu
hybnosti, `(r, θ) = const, ilustruje veličina
gtt = −
`
,
(75)
V (ϕ)
která má rozměr délky a v práci [4] je nazývána „von Zeipelův poloměr“ i s vědomı́m, že
jejı́ přı́má souvislost s tzv. von Zeipelovými cylindry, tj. plochami konstantnı́ho specifického
momentu hybnosti ` a konstantnı́ úhlové rychlosti Ω, existuje pouze ve statickém prostoročase,
kde plochy R = const a von Zeipelovy cylindry splývajı́. 11
Je-li spin Kerrovy černé dı́ry a < ac(bh) , jsou plochy konstantnı́ orbitálnı́ rychlosti vůči LNRF
otevřené, majı́cı́ vesměs cylindrickou topologii s tı́m, že poslednı́ cylindrická plocha je kritická
– sebeprotı́najı́cı́, a zdánlivě tak jejich struktura připomı́ná strukturu von Zeipelových cylindrů.
V přı́padě velmi rychle rotujı́cı́ Kerrovy černé dı́ry, a > a c(bh) (v přı́padě stacionárnı́ch toroidů:
a > ac(tori) ), a pro vhodně zvolený parametr ` se struktura ekvirychlostnı́ch ploch významně
změnı́, nebot’ společně s druhou kritickou plochou se v úzké oblasti mezi oběma kritickými
body (odpovı́dajı́cı́mi lokálnı́m maximům V (ϕ) ) vyskytujı́ uzavřené – toroidálnı́ – plochy, jejichž
střed ležı́ v mı́stě lokálnı́ho minima V (ϕ) ; poslednı́ mezně uzavřená plocha je vždy kritická a
R≡
11 Abramowicz ukázal, že v přı́padě barotropnı́ diferenciálně rotujı́cı́ kapaliny plochy konstantnı́ho specifického
momentu hybnosti ` a konstantnı́ úhlové rychlosti Ω splývajı́ [Abramowicz, 1971]. Navı́c topologie ploch, ležı́cı́ch
nad kritickou – sebeprotı́najı́cı́ – plochou, je podobně jako v newtonovské mechanice cylindrická [Abramowicz,
1974].
28
Relativistické disky kolem kompaktnı́ch objektů
3
3.0
3.9
5.5
7.5
4.6
3.5
3.6
0.4
6.5
2
3.8
0.2
r cos Θ
r cos Θ
1
0
3.84
¡
¡
ª
¡
0
3.81219
3.82
3.82
3.84
-1
-0.2
-2
3.8
a = 0.99
-3 ` = 2.18 3.0
1
2
4.6
3.5
3.9
6.5
5.5
3
4
r sin Θ
-0.4
7.5
5
6
7
a = 0.999 96
` = 2.004 7
0.8
1
1.2
r sin Θ
1.4
0.6
3.6
3.4
0.4
3.6
0.2
3.83
0.2
¡
3.81068
3.82
3.82 3.84
r cos Θ
¡
¡
ª
3.76
0.4
3.8
0
1.6
(b)
(a)
r cos Θ
3.6
3.80286
3.81
0
3.84
3.9
-0.2
-0.2
3.8
-0.4
a = 0.999 96
.
` = 2.005 09
0.8
1
1.2
r sin Θ
(c)
a = 0.999 96
` = 2.007
-0.6
3.6
1.4
-0.4
1.6
0.8
1
1.2
3.6
3.76
3.4
1.4
r sin Θ
1.6
1.8
2
(d)
Obr. 14. Profily ekvirychlostnı́ch ploch v tlusté části akrečnı́ho disku s konstantnı́m rozloženı́m specifického momentu hybnosti blı́zkého hodnotě na mezně stabilnı́ orbitě. Tmavá oblast znázorňuje Kerrovu
černou dı́ru s rotačnı́m parametrem a, čárkovaně je vyznačena hranice ergosféry, vystı́novaná oblast
odpovı́dá kapalinovému disku se specifickým momentem hybnosti `, křı́ž označuje centrum disku.
Ekvirychlostnı́ plochy jsou reprezentovány hodnotou veličiny R (75), centrálnı́ prstenec toroidálnı́ch
ploch je znázorněn černým bodem. (a) V disku existujı́ pouze cylindrické ekvirychlostnı́ plochy, jediná
.
.
kritická plocha, Rc = 3.178, je otevřená. (b) Vnitřnı́ kritická plocha, Rc(in) = 3.832, uzavı́rá toroidálnı́
.
.
plochy kolem prstence, Rring = 3.847, druhá kritická plocha, Rc(out) = 3.812, je otevřená. (c) Jediná
.
kritická plocha, Rc = 3.811, se dvěma kritickými body obepı́najı́cı́ oblast toroidálnı́ch ploch kolem
.
.
prstence, Rring = 3.840. (d) Vnitřnı́ kritická plocha, Rc(in) = 3.668, je otevřená; toroidálnı́ plochy
.
.
kolem prstence, Rring = 3.814, jsou uzavřeny vnějšı́ kritickou plochou, Rc(out) = 3.803.
při vhodně zvolené hodnotě parametru ` může obsahovat oba kritické body. Obr. 14 ukazuje
rozloženı́ toroidálnı́ch ekvirychlostnı́ch ploch v blı́zce Keplerovském akrečnı́m disku, jehož
vnitřnı́ výdut’je reprezentována kapalinovým toroidem (světle stı́novaná oblast) s konstantnı́m
2. Aschenbachův efekt
29
rozloženı́m specifického momentu hybnosti blı́zkého hodnotě na mezně stabilnı́ orbitě daného
černoděrového prostoročasu.
Závěrem dodejme, že Aschenbachův efekt a s nı́m spojené toroidálnı́ ekvirychlostnı́ plochy
se vyskytujı́ pouze uvnitř ergosféry velmi rychle rotujı́cı́ Kerrovy černé dı́ry.
2.1 Shrnutı́
Aschenbachův efekt, projevujı́cı́ se změnou znaménka radiálnı́ho gradientu orbitálnı́ rychlosti
akrečnı́ho disku měřené v lokálně nerotujı́cı́ch soustavách, má tyto stěžejnı́ vlastnosti:
(i) podmı́nkou výskytu je velmi rychle rotujı́cı́ Kerrova černá dı́ra; pomocı́ předefinovaného
rotačnı́ho parametru 1 − a musı́ spin černé dı́ry splňovat:
(a) 1 − a < 4.7 × 10−3 pro Keplerovské disky,
(b) 1 − a < 2.1 × 10−4 pro kapalinové toroidy s `(r, θ) = const,
(ii) efekt nastává v úzké oblasti v blı́zkosti mezně stabilnı́ orbity a nepřekročı́ hranici ergosféry,
(iii) rozdı́l lokálnı́ho maxima a lokálnı́ho minima orbitálnı́ rychlosti v LNRF dosahuje maximálnı́ hodnoty pro extrémnı́ černou dı́ru a to:
(a) ∆V (ϕ) ≈ 0.07 c pro Keplerovské disky,
(b) ∆V (ϕ) ≈ 0.02 c pro kapalinové toroidy s `(r, θ) = `ms ,
(iv) pro kapalinové toroidy s `(r, θ) = const se projevuje existencı́ toroidálnı́ch ekvirychlostnı́ch ploch lokálně měřené orbitálnı́ rychlosti ohraničených kritickou – sebeprotı́najı́cı́ –
plochou,
(v) charakteristické frekvence spojené s možnými lokálnı́mi nestabilitami, generovanými
Aschenbachovým efektem, přepočtené pro statického pozorovatele v radiálnı́m nekonečnu,
dosahujı́ těchto maximálnı́ch hodnot:
(a) f˜∞ ≈ 97 (M/M¯ )−1 Hz pro Keplerovské disky,
(b) f˜∞ ≈ 24 (M/M¯ )−1 Hz pro kapalinové toroidy s `(r, θ) = `ms .
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[Stuchlı́k et al., 2004] Stuchlı́k, Z., Slaný, P., and Török, G. (2004). Orbital velocity gradient
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Astronomy and Astrophysics, 436(1):1.
PŮVODNÍ ČLÁNKY
Zbývajı́cı́ část disertačnı́ práce tvořı́ původnı́ časopisecké publikace v následujı́cı́m pořadı́:
[1] Z. Stuchlı́k, P. Slaný and S. Hledı́k: Equilibrium configurations of perfect fluid orbiting
Schwarzschild–de Sitter black holes, Astronomy and Astrophysics 363, 425 (2000).
[2] Z. Stuchlı́k and P. Slaný: Equatorial circular orbits in the Kerr–de Sitter spacetimes,
Physical Review D 69, 064001 (2004).
[3] P. Slaný and Z. Stuchlı́k: Relativistic thick discs in the Kerr–de Sitter backgrounds, přijato
k publikovánı́ v Classical and Quantum Gravity.
[4] Z. Stuchlı́k, P. Slaný, G. Török and M. A. Abramowicz: Aschenbach effect: Unexpected
topology changes in the motion of particles and fluids orbiting rapidly rotating Kerr black
holes, Physical Review D 71, 024037 (2005).
Články [1, 2, 4] jsou uvedeny v podobě, v nı́ž byly publikovány v konkrétnı́m časopise, článek
[3] je prezentován v „preprintovém“ stylu časopisu Classical and Quantum Gravity.
32
Astron. Astrophys. 363, 425–439 (2000)
ASTRONOMY
AND
ASTROPHYSICS
Equilibrium configurations of perfect fluid orbiting
Schwarzschild–de Sitter black holes
Z. Stuchlı́k, P. Slaný? , and S. Hledı́k
Silesian University, Department of Physics, Faculty of Philosophy and Science, Bezručovo nám. 13, Opava, Czech Republic
([email protected]; [email protected]; [email protected])
Received 30 May 2000 / Accepted 15 September 2000
Abstract. The hydrodynamical structure of perfect fluid orbiting Schwarzschild–de Sitter black holes is investigated for
configurations with uniform distribution of angular momentum
density. It is shown that in the black-hole backgrounds admitting
the existence of stable circular geodesics, closed equipotential
surfaces with a cusp, allowing the existence of toroidal accretion disks, can exist. Two surfaces with a cusp exist for the
angular momentum density smaller than the one corresponding
to marginally bound circular geodesics; the equipotential surface corresponding to the marginally bound circular orbit has
just two cusps. The outer cusp is located nearby the static radius where the gravitational attraction is compensated by the
cosmological repulsion. Therefore, due to the presence of a repulsive cosmological constant, the outflow from thick accretion
disks can be driven by the same mechanism as the accretion
onto the black hole. Moreover, properties of open equipotential
surfaces in vicinity of the axis of rotation suggest a strong collimation effects of the repulsive cosmological constant acting on
jets produced by the accretion disks.
Key words: accretion, accretion disks – black hole physics –
gravitation – relativity – galaxies: quasars: general
1. Introduction
Recent observations give strong evidence that the energy
sources of quasars and active galactic nuclei are accretion
disks around central massive black holes (Abramowicz & Percival 1997; Blandford 1990). Similar, scaled down, accretion
disks appear in some extraordinary galactic binary systems containing a black hole (or a neutron star). In the accretion disks,
the potential gravitational energy of matter orbiting the central
black hole is liberated and transferred into heat, due to viscous
stresses acting against shearing motion, and radiated (at least
partly) away. During the process, angular momentum of the accreting matter has to be transported outwards.
It is well known that at low accretion rates the pressure is
negligible, and the accretion disk is geometrically thin. Its baSend offprint requests to: Z. Stuchlı́k
?
Present address: Technical University at Ostrava, Department of
Physics, Faculty of Mining, Ostrava, Czech Republic
sic properties are determined by the circular geodesic motion
in the black-hole background. The radius rms of the marginally
stable circular orbit represents the inner edge of the Keplerian
disks, since matter, following quasikeplerian orbits down to rms ,
falls freely into the black hole from this radius (Novikov &
Thorne 1973; Stoeger 1976). At high accretion rates, the pressure is relevant, and the accretion disk must be geometrically
thick. Its basic properties are determined by equipotential surfaces of test perfect fluid (i.e., perfect fluid that does not alter
the black-hole geometry) rotating in the black-hole background.
The accretion is possible, if a toroidal equilibrium configuration
of the test fluid containing a critical, self-crossing equipotential
surface can exist in the background. The cusp of this equipotential surface corresponds to the inner edge of the disk, and the
accretion inflow of matter into the black hole is possible due to
a mechanical non-equilibrium process, i.e., because of matter
slightly overcoming the critical equipotential surface. The pressure gradients push the inner edge of the thick disks under the
radius rms (Kozlowski et al. 1978; Abramowicz et al. 1978).
The simplest, but quite illustrative case of the equipotential surfaces of the test fluid can be constructed for the configurations with uniform distribution of the angular momentum density. This case is fully governed by the geometry of the
spacetime, however, it contains all the characteristic features of
more complex cases of the rotation of the fluid (Jaroszyński et
al. 1980). Moreover, this case is also very important physically
since it corresponds to marginally stable equilibrium configurations (Seguin 1975).
The equipotential surfaces were analyzed for both
Schwarzschild and Kerr black-hole spacetimes. The critical
closed surfaces with a cusp can exist for angular momentum density higher (lower) than the one corresponding to the marginally
stable (bound) circular geodesic, and the location of the cusp
shifts from rms to the radius of the marginally bound geodesic
orbit rmb . The cusp close to the horizon enables the inflow of
matter into the black hole. However, the character of the equilibrium configurations does not allow outflow of matter and
transfer of the angular momentum for disks around isolated
black holes. In binary systems the outflow is possible through
the Lagrange point L2 – see, e.g., Novikov & Thorne (1973).
Very recently, a wide variety of cosmological observations
(measurements of the present value of the Hubble parameter, de-
426
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
tails of the anisotropy of the cosmic relic radiation, statistics of
gravitational lensing of quasars, and high-redshift supernovae)
suggest a non-zero, repulsive cosmological constant (Krauss &
Turner 1995; Ostriker & Steinhardt 1995; Krauss 1998). Therefore, it is interesting to clarify the influence of the repulsive
cosmological constant on astrophysically relevant properties of
black-hole spacetimes.
Here, we shall show that in the field of black-hole spacetimes with a repulsive cosmological constant the outflow of
matter from the accretion disk is possible, because equipotential surfaces with an outer cusp in vicinity of the so called
static radius can exist (beside the critical surfaces with the inner cusp nearby the horizon), if the mass of the black hole is
small enough to admit existence of the stable circular geodesics
(Stuchlı́k & Hledı́k 1999). Moreover, if the uniform angular momentum density of the equilibrium configuration corresponds
to the marginally bound orbit of the background, the critical
equipotential surface has both the inner and outer cusps. In this
situation, any mechanical non-equilibrium in the thick disk leads
to both inflow into the black hole, and outflow from the disk near
the static radius.
The plan of this paper is following. In Sect. 2, the basic formulae for the equilibrium configurations of test perfect fluid in a given stationary and axially symmetric background are summarized, following the papers of Abramowicz
and coworkers (Kozlowski et al. 1978; Abramowicz et al. 1978;
Jaroszyński et al. 1980). In Sect. 3, the equipotential surfaces of
the marginally stable configurations (having a uniform distribution of angular momentum density) of the test perfect fluid are
determined for the Schwarzschild–de Sitter black-hole spacetimes. For completeness, we include also discussion of the case
of the Schwarzschild–anti-de Sitter spacetimes with an attractive cosmological constant. In Sect. 4, some concluding remarks
are presented, and astrophysical consequences of the presented
results are pointed out. We shall use the geometric system of
units (c = G = 1), if not stated otherwise.
We shall consider test perfect fluid rotating in the φ direction.
Its four velocity vector field U µ has, therefore, only two nonzero components
U µ = (U t , U φ , 0, 0),
(2)
which can be functions of the coordinates r, θ. The stress-energy
tensor of the perfect fluid is
T µν = (p + )U µ Uν + p δνµ ,
(3)
where and p denote the total energy density and the pressure
of the fluid. The rotating fluid can be characterized by the vector
fields of the angular velocity Ω, and the angular momentum per
unit mass (angular momentum density) `, defined by
Ω=
Uφ
,
Ut
`=−
Uφ
.
Ut
(4)
These vector fields are related by
Ω=−
gtφ + `gtt
.
gφφ + `gtφ
(5)
In static spacetimes (gtφ = 0), the relation (5) reduces to a
simple formula
gtt
Ω
=−
.
`
gφφ
(6)
The surfaces of constant ` and Ω are called von Zeipel’s
cylinders. The family of von Zeipel’s cylinders does not depend on the assumed rotation law of the fluid, ` = `(Ω), in the
static spacetimes, but it will depend on the rotation law in the
/ 0) (Kozlowski et al. 1978).
stationary spacetimes (with gtφ =
Projecting the energy conservation law T µν;ν = 0 onto the
hypersurface orthogonal to the four velocity U µ by the projection tensor hµν = gµν + Uµ Uν , we obtain the relativistic Euler
equation in the form
Ω ∂µ `
∂µ p
= −∂µ (ln Ut ) +
,
p+
1 − Ω`
(7)
where
2
gtφ
− gtt gφφ
.
gφφ + 2`gtφ + `2 gtt
2. Boyer’s condition for equilibrium configurations
of test perfect fluid
(Ut )2 =
We briefly summarize the well known results of a general theory of the equipotential surfaces inside any relativistic, differentially rotating, perfect fluid body (Boyer 1965; Abramowicz 1974), applied to test configurations of perfect fluid rotating
in the stationary and axially symmetric spacetimes (Kozlowski
et al. 1978; Abramowicz et al. 1978; Jaroszyński et al. 1980).
In the standard coordinate system the spacetimes are described
by the line element
The solution of the relativistic Euler equation can be given
by Boyer’s condition determining the surfaces of constant
pressure through the “equipotential surfaces” of the potential
W (r, θ) by the relations (Abramowicz et al. 1978)
Z p
dp
= Win − W,
(9)
p
+
0
Z `
Ω d`
;
(10)
Win − W = ln(Ut )in − ln(Ut ) +
1
− Ω`
`in
ds2 = gtt dt2 + 2gtφ dtdφ + gφφ dφ2 + grr dr2 + gθθ dθ2 , (1)
where the metric coefficients depend neither on the time coordinate, t, nor the azimuthal coordinate, φ, i.e., the spacetimes
contain timelike and azimuthal Killing vector fields ∂/∂t and
∂/∂φ.
(8)
the subscript “in” refers to the inner edge of the disk. For an
alternative definition of Boyer’s condition see (Abramowicz
et al. 1978; Fishbone & Moncrief 1976; Fishbone 1977). The
equipotential surfaces are determined by the condition
W (r, θ) = const,
(11)
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
and in a given spacetime can be found from Eq. (10), if a rotation law Ω = Ω(`) is given. The surfaces of constant pressure
p(r, θ) = const are given by Eq. (9). The structure of thick
accretion disks can be obtained also in the framework of a very
practical and accurate Newtonian model for the gravitational
field of a non-rotating black hole, known as the Paczyński–Wiita
potential (Paczyński & Wiita 1980; Abramowicz et al. 1980).
3. Equipotential surfaces
of the marginally stable configurations
orbiting Schwarzschild–de Sitter black holes
Equilibrium configurations of test perfect fluid rotating around
an axis of rotation in a given spacetime are determined by
the equipotential surfaces, where the gravitational and inertial
forces are just compensated by the pressure gradient. (In an axially symmetric spacetime, the axis of rotation coincides with
the axis of symmetry of the spacetime, while in a spherically
symmetric spacetime the axis of rotation can be any radial line;
usually, the coordinate system is chosen so that the rotation axis
corresponds to θ = 0.)
The equipotential surfaces can be closed or open. Moreover, there is a special class of critical, self-crossing surfaces
(with a cusp), which can be either closed or open. The closed
equipotential surfaces determine stationary equilibrium configurations. The fluid can fill any closed surface—at the surface of
the equilibrium configuration pressure vanish, but its gradient is
non-zero (Kozlowski et al. 1978). On the other hand, the open
equipotential surfaces are important in dynamical situations,
e.g., in modeling jets (Lynden-Bell 1969; Blandford 1987). The
critical, self-crossing closed equipotential surfaces Wcusp are
important in the theory of thick accretion disks, because accretion onto the black hole through the cusp of the equipotential
surface located in the equatorial plane is possible due to the
Paczyński mechanism.
According to Paczyński, the accretion into the black hole is
driven through the vicinity of the cusp due to a little overcoming
of the critical equipotential surface, W = Wcusp , by the surface
of the disk. The accretion is thus driven by a violation of the
hydrostatic equilibrium, rather than by viscosity of the accreting
matter (Kozlowski et al. 1978).
It is well known that all characteristic properties of the
equipotential surfaces for a general rotation law are reflected
by the equipotential surfaces of the simplest configurations
with uniform distribution of the angular momentum density
` – see Jaroszyński et al. (1980). Moreover, these configurations are very important astrophysically, being marginally stable
(Seguin 1975). Under the condition
`(r, θ) = const,
(12)
holding in the rotating fluid, a simple relation for the equipotential surfaces follows from Eq. (10):
W (r, θ) = ln Ut (r, θ),
(13)
with Ut (r, θ) being determined by ` = const, and the metric
coefficients only.
427
The equipotential surfaces are described by the formula θ =
θ(r), which can be given by the differential equation
dθ
∂p/∂r
=−
,
dr
∂p/∂θ
(14)
which for the configurations with ` = const reduces to
∂Ut /∂r
dθ
.
=−
dr
∂Ut /∂θ
(15)
The influence of a non-zero cosmological constant on character of the equipotential surfaces of the marginally stable configurations rotating around a black hole will be examined in
the simplest case of Schwarzschild–de Sitter spacetimes corresponding to a repulsive cosmological constant, Λ > 0. (For
completeness, we briefly discuss the case of Schwarzschild–
–anti-de Sitter spacetimes corresponding to an attractive cosmological constant, Λ < 0.)
In the standard Schwarzschild coordinates, the non-zero
metric coefficients of the Schwarzschild–(anti)-de Sitter spacetimes are
−1
= (1 − 2r−1 − yr2 ),
− gtt = grr
gθθ = r2 ,
gφφ = r2 sin2 θ.
(16)
(17)
(18)
Here, the radial coordinate r is expressed in units of the mass
parameter M , and the dimensionless cosmological constant parameter
y = 13 ΛM 2
(19)
is introduced. It should be stressed that a static region exists in
the Schwarzschild–de Sitter spacetimes with subcritical values
of
y < yc =
1
27 ;
(20)
of course, the equilibrium configurations are possible only in
these spacetimes. Now, the equipotential surfaces are given by
the formulae
(1 − 2r−1 − yr2 )1/2 r sin θ
W (r, θ) = ln 1/2
r2 sin2 θ − (1 − 2r−1 − yr2 )`2
(21)
and
r(1−yr3 ) sin2 θ − (1−2r−1 −yr2 )2 `2 r
dθ
;
= tan θ
dr
(r−2−yr3 )2 `2
(22)
for y = 0 these relations reduce to the well known
Schwarzschild formulae (Jaroszyński et al. 1980).
The best insight into the nature of the ` = const configurations can be obtained by the examination of the behavior of the
potential W (r, θ) in the equatorial plane (θ = π/2). There are
two reality conditions of W (r, θ = π/2):
1 − 2r−1 − yr2 ≥ 0,
r2 − (1 − 2r−1 − yr2 )`2 ≥ 0.
(23)
(24)
428
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
1. x 1015
1. x 1012
r
1. x 109
1. x 106
1000
1
1. x 10-25
1. x 10-19
1. x 10-13
y
The first condition is identical with the condition for the static
regions (located between the black-hole and cosmological horizons); the second condition can be expressed in the form
`2 ≤ `2ph (r; y) ≡
r3
.
r − 2 − yr3
(25)
The function `2ph (r; y) is the effective potential of the photon
geodesic motion; recall that ` ≡ Uφ /Ut corresponds to the
definition of the impact parameter for photon’s geodesic motion
– see Stuchlı́k & Hledı́k (1999). Further, the condition of the
local extrema of the potential W (r, θ = π/2) is identical with
the condition of vanishing of the pressure gradient (∂Ut /∂r =
0, ∂Ut /∂θ = 0). Since at the equatorial plane there is ∂Ut /∂θ =
0 independently of the ` = const, and
∂Ut (r, θ = π/2)
∂r r(1−yr3 ) − (1−2r−1 −yr2 )2 `2
,
=
3/2
(1−2r−1 −yr2 )1/2 [r2 − (1−2r−1 −yr2 )`2 ]
(26)
we arrive at the condition
`2 = `2K (r; y) ≡
3
3
r (1 − yr )
.
(r − 2 − yr3 )2
(27)
The extrema of W (r, θ = π/2) correspond to the points, where
the fluid moves along a circular geodesic, since `2K (r; y) corresponds to the distribution of the angular momentum density of
the circular geodesic orbits. Clearly,
Wextr (r, θ = π/2; y) = ln Ec (r, y),
(28)
where
Ec (r, y) =
2
1 − − yr2
r
−1/2
3
1−
r
1. x 10-7
0.1
motion (Stuchlı́k & Hledı́k 1999).) The most important properties of the potential W (r, θ) are determined by its behavior
at the equatorial plane, and, especially, by the properties of the
functions `2ph (r; y), and `2K (r, y). Discussion of these properties enables us to give a classification of the Schwarzschild–
–(anti)-de Sitter spacetimes according to the properties of the
equipotential surfaces of test perfect fluid. We shall separate the
discussion to the case of the Schwarzschild–de Sitter (y > 0),
and Schwarzschild–anti-de Sitter spacetimes (y < 0). For the
pure Schwarzschild spacetime (y = 0) the analysis can be found
in (Kozlowski et al. 1978).
3.1. Schwarzschild–de Sitter black holes
If y > 0, the function `2ph (r, y) diverges at the black-hole horizon, rh , and the cosmological horizon, rc , determined by equality in the condition (23). The horizons are given by the relations
π+ξ
2
,
rh = √ cos
3
3y
π−ξ
2
,
rc = √ cos
3
3y
is the specific energy of the circular geodesics. (Recall that the
specific energy of circular geodesics corresponds to the local
extrema of the effective potential Veff (r; `, y) of the geodesic
(30)
(31)
where
p ξ = cos−1 3 3y .
(32)
The radii of the horizons are illustrated in Fig. 1. The local
minimum of `2ph (r, y) is located at rph = 3, independently of y,
and determines the unstable photon circular geodesic with the
impact parameter
`2ph(c) = `2ph(min) (y) ≡
(29)
Fig. 1. Characteristic radii of the
Schwarzschild–de Sitter spacetimes as
functions of the parameter y. The black
hole (rh ) and cosmological (rc ) horizons
are given by bold solid lines, the static
radius (rs ) by bold dotted line, the radii of
marginally stable orbits (rms(i) and rms(o) )
by thin dashed lines, and marginally bound
orbits (rmb(i) and rmb(o) ) by thin solid
lines.
27
.
1 − 27y
(33)
The function `2K (r; y), determining the Keplerian (geodesic)
circular orbits, has a zero point at the so called static radius
rs (y) given by
rs = y −1/3 ,
(34)
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
429
10000
l
1000
100
10
1
1. x 10-25
1. x 10-19
1. x 10-13
y
1. x 10-7
0.1
Fig. 2. The angular momentum density of
the marginally stable (`ms(i) and `ms(o) ,
solid line) and marginally bound (`mb ,
dashed line) orbits as functions of the parameter y of the Schwarzschild–de Sitter
spacetimes. Note that lim `mb = 4 =
/
y→0
√
lim `ms(i) = 3 3/2, lim `ms(o) = +∞.
y→0
y→0
1
0.99
0.98
E
0.97
0.96
0.95
0.94
0.93
1. x 10-25
1. x 10-19
1. x 10-13
y
1. x 10-7
0.1
Fig. 3. The specific energy of the marginally
stable (Ems(i) and Ems(o) , solid line) and
marginally bound (Emb , bold dotted line)
orbits as functions of the parameter y
of the Schwarzschild–de Sitter spacetimes.
Note that lim Ems(o) = lim Emb = 1,
y→0
y→0
√
lim Ems(i) = 2 2/3.
y→0
and it is not well defined at r > rs , being negative there. At the
static radius (unstable) stationary equilibrium of test particles is
possible because the gravitational attraction of the black hole is
just compensated by the cosmological repulsion there.
The function `2K (r; y) diverges at the black-hole horizon:
2
`K (r → rh , y) → +∞; at the cosmological horizon, there is
`2K (r → rc , y) → −∞. Since
r2 [r − 6 + yr3 (15 − 4r)]
∂`2K
=
,
∂r
(r − 2 − yr3 )3
(35)
the local extrema of `2K (r, y) are given by the condition
r−6
,
y = yms (r) ≡ 3
r (4r − 15)
ye = 1/118125 = 1/(33 54 7) ∼ 0.00000846.
(36)
determining the marginally stable circular geodesics. The local
maximum of yms (r) gives the critical value of the parameter y
admitting stable circular orbits
yms = 12/154 ∼ 0.000237.
If y < yms , there exists an inner (outer) marginally stable
circular geodesic at rms(i) (rms(o) ), see Fig. 1. The angular
momentum density of the marginally stable orbits `ms(i) (y),
and `ms(o) (y), is simultaneously determined by Eqs. (27) and
(36)—see Fig. 2. The specific energy of these orbits Ems(i) (y),
and Ems(o) (y), is simultaneously determined by Eqs. (29) and
(36)—see Fig. 3. There is other special value of y, corresponding to the situation, where the value of the minimum of `2ph (r; y)
equals to the maximum of `2K (r; y). We denote this value ye . It
can be found that
(37)
(38)
In the Schwarzschild–de Sitter spacetimes, there is another
important class of circular geodesics—namely the marginally
bound orbits. These orbits exist in the Schwarzschild–de Sitter
spacetimes admitting existence of the stable circular orbits, i.e.,
spacetimes with y < yms . In these spacetimes, there exists an
inner, rmb(i) (outer, rmb(o) ), marginally bound orbit close to the
430
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
50
y = 5 x 10-6 < 1/118125
30
20
l2K
l2mb
0.4
0.6
0.8
50
l2K
1
1.2
log r
l2mb
10
1.4
1.6
0.4
b
0.6
0.8
1
log r
1.2
50
y = 2 x 10-5 > 1/118125
l2ph
40
l2
l2K
1.4
1.6
y = 10-3
l2ph
40
30
20
30
20
10
a
y = 1/118125
l2ph
40
l2
l2
40
l2
50
l2ph
30
20
l2K
l2mb
10
c
0.4
0.6
0.8
1
log r
1.2
10
1.4
1.6
d
0.4
0.6
0.8
log r
1
1.2
1.4
Fig. 4a–d. Behavior of the functions `2ph (r; y) and `2K (r; y) in the four qualitatively different cases determining the four classes of the
Schwarzschild–de Sitter spacetimes with different properties of the equipotential surfaces (both r and `2 are given in units of M ). Figures
a–d reflect subsequently the cases 0 < y < ye , y = ye , ye < y < yms , and yms < y < yc . In the shaded region, the equipotential surfaces
are not defined in the equatorial plane of the spacetime, defined by the axis of rotation of the perfect fluid. The descending parts of the function `2K (r; y) determine the cusps, while the growing parts determine central rings of the equilibrium configurations. The dotted line (`2ph(c) )
determines the impact parameter of the photon circular geodesic at r = 3.
black-hole horizon (static radius). These orbits are defined by
the condition
Emb (rmb(i) , `mb ) = Emb (rmb(o) , `mb ),
(39)
and are determined by an appropriate numerical procedure
(see Figs. 1–3). In the Schwarzschild spacetime (y = 0) the
marginally bound orbit is located at rmb = 4, and Emb = 1—
it is because the effective potential of the geodesic motion
Veff → 1 at r → ∞ independently of the angular momentum
density in the Schwarzschild spacetime. In the Schwarzschild–
–de Sitter spacetimes with yms < y < yc the marginally bound
circular orbits are not defined because only unstable circular
orbits exist in these spacetimes; particles from them can always
escape to infinity.
We can distinguish four qualitatively different cases of the
behavior of the functions `2ph (r; y), `2K (r; y) which give four
classes of the Schwarzschild–de Sitter black holes with different
character of the equipotential surfaces of the rotating perfect
fluid. These four classes are defined according to values of the
cosmological parameter y in the following way:
(A)
(B)
(C)
(D)
0 < y < ye ,
y = ye ,
ye < y < yms ,
yms ≤ y < yc .
For these classes, the typical behavior of the functions `2ph (r; y),
`2K (r; y), with y fixed, is given in Figs. 4(a)–(d). For completeness, the corresponding value of `mb (y) is exhibited in these
figures. Note that the descending parts of the curve `2K (r; y)
(with y fixed) correspond to the unstable circular geodesics,
while its growing part (if it exists) corresponds to the stable circular geodesics. The extrema of `2K (r; y), if they exist, have an
important role: the minimum `ms(i) , at rms(i) , determines the
inner marginally stable circular geodesic, while the maximum
`ms(o) , at rms(o) , determines the outer marginally stable circular
geodesic.
Properties of the equipotential surfaces can be established
easily, using the behavior of the potential W (r, θ) in the equatorial plane. The properties of the potential W (r, θ = π/2; y)
are closely related to the properties of the effective potential of the geodesic motion, and at their local extrema, located at the same radii, the condition (28) is satisfied. Further,
W (r; θ = π/2, y) → −∞, if r → rh or r → rc . The topological properties of the equipotential surfaces can be directly
inferred from the properties of the potential W (r, θ = π/2; y).
The local extrema of the potential W (r, θ = π/2; y) are determined by the condition
`2 = `2K (r; y);
(40)
therefore, at the radii determined by the local extrema of
W (r, θ = π/2; y), perfect fluid follows free, geodesic circular orbits. The maxima of the potential are determined by the
descending part of `2K (r; y), they correspond to the cusps of
the equipotential surfaces, and the matter moves along an unstable geodesic orbit at the corresponding radii. The minima
of the potential are determined by the rising part of `2K (r; y),
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
they correspond to the central rings of the equilibrium configurations, and the matter moves along a stable geodesic orbit at
the corresponding radii.
Now, we give a complete survey of the behavior of the
equipotential surfaces, and the related potential W (r, θ =
π/2; y). We start with the astrophysically most important case.
(A) 0 < y < ye . From Fig. 4a, we obtain nine qualitatively different cases of the behavior of the potential
W (r, θ = π/2), and corresponding nine qualitatively different families of the equipotential surfaces, according to
the values of ` = const. (In the following, we consider
` > 0 only. This can be done due to the symmetry of the
spacetimes under consideration.)
(I) ` < `ms(i) . Open surfaces only; no disks are possible.
Surface with the outer cusp exists. (Fig. 5a)
(II) ` = `ms(i) . An infinitesimally thin, unstable ring located at rms(i) exists. An open surface with the outer
cusp exists. (Fig. 5b)
(III) `ms(i) < ` < `mb . Closed surfaces exist. Many equilibrium configurations without cusps are possible, and
one with the inner cusp. An open surface with the outer
cusp exists. (Fig. 5c)
(IV) ` = `mb . Many equilibrium configurations without
cusps are possible. There is an equipotential surface with
both the inner and outer cusps. Now, the mechanical nonequilibrium causes an inflow into the black hole, and an
outflow from the disk, with the same efficiency; it is the
most interesting new feature of the accretion processes
caused by the presence of a repulsive cosmological constant. (Fig. 5d)
(V) `mb < ` < `ph(c) . Equilibrium configurations are possible because closed equipotential surfaces exist. However, accretion into the black hole is impossible because
the equilibrium configurations (closed surfaces) have no
inner cusp; the inner cusp has an open equipotential surface. The outer cusp belongs to a closed surface, and the
outflow from the disk is possible. (Fig. 5e)
(VI) ` = `(ph(c)) . The potential W (r, θ = π/2; y) diverges
at the photon circular orbit located at r = 3, and the inner
cusp disappears. The closed equipotential surfaces still
exist, with the most extended one containing the outer
cusp that enables outflow from the disk. (Fig. 5f)
(VII) `ph(c) < ` < `ms(o) . In the region defined by
`2ph (r; y), the equipotential surfaces cannot reach the
equatorial plane. The closed equipotential surfaces exist, one with the outer cusp. (Fig. 5g)
(VIII) ` = `ms(o) . An infinitesimally thin, unstable ring
located at rms(o) exists (the center, and the outer cusp
coalesce). (Fig. 5h)
(IX) ` > `ms(o) . Open equipotential surfaces exist only.
There is no cusp in this case. (Fig. 5i).
(B) y = ye . For this special value of y (Fig. 4b), we still obtain
the families of equipotential surfaces given by (A-I)–(A-V)
and (A-IX). However, the case (A-VII) disappears, and the
cases (A-VI) and (A-VIII) coalesce, giving the case
431
(X) ` = `ph(c) = `ms(o) . The inner cusp just disappears,
while the outer cusp coalesce with the center. (Fig. 5j)
(C) ye < y < yms . From Fig. 4c it follows that the intervals
of `, and the families of equipotential surfaces (A-I)–(A-IV)
remain. The following new intervals of the angular momentum density must be introduced.
(XI) `mb < ` < `ms(o) . This case is equivalent to the case
(A-V).
(XII) ` = `ms(o) . There is the inner cusp of an open
equipotential surface, but the center and the outer cusp
coalesce—this corresponds to an infinitesimally thin unstable ring, located at rms(o) . (Fig. 5k)
(XIII) `ms(o) < ` < `ph(c) . There are open surfaces only,
one being with the inner cusp. (Fig. 5l)
(XIV) ` ≥ `ph(c) . This case corresponds to the case (A-IX).
(D) yms ≤ y < yc . For this interval of y, the function
`2K (r; y) is descending everywhere (see Fig. 4d). Only maxima of the potential W (r, θ = π/2; y) are possible (if
`2 < `2ph(c) ), and open equipotential surfaces can exist
only. Equilibrium configurations corresponding to toroidal
disks are not possible. This is quite natural result, since in
the spacetimes under consideration stable circular geodesics
cannot exist. Now, there are only two different intervals of
the parameter `.
(XV) ` < `ph(c) . This family of equipotential surfaces corresponds to the case (A-I).
(XVI) ` ≥ `ph(c) . This family of equipotential surfaces
corresponds to the case (A-IX).
Values of the potential at the central ring and the cusps (provided they exist) are given in Table 1. Note that the maximum
difference between the values of the potential W on the boundary and at the center of the toroidal disk in the Schwarzschild
spacetime is ∆W = 0.0431 (Abramowicz et al. 1978). Comparing this with the value ∆Wi = ∆Wo = 0.0309 from Table 1
characterizing the limiting accretion disk with ` = `mb , we can
conclude that the presence of a repulsive cosmological constant
makes the structure of the disk ‘smoother’.
The Schwarzschild case y = 0 was discussed in (Kozlowski
et al.1978) and will not be repeated here. We only mention
that the critical self-crossing surface for the marginally bound
configurations Wcusp (` = `mb ; y = 0) = 0, while Wcusp (` =
`mb , y > 0) < 0.
3.2. Schwarzschild–anti-de Sitter black holes
If y < 0, the function `2ph (r, y) diverges at infinity, and at the
black-hole horizon given by the relation
"
1/2 #1/3
1
1
1
−
rh = − +
y
y2
27y 3
"
1/2 #1/3
1
1
1
+ − −
−
.
(41)
y
y2
27y 3
The local minimum of `2ph (r; y) is again located at rph = 3,
and the impact parameter of the corresponding photon cir-
432
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
(a)
y = 10−6
3
` = 2.34
(b)
y = 10−6
3
` = 3.67403603
- 10.0
- 10.0
2
2
0.01
cusp
- 0.3
-0.04
0
- 1
-0.06
-0.08
-0.1
-0.12
- 2
-0.25
r ) cos
-0.02
( log
-0.2
- 0.025
1
- 0.03
- 0.09
W(r, π/2)
r ) cos
-0.15
( log
W(r, π/2)
-0.1
θ
0
1
cusp2
- 0.01
θ
0
-0.05
- 0.06
- 0.06
- 0.01
0.01
- 0.046
cusp1
cent
0
- 1
- 2
-0.14
0
0.5
1
1.5
2
- 3
2.5
(c)
y = 10−6
0
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
log r
1
1.5
2
- 3
2.5
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
log r
3
` = 3.84
0.5
(d)
y = 10−6
3
` = 3.93920702
- 10.0
2
-0.02
-0.06
- 1
- 2
-0.08
0
0.5
1
1.5
2
(e)
-0.06
0
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
0.5
1
1.5
2
- 3
2.5
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
3
` = 4.18
(f)
- 10.0
2
y = 10−6
` = 5.19622257
- 10.0
2
- 0.06
cusp1
- 0.06
0.01
cusp2
θ
0.0
θ
cent
- 1
log r
3
y = 10−6
- 0.035
0
- 2
-0.08
- 3
2.5
log r
cusps
- 0.025
-0.04
( log
- 0.04
cent
W(r, π/2)
r ) cos
cusp1
-0.04
( log
W(r, π/2)
1
- 0.022
0
- 0.01
θ
0
0.1
1
-0.02
- 0.06
0.01
r ) cos
θ
- 0.01
0
- 10.0
2
- 0.06
cusp2
0.5
0.9
- 0.01
1
1
cusp
0
W(r, π/2)
-0.025
- 1
-0.05
( log
r ) cos
cent
0
0
( log
W(r, π/2)
0.025
r ) cos
- 0.025
0.05
-0.02
-0.04
- 2
- 0.018
0
cent
- 1
- 2
-0.075
-0.06
- 3
- 3
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
log r
0
(h)
(g)
y = 10
` = 5.9
1
1.5
2
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
2.5
log r
3
−6
0.5
y = 10−6
3
` = 7.12938367
- 10.0
2
2
- 10.0
- 0.06
0.4
- 0.06
cusp
( log
0
- 0.01
- 1
- 0.02
- 2
- 0.03
θ
cent
W(r, π/2)
r ) cos
- 0.015
0
0.1
0.05
1
- 0.01
0
-0.05
( log
θ
W(r,
π /2)
0.01
cusp
r ) cos
0.01
0.8
1
0.02
0
0.0
cent
- 1
-0.1
- 2
-0.15
- 0.04
- 3
0
0.5
1
log
1.5
r
2
2.5
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
1
1.5
2
log r
2.5
3
- 3
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
Fig. 5a–h. Equipotential surfaces (meridional sections) for the marginally stable (` = const) configurations of test perfect fluid orbiting the
Schwarzschild–de Sitter black-holes, and the related potential W (r, θ = π/2; y). The radial coordinate is expressed in units of M ; the logarithmic
scale is used, in order to cover whole the range between the inner and outer cusps. The central black hole is shaded. The sequence of figures a–l
covers all the possibilities of the behavior of the equipotential surfaces for black holes in spacetimes with a repulsive cosmological constant. The
sequence a–i gives successively all the possibilities for the behavior of the equipotential surfaces in the spacetimes of class A, with 0 < y < ye ,
which is the astrophysically most plausible class. For the spacetimes of the classes B–D, the relevant sequences of the equipotential surfaces are
determined in the text. The cusps of the toroidal disks correspond to the local maxima of W (r, θ = π/2), the central rings correspond to their
local minima. The dashed lines give asymptotics of W (r, θ = π/2), and determine the interval of radii where the equipotential surfaces cannot
reach the equatorial plane.
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
433
3
(i)
y = 10−6
` = 8.11
(j) y = 1/118125
- 10.0
2
` = 5.19674637
θ
r ) cos
0
0.2
( log
-0.02
0
- 1
-0.03
- 2
1.25
1.5
1.75
2
2.25
(k)
y = 2 × 10−5
0
0 0.5
1 1.5
2 2.5
( log
r ) sin
θ
` = 4.62683526
0
cent
- 1
- 2
- 3
2.5
log r
- 0.02
- 0.2
- 0.4
-0.04
1
0.01
cusp
0.4
π /2)
-0.01
( log
W(r, π/2)
0
1
- 0.07
0.0
1
W(r,
θ
r ) cos
0.01
- 10.0
1.5
- 0.012
0.02
2
- 0.06
0.5
0.5
3
1
1.5
r
log
(l)
2
y = 2 × 10−5
2
0 0.5
( log
2.5
` = 4.98
1 1.5
2 2.5
r ) sin
θ
2
- 10.0
- 10.0
- 0.09
cusp
5.0
θ
θ
cusp1
1
- 0.09
1
10.0
0.8
cent
0
0.4
W(r, π/2)
( log
W(r, π/2)
r ) cos
0.6
0.1
0
- 0.03
cusp2
- 1
-0.1
0.0
0
0.2
( log
r ) cos
0.0
0.2
0
- 1
-0.2
-0.4
-0.2
- 2
- 2
-0.6
0
0.5
1
1.5
2
0
log r
0.5
( log
1 1.5
r ) sin
2
0
θ
0.5
1
1.5
log r
2
0
0.5
( log
1 1.5
r ) sin
2
θ
Fig. 5i–l.
50
y = -10-6
l2ph
l2
40
30
20
l2K
10
0.4
0.6
0.8
1
1.2
log r
1.4
1.6
Fig. 6. Behavior of the functions `2ph (r; y) and `2K (r; y) for the
Schwarzschild–anti-de Sitter spacetimes, given for y = −10−6 (both r
and `2 are given in units of M ). The dotted line determines `2ph(c) , as in
Fig. 4. It is qualitatively similar to the pure Schwarzschild case (y = 0),
and it has the same character for all y < 0. In the shaded region, the
equipotential surfaces are not defined in the equatorial plane.
cular geodesic is given by Eq. (33). If y < 0, there is no
zero point of `2K (r, y) and `2K (r → rh , y) → +∞, `2K (r →
∞, y) → +∞. Now, Eq. (36) determines only one marginally
stable circular geodesic, close to the horizon. On the other hand,
in the Schwarzschild–anti-de Sitter spacetimes the notion of
marginally bound circular geodesic ceases any meaning because
particles from the unstable circular orbits never escape to infinity, since the effective potential diverges at infinity for each value
of the angular momentum density (Stuchlı́k & Hledı́k 1999).
If y < 0, the behavior of the functions `2ph (r; y) and `2K (r; y)
is qualitatively the same as in the Schwarzschild case. It is illustrated in Fig. 6. The function `2K (r; y) has a minimum `ms at rms
corresponding to the marginally stable circular geodesic. The
unstable geodesics are given by the descending part of `2K (r; y),
while the stable are given by the rising part.
Now, it is immediately clear that for all of the
Schwarzschild–anti-de Sitter spacetimes we always obtain four
possible cases of the behavior of the potential W (r, θ = π/2; y)
and four corresponding families of the equipotential surfaces;
notice that W (r, θ = π/2, y) → ∞ as r → ∞. These cases are
given by the following intervals of `:
(I) ` < `ms . There are open equipotential surfaces only.
(Fig. 7a)
(II) ` = `ms . An infinitesimally thin unstable ring is located at
rms . (Fig. 7b)
(III) `ms < ` ≤ `ph(c) . Closed equipotential surfaces exist, one
with the cusp that enables accretion from the toroidal disk
into the black hole. (Fig. 7c)
(IV) ` > `ph(c) . Closed equipotential surfaces exist, but no
with a cusp at the equatorial plane. In vicinity of the horizon (in region limited by radii determined by the equation
`2ph (r; y) = `2 ) the equipotential surfaces cannot cross the
equatorial plane. (Fig. 7d)
Values of the potential at the cusp and the central ring (provided they exist) are given in Table 2.
434
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
(a)
y = −10−6
3
` = 2.18
(b)
0.4
y = −10−6
3
` = 3.67443285
0.4
0.05
0.07
2
2
0.0
0.0
- 0.025
0.1
θ
θ
0.1
1
1
- 0.022
-0.05
- 1
- 2
-0.1
r ) cos
0
-0.05
( log
0
0
( log
W(r, π/2)
0.05
- 0.3
W(r, π/2)
r ) cos
- 0.08
0.05
-0.1
0.5
1
1.5
2
2.5
(c)
y = −10−6
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
log r
3
3
` = 4.12
- 1
- 3
0
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
log r
cusp
cent
- 2
-0.15
- 3
0
- 0.4 - 0.045
0
(d)
0.4
2
y = −10
−6
` = 5.8
0.4
2
cusp
1
0.2
cent
0.15
-0.05
- 1
0.05
- 2
-0.1
0
cent
0.1
W(r,
( log
0
( log
0
- 0.013
r ) cos
- 0.03
π /2)
r ) cos
0.05
0.0
θ
θ
- 0.01
1
0.1
W(r, π/2)
0.07
0
- 1
- 2
- 0.05
- 3
0
0.5
1
1.5
2
2.5
log r
- 3
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
0
0.5
1
1.5
log
2
2.5
3
r
0 0.5 1 1.5 2 2.5 3
( log
r ) sin
θ
Fig. 7a–d. Equipotential surfaces (meridional sections) for the marginally stable (` = const) configurations of test perfect fluid orbiting the
Schwarzschild–anti-de Sitter black holes, and the related potential W (r, θ = π/2, y), given for y = −10−6 . The behavior of the equipotential
surfaces has the same character for all y < 0. There are four possibilities described in the text. We express the radial coordinate in units of M ,
and use the logarithmic scale. The central black hole is shaded. Notice the special shape of the equipotential surfaces with a cusp, resembling a
falling wave. The dashed lines give asymptotics of W (r, θ = π/2), and determine the interval of radii where the equipotential surfaces cannot
reach the equatorial plane.
Table 1. Radii of the inner cusp (rcu(i) ), outer cusp (rcu(o) ), and the central ring (rcent ), the corresponding values of the potential (Wcu(i) ,
Wcu(o) , Wcent ), and the differences (∆Wi = Wcu(i) − Wcent , ∆Wo = Wcu(o) − Wcent ) for the equilibrium configurations with ` = const in
the Schwarzschild–de Sitter spacetimes. (Radii and ` are in units of mass parameter M , while W and ∆W are in units of c2 .)
y
`
rcu(i)
rcent
rcu(o)
10−6
10−6
10−6
10−6
10−6
10−6
10−6
10−6
10−6
10−6
2 × 10−5
2 × 10−5
10−3
1/118125
2.3400
3.6740
3.8400
3.9392
3.9900
4.1800
5.1962
5.9000
7.1294
8.1100
4.6268
4.9800
2.1800
5.1967
none
6.00195
4.41949
4.13295
4.02028
3.70860
3
none
none
none
3.29004
3.09207
none
3
none
6.00195
8.82178
9.87591
10.3897
12.2516
22.6904
31.4330
62.1768
none
22.2247
none
none
30
98.2201
95.3473
94.8669
94.5644
94.4048
93.7790
89.4730
85.0665
62.1768
none
22.2247
none
9.06615
30
Wcu(i)
Wcent
Wcu(o)
none
−0.05895
−0.03464
−0.01443
−0.00251
0.05255
∞
none
none
none
0.27754
0.71569
none
∞
none
−0.05895
−0.04940
−0.04537
−0.04358
−0.03797
−0.02193
−0.01664
−0.01197
none
−0.03271
none
none
−0.02451
−0.01496
−0.01454
−0.01448
−0.01443
−0.01441
−0.01433
−0.01378
−0.01328
−0.01197
none
−0.03271
none
−0.15977
−0.02451
4. Conclusions
The new phenomena in the structure of equilibrium configurations of test perfect fluid, caused by the presence of a repulsive
cosmological constant, can be summarized in the following way.
1. There is always an equipotential surface with a cusp for
` = 0. It is always an open surface.
∆Wi
∆Wo
none
0
0.01476
0.03094
0.04107
0.09052
∞
none
none
none
0.31025
none
none
∞
none
0.04410
0.03492
0.03094
0.02917
0.02364
0.00815
0.00337
0
none
0
none
none
0
2. The position of the outer cusp of the equipotential
surface
p
with ` = 0 is just at rcusp (` = 0) = rs = 3 1/y. The value
of the potential at the cusp is given by
Wmax (rs , ` = 0, y) = ln Ec (rs , ` = 0, y).
(42)
Because
1/2
,
Ec (r = rs , ` = 0, y) = 1 − 3y 1/3
(43)
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
435
Table 2. Radii of the cusp and central ring, the corresponding values of the potential and their difference for equilibrium configurations with
` = const in the Schwarzschild–anti-de Sitter spacetimes.
y
−6
−10
−10−6
−10−6
−10−6
`
rcu(i)
rcent
rcu(o)
2.1800
3.6744
4.1200
5.8000
none
5.99806
3.79293
none
none
5.99806
11.6194
28.4736
none
none
none
none −0.05884 −0.05884
none
0.03332 −0.03940
none
none
−0.01629
we find
Wcusp (` = 0) =
Wcu(i)
Wcent
1
ln 1 − 3y 1/3 .
2
(44)
3. The accretion disks around black holes can exist when an
inner cusp will appear near the black-hole horizon, in additionpto the outer cusp, located nearby the static radius
rs = 3 1/y.
4. Closed equipotential surfaces, necessary for the existence of
toroidal accretion disks, can exist for ` ∈ (`ms(i) , `ms(o) ).
Here `ms(i) (`ms(o) ) corresponds to the local minimum (maximum) of the function `2K (r; y), giving the minimum (maximum) value ` of stable circular geodesic (Keplerian) orbits. The closed surfaces can exist in the spacetimes with
y < yms ∼ 0.000237.
5. Accretion onto the central black hole by the Paczyński
mechanism is possible, if ` ∈ (`ms(i) , `mb ); The value `mb
corresponds to the marginally bound circular geodesics.
Now, they are determined nontrivially: by the condition
that for ` = `mb both W (r, θ = π/2) and the effective
potential of geodesic motion Veff (r; `mb , y) have two local maxima with the same value (recall that Wmax (r, θ =
π/2) = ln Veff(max) there). In this case, outflow from
the accretion disk is possible through both cusps, if the
mechanical equilibrium is destroyed for both the cusps,
i.e., if both equipotential surfaces with cusp are filled up:
W > Wcusp(o) > Wcusp(i) . If Wcusp(i) < W < Wcusp(o) ,
the accretion flow is directed down the black hole only.
6. We stress that for ` = `mb , the equipotential surface with
W = Wmax (rmb(i) , y) = Wmax (rmb(o) , y) has two cusps.
The mass outflow due to mechanical non-equilibrium, i.e.,
overfilling of the (both-sided) marginally closed equipotential surface, is equally efficient for the inflow down the black
hole and the outflow near the static radius. Of course, we
could expect significant differences in details of the accretion inflow near the black-hole horizon, and the outflow near
the static radius.
7. The outer cusp of the configuration with ` = `mb , and
Wcusp = ln Emb , i.e., the limiting equilibrium configuration which enables accretion into the Schwarzschild–de Sitter black holes, is located at r = rmb(o) . It is quite interesting that such configurations will approach the static radius,
however, they cannot exceed the static radius (rmb(o) → rs
if y → 0). Notice that `mb ∼ `ms(i) (Fig. 2), while
Emb ∼ Ems(o) (Fig. 3).
8. For ` ∈ (`mb , `ms(o) ), the accretion flow down the hole is
“switched-off”, because an open self–crossing equipotential
Wcu(o)
∆Wi
∆Wo
none
none
none
none
none
0
0.07272
none
none
none
none
none
surface with W = Wcusp(i) appears under the inner edge of
the toroidal configuration in the equatorial plane. However,
the outflow through the cusp near the static radius can still
occur due to a possible mechanical non-equilibrium.
9. Toroidal structures of equipotential surfaces, leading to
equilibrium configurations of perfect fluid, cannot exist just
if ` > `ms(o) . Then, an inner cusp, nearby the blackhole horizon, still exists for equipotential surfaces with
W = Wcusp(i) > 0. However, these equipotential surfaces are always open, and can exist in spacetimes with
ye < y < yc .
10. The behavior of the open equipotential surfaces along the
axis of rotation gives an important effect—the surfaces become significantly narrower while approaching the static radius and the cosmological horizon. This behavior suggests
a strong collimation effect on jets, caused by the influence
of a repulsive cosmological constant.
In the case of Schwarzschild–anti-de Sitter spacetimes the
situation is different. The presence of an attractive cosmological constant brings no qualitatively new phenomena in comparison with the Schwarzschild case, concerning the character of
the equilibrium configurations related to accretion disks. Notice, however, the special shape (resembling a falling wave) of
the closed equipotential surfaces which manifests in an illustrative way the interplay of the gravitational, cosmological, and
centrifugal forces. Moreover, there exist no open equipotential
surfaces around the rotation axis in these spacetimes.
From the astrophysical point of view, the most important
phenomena were discovered in spacetimes with a repulsive cosmological constant, if they admit stable circular geodesic orbits.
The first is the presence of an outer cusp of toroidal disks nearby
the static radius which enables outflow of mass and angular momentum from the accretion disks by the Paczyński mechanism,
i.e., due to a violation of the hydrostatic equilibrium. This is the
same mechanism that drives the accretion into the black hole
through the inner cusp. (Recall that outflow from toroidal disks
around a Schwarzschild or Kerr black hole by the Paczyński
mechanism is impossible because no outer cusp of toroidal
disks exists in the asymptotically flat black-hole spacetimes
(Kozlowski et al. 1978; Abramowicz et al. 1978; Jaroszyński
et al. 1980).) The second is the possibility of strong collimation
effects on jets escaping along the rotation axis of toroidal disks
following the open equipotential surfaces that are narrowing
strongly when approaching the static radius (and the cosmological horizon). We give an explicit illustration of these two
principally new phenomena caused by the repulsive cosmolog-
436
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
collimated jet
jet
accretion disk @
@
@@
outer cusp
*
static radius
Thick accretion disk around
a Schwarzschild black hole,
y = 0,
`(r, ϑ) = 3.96 < `mb
Thick accretion disk around
a Schwarzschild–de Sitter
black hole, y = 10−6,
`(r, ϑ) = const = `mb
ical constant in Fig. 8. Of course, both of those very interesting
phenomena deserve further, more detailed studies. Further, the
runaway instability of the toroidal disks with respect to the outflow through the outer cusp, and the influence of self–gravitation
on their structure, have to be examined. We plan these studies
in near future.
It is interesting to find astrophysically plausible situations
in which these two phenomena could be relevant. We should
consider their role in
(a) quasars and active galactic nuclei during the present period
of expansion of the Universe,
Fig. 8. The structure of an accretion disk
with a jet is compared in meridian sections.
The radial coordinate is expressed in units
of M , but the logarithmic scale is not used
here, since we are interested in the regions
near the static radius where both the outer
cusp and the collimation effect are evident.
(b) accretion processes onto primordial black holes during the
very early stages of expansion of the Universe, when phase
transitions connected to symmetry breaking of physical interactions due to Higgs mechanism (e.g., the breaking of
electroweak interactions) could take place, and the effective cosmological constant can have values in many orders
exceeding its present value (Kolb & Turner 1990).
Recent cosmological observations give strong indications
that the present value of the vacuum energy density is
(Krauss 1998)
ρvac(0) ≈ 0.65ρcrit(0)
(45)
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
437
Table 3. Characteristic radii of the Schwarzschild–de Sitter black-hole spacetimes (in units of mass parameter M ). The parameter S =
rmb(o) /rmb(i) determines the relative extension of the toroidal accretion disks with ` = const = `mb . The table ends at y = yms ∼ 0.000237,
corresponding to the marginal spacetime allowing stable circular geodesics. In spacetimes with y > yms , stable circular geodesics are not
allowed, and both thick and thin accretion disks cannot exist.
y
rh
rc
rs
rms(i)
rms(o)
rmb(i)
rmb(o)
S
1 × 10−30
1 × 10−29
1 × 10−28
1 × 10−27
1 × 10−26
1 × 10−25
1 × 10−24
1 × 10−23
1 × 10−22
1 × 10−21
1 × 10−20
1 × 10−19
1 × 10−18
1 × 10−17
1 × 10−16
1 × 10−15
1 × 10−14
1 × 10−13
1 × 10−12
1 × 10−11
1 × 10−10
1 × 10−9
1 × 10−8
1 × 10−7
1 × 10−6
5 × 10−6
8.466 × 10−6
1 × 10−5
2 × 10−5
1 × 10−4
2 × 10−4
2.370 × 10−4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2.0001
2.0002
2.0008
2.0016
2.0019
1 × 1015
3.16 × 1014
1 × 1014
3.16 × 1013
1 × 1013
3.16 × 1012
1 × 1012
3.16 × 1011
1 × 1011
3.16 × 1010
1 × 1010
3.16 × 109
1 × 109
3.16 × 108
1 × 108
3.16 × 107
1 × 107
3.16 × 106
1 × 106
3.16 × 105
1 × 105
3.16 × 104
1 × 104
3.16 × 103
1 × 103
4.46 × 102
3.43 × 102
3.16 × 102
2.23 × 102
1 × 102
70
64
1 × 1010
4.64 × 109
2.15 × 109
1 × 109
4.64 × 108
2.15 × 108
1 × 108
4.64 × 107
2.15 × 107
1 × 107
4.64 × 106
2.15 × 106
1 × 106
4.64 × 105
2.15 × 105
1 × 105
4.64 × 104
2.15 × 104
1 × 104
4.64 × 103
2.15 × 103
1 × 103
4.64 × 102
2.15 × 102
1 × 102
58.5
49.1
46.4
36.8
21.5
17.1
16.2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6.002
6.01
6.02
6.02
6.04
6.24
6.72
7.50
6.30 × 109
2.92 × 109
1.36 × 109
6.30 × 108
2.92 × 108
1.36 × 108
6.30 × 107
2.92 × 107
1.36 × 107
6.30 × 106
2.92 × 106
1.36 × 106
6.30 × 105
2.92 × 105
1.36 × 105
6.30 × 104
2.92 × 104
1.36 × 104
6.30 × 103
2.92 × 103
1.36 × 103
6.30 × 102
2.92 × 102
1.36 × 102
62.2
36.0
30.0
28.3
22.2
12.2
8.89
7.50
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4.001
4.003
4.006
4.012
4.026
4.058
4.132
4.247
4.306
4.328
4.446
5.082
6.097
7.568
1 × 1010
4.64 × 109
2.15 × 109
1 × 109
4.64 × 108
2.15 × 108
1 × 108
4.64 × 107
2.15 × 107
1 × 107
4.64 × 106
2.15 × 106
1 × 106
4.64 × 105
2.15 × 105
1 × 105
4.64 × 104
2.15 × 104
1 × 104
4.64 × 103
2.15 × 103
9.95 × 102
4.59 × 102
2.10 × 102
9.46
53.0
43.5
40.8
31.1
15.3
9.84
7.568
1.05 × 109
4.87 × 108
2.26 × 108
1.05 × 108
4.87 × 107
2.26 × 107
1.05 × 107
4.87 × 106
2.26 × 106
1.05 × 106
4.87 × 105
2.26 × 105
1.05 × 105
4.87 × 104
2.26 × 104
1.05 × 104
4.87 × 103
2.26 × 103
1.05 × 103
4.87 × 102
2.26 × 102
1.05 × 102
48.6
22.5
10.4
5.98
4.99
4.70
3.68
1.96
1.32
1
with present values of the critical energy density ρcrit(0) , and
Hubble parameter H0 given by
ρcrit(0) =
3H02
,
8π
H0 = 100h km s−1 Mpc−1 .
(46)
Taking value of the dimensionless parameter h ∼ 0.7, we arrive at the present value of the “relict” repulsive cosmological
constant
Λ0 = 8πρvac(0) ≈ 1.1 × 10−56 cm−2 .
(47)
Having this value of Λ0 , we can determine the mass parameter
of the spacetime corresponding to any given value of y, and
all the relevant parameters of the equilibrium configurations.
The results concerning the important radii characterizing the
Schwarzschild–de Sitter spacetimes with Λ = Λ0 are summarized in Table 3 and Table 4.
We can clearly see that the relict cosmological constant
Λ0 ∼ 1.1 × 10−56 cm−2 puts a natural limit on the size of equilibrium configurations rotating around black holes. In fact, the
outer edge of the accretion disks, where the outflow goes through
the outer cusp of the toroidal structure, is located nearby the
static radius. It is quite interesting that for black holes of masses
∼ 108 M –109 M , corresponding to black holes located in the
central parts of quasars and active galactic nuclei, the outer edge
of the largest accretion disks is located at rmb(o) ∼ 50–100 kpc,
and is comparable with maximum extension of large galaxies.
Note that extension of quasikeplerian, thin accretion disks is
limited by the outer marginally stable circular orbit; if y is small
enough (y ≤ 10−8 ), it can be shown that
rms(o) ∼ 0.63rs ,
(48)
and dimensions of these disks are comparable to the static radius, too. Therefore, the relict repulsive cosmological constant
can radically influence the behavior of accretion disks in large
galaxies with active nuclei, and can even be connected to the
limit of extension of these large galaxies.
Moreover, it is clear that the collimation effect of the repulsive cosmological constant could be relevant in these situations,
438
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
Table 4. Mass parameter and the radius rmb(o) determining the outer edge of toroidal disks with ` = const = `mb in the Schwarzschild–
–de Sitter black-hole spacetimes, given for (a) the relict repulsive cosmological constant indicated by recent cosmological observations Λ0 ∼
0.65Λcrit(0) ∼ 1.1 × 10−56 cm−2 , (b) the primordial effective cosmological constant Λew ∼ 0.028 cm−2 , and (c) the other possible primordial
effective cosmological constant Λqc ∼ 2.8 × 10−10 cm−2 .
y
1 × 10−30
1 × 10−29
1 × 10−28
1 × 10−27
1 × 10−26
1 × 10−25
1 × 10−24
1 × 10−23
1 × 10−22
1 × 10−21
1 × 10−20
1 × 10−19
1 × 10−18
1 × 10−17
1 × 10−16
1 × 10−15
1 × 10−14
1 × 10−13
1 × 10−12
1 × 10−11
1 × 10−10
1 × 10−9
1 × 10−8
1 × 10−7
1 × 10−6
5 × 10−6
8.5 × 10−6
0.00001
0.00002
0.0001
0.0002
0.00024
Λ0
M
[M ]
1.1 × 108
3.5 × 108
1.1 × 109
3.5 × 109
1.1 × 1010
3.5 × 1010
1.1 × 1011
3.5 × 1011
1.1 × 1012
3.5 × 1012
1.1 × 1013
3.5 × 1013
1.1 × 1014
3.5 × 1014
1.1 × 1015
3.5 × 1015
1.1 × 1016
3.5 × 1016
1.1 × 1017
3.5 × 1017
1.1 × 1018
3.5 × 1018
1.1 × 1019
3.5 × 1019
1.1 × 1020
2.5 × 1020
3.2 × 1020
3.5 × 1020
5 × 1020
1.1 × 1021
1.6 × 1021
1.7 × 1021
Λew
Λqc
rmb(o)
[kpc]
M
[g]
rmb(o)
[cm]
M
[g]
rmb(o)
[cm]
56
82
130
170
250
360
530
780
1100
1700
2500
3600
5300
7800
11000
17000
25000
36000
53000
78000
110000
170000
240000
350000
500000
630000
670000
690000
740000
810000
740000
620000
1.4 × 1014
4.4 × 1014
1.4 × 1015
4.4 × 1015
1.4 × 1016
4.4 × 1016
1.4 × 1017
4.4 × 1017
1.4 × 1018
4.4 × 1018
1.4 × 1019
4.4 × 1019
1.4 × 1020
4.4 × 1020
1.4 × 1021
4.4 × 1021
1.4 × 1022
4.4 × 1022
1.4 × 1023
4.4 × 1023
1.4 × 1024
4.4 × 1024
1.4 × 1025
4.4 × 1025
1.4 × 1026
3.1 × 1026
4 × 1026
4.4 × 1026
6.2 × 1026
1.4 × 1027
2 × 1027
2.1 × 1027
0.00011
0.00016
0.00024
0.00034
0.00048
0.0007
0.001
0.0015
0.0022
0.0032
0.0048
0.007
0.01
0.015
0.022
0.032
0.048
0.07
0.1
0.15
0.22
0.32
0.47
0.68
0.97
1.2
1.3
1.3
1.4
1.6
1.4
1.2
1.4 × 1018
4.4 × 1018
1.4 × 1019
4.4 × 1019
1.4 × 1020
4.4 × 1020
1.4 × 1021
4.4 × 1021
1.4 × 1022
4.4 × 1022
1.4 × 1023
4.4 × 1023
1.4 × 1024
4.4 × 1024
1.4 × 1025
4.4 × 1025
1.4 × 1026
4.4 × 1026
1.4 × 1027
4.4 × 1027
1.4 × 1028
4.4 × 1028
1.4 × 1029
4.4 × 1029
1.4 × 1030
3.1 × 1030
4 × 1030
4.4 × 1030
6.2 × 1030
1.4 × 1031
2 × 1031
2.1 × 1031
1.1
1.6
2.4
3.4
4.8
7.0
10
15
22
32
48
70
100
150
220
320
480
700
1000
1500
2200
3200
4700
6800
9700
12000
13000
13000
14000
16000
14000
12000
because the largest observed jets extend to distances ∼ 200 kpc
(Blandford 1990), exceeding dimensions of the “seed” galaxy
(comparable to the static radius).
It is well known (Carroll & Ostlie 1996) that dimensions of
large galaxies, of both spiral and elliptical type, are in the interval
50–100 kpc, while the extremely large elliptical galaxies of cD
type extend up to 1000 kpc. Thus, we can conclude that toroidal
disks around a central hole of mass M ∼ 109 M have sizes
comparable with the large galaxies and can be related to sizelimits on these galaxies. On the other hand, such disks are well
inside the cD elliptical galaxies; in order to obtain an accretion
disk of dimension ∼ 1000 kpc, mass parameter of the central
black hole have to be ∼ 1012 M .
Of course, if the mass of a protogalactic disk related to a
quasar is higher than the mass of the central black hole, the
self-gravitational effects of the disk itself have to be taken into
consideration. Nevertheless, we can expect that even in the sit-
uation like this the repulsive cosmological constant keeps the
presence of the outer cusp enabling outflows of matter from the
disk. On the other hand, the collimation effect on jets could be
efficient even for small toroidal disks, with outer edge located
deeply under the static radius. In such disks the self-gravitational
effects could usually be neglected.
In the case of accretion onto primordial black holes in the
very early universe, with assumed high values of repulsive cosmological constant, we can expect even stronger effects. Considering the electroweak phase transition at Tew ∼ 100 GeV,
we obtain an estimate of the primordial effective cosmological
constant
Λew ∼ 0.028 cm−2 .
(49)
Considering the quark confinement at Tqc ∼ 1 GeV, we obtain
an estimate of the primordial cosmological constant
Λqc ∼ 2.8 × 10−10 cm−2 .
(50)
Z. Stuchlı́k et al.: Perfect fluid orbiting Schwarzschild–de Sitter black holes
It follows from the Table 4 that the accretion onto primordial black holes of mass M > Mew ∼ 2 × 1027 g, and
M > Mqc ∼ 2 × 1031 g, respectively, is then forbidden in the
disk regime because no equilibrium configurations of perfect
fluid are allowed in the corresponding Schwarzschild–de Sitter
backgrounds. Of course, the accretion can be realized in quasispherical regime in these spacetimes, however, its character
represents an open problem.
Acknowledgements. This work has been supported by the GAČR Grant
No.202/99/0261, by the Committee for Collaboration of Czech Republic with CERN and by the Bergen Computational Physics Laboratory
project, an EU Research Infrastructure at the University of Bergen,
Norway, supported by the European Community – Access to Research
Infrastructure Action of the Improving Human Potential Programme.
Two of authors (Z. S. and S. H.) would like to acknowledge the perfect
hospitality and excellent working conditions at the CERN’s Theory
Division and the Institute of Physics of the University of Bergen.
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PHYSICAL REVIEW D 69, 064001 共2004兲
Equatorial circular orbits in the Kerr–de Sitter spacetimes
Zdeněk Stuchlı́k* and Petr Slaný†
Department of Physics, Faculty of Philosophy and Science, Silesian University at Opava Bezručovo nám. 13,
746 01 Opava, Czech Republic
共Received 18 June 2003; published 1 March 2004兲
Equatorial motion of test particles in Kerr–de Sitter spacetimes is considered. Circular orbits are determined,
their properties are discussed for both black-hole and naked-singularity spacetimes, and their relevance for thin
accretion disks is established. The circular orbits constitute two families that coalesce at the so-called static
radius. The orientation of the motion along the circular orbits is, in accordance with case of asymptotically flat
Kerr spacetimes, defined by relating the motion to the locally nonrotating frames. The minus-family orbits are
all counterrotating, while the plus-family orbits are usually corotating relative to these frames. However, the
plus-family orbits become counterrotating in the vicinity of the static radius in all Kerr–de Sitter spacetimes,
and they become counterrotating in the vicinity of the ring singularity in Kerr–de Sitter naked-singularity
spacetimes with a low enough rotational parameter. In such spacetimes, the efficiency of the conversion of the
rest energy into heat energy in the geometrically thin plus-family accretion disks can reach extremely high
values exceeding the efficiency of the annihilation process. The transformation of a Kerr–de Sitter naked
singularity into an extreme black hole due to accretion in the thin disks is briefly discussed for both the
plus-family and minus-family disks. It is shown that such a conversion leads to an abrupt instability of the
innermost parts of the plus-family accretion disks that can have strong observational consequences.
DOI: 10.1103/PhysRevD.69.064001
PACS number共s兲: 04.25.⫺g, 04.20.Dw, 04.70.Bw, 98.62.Mw
I. INTRODUCTION
It is commonly accepted that the energy sources of quasars and active galactic nuclei are accretion disks around
central massive black holes 关1,2兴. The basic properties of
geometrically thin accretion disks 共with negligible pressure兲
are determined by the circular geodesic motion in the blackhole background 关3兴. The basic properties of geometrically
thick disks are determined by the equilibrium configurations
of a perfect fluid orbiting in the black-hole background; however, the geodesic structure of the background is relevant
also for the properties of the thick disks 关4兴.
According to the cosmic censorship hypothesis 关5兴 and
the uniqueness theorems for black holes 关6兴, the result of the
gravitational collapse of a sufficiently massive rotating object is a rotating Kerr black hole, rather than a Kerr naked
singularity; further, the laws of black-hole thermodynamics
forbid conversion of black holes into a naked singularity.
However, although the cosmic censorship is a very plausible
hypothesis, there is some evidence against it. Naked singularities arise in various models of spherically symmetric collapse 共e.g., 关7兴兲. In modeling the collapse of rotating stars, it
was pointed out that, although mass shedding and gravitational radiation will reduce the angular momentum of the star
during collapse, it will not in general be reduced to the value
that corresponds to a Kerr black hole 关8兴. The 2D numerical
models 关9兴 imply that a rotating, collapsing supermassive
object will not always dissipate enough angular momentum
to form a Kerr black hole, but a Kerr-like naked singularity
has to be expected to develop from objects rotating rapidly
enough. Candidates for the formation of naked Kerr geom-
*Electronic address: [email protected]
†
Electronic address: [email protected]
0556-2821/2004/69共6兲/064001共21兲/$22.50
etry with a ring singularity from the collapse of rotating stars
were found in the scenario of 关10兴. The numerical models of
the collapse of collisionless gas spheroids also results in
strong candidates for the formation of naked singularities
关11兴.
Because Penrose’s cosmic censorship hypothesis 关5兴 is far
from being proved, naked singularity spacetimes related to
black-hole spacetimes with a nonzero charge and/or rotational parameter could still be considered conceivable models of quasars and active galactic nuclei and deserve some
attention. Of particular interest are those effects that could
distinguish a naked singularity from black holes.
Test particle motion and test fields were extensively studied for Kerr black-hole spacetimes 关12–21兴. Gravitational
radiation of particles moving in the field of a Kerr black hole
were discussed in 关22兴, the motion of spinning test particles
was discussed in 关23兴. For a detailed review see the books of
Chandrasekhar 关24兴 and Frolov and Novikov 关25兴. However,
Kerr naked singularities were also studied widely. Their repulsive effects and causality-violating regions were investigated by de Felice and co-workers 关26 –28兴, the equatorial
circular geodesics and motion of spherical shell of incoherent dust were investigated in 关29兴, the collimation effect of
the region nearby the ring singularity was treated in 关30兴, and
the motion of spinning test particles was discussed in 关23兴.
Chandrasekhar 关24兴 also devotes some attention to the effects
of Kerr naked singularities, saying ‘‘considerable interest attaches to knowing the sort of things space-times with naked
singularities are and whether there are any essential differences in the manifestations of space-times with singularities
concealed behind event horizons.’’ We follow Chandrasekhar’s approach.
All recently available data from a wide variety of cosmological tests indicate convincingly that in the framework of
the inflationary cosmology a nonzero, although very small,
69 064001-1
©2004 The American Physical Society
PHYSICAL REVIEW D 69, 064001 共2004兲
Z. STUCHLÍK AND P. SLANÝ
repulsive cosmological constant ⌳⬎0 has to be invoked in
order to explain the dynamics of the recent Universe 关31,32兴.
The presence of a repulsive cosmological constant changes
substantially the asymptotic structure of the black-hole 共or
naked-singularity兲 backgrounds, as they become asymptotically de Sitter spacetimes, not flat spacetimes. In such spacetimes, an event horizon always exists behind which the geometry is dynamic; we call it a cosmological horizon.
Therefore, it is relevant to clarify the influence of the repulsive cosmological constant on the astrophysically interesting
properties of the black-hole or naked-singularity background.
For these purposes, analysis of the geodesic motion of test
particles and photons is among the most important techniques. 共It could be noted that the optical reference geometry
introduced by Abramowicz, Carter, and Lasota 关52兴 reflects
in an illustrative and intuitive way some hidden properties of
the geodesic motion 关33–35兴.兲 Of particular interest are circular geodesics being relevant for the accretion disks.
Properties of the geodesic motion in the Schwarzschild–
共anti-兲de Sitter and Reissner–Nordström–共anti-兲de Sitter
spacetimes were discussed in 关36,37兴. Properties of the circular orbits of test particles show that due to the presence of
a repulsive cosmological constant, the thin disks have not
only an inner edge determined 共approximately兲 by the radius
of the innermost stable circular orbit, but also an outer edge
given by the radius of the outermost stable circular orbit,
located nearby the so-called static radius, where the gravitational attraction of a black hole 共naked singularity兲 is just
compensated for by the cosmological repulsion.
A similar analysis of the equilibrium configurations of a
perfect fluid orbiting the Schwarzschild–de Sitter black-hole
backgrounds allowing the existence of stable circular orbits,
which is a necessary condition for the existence of accretion
disks, shows that also thick accretion disks have both the
inner and outer edges located nearby the inner 共outer兲 marginally bound circular geodesic. The accretion through the
inner cusp and the outflow of matter through the outer cusp
of the equilibrium configurations are driven by the Paczyński
mechanism. It is a mechanical nonequilibrium process when
the matter of the disk slightly overfills the critical equipotential surface with two cusps and thus violates the hydrostatic
equilibrium 关38兴.
In the case of Reissner–Nordström–共anti-兲de Sitter backgrounds 关37兴, the discussion has been enriched for the case
of the naked-singularity spacetimes—it was shown that even
two separated regions of stable circular orbits are allowed for
the naked-singularity spacetimes with spacetime parameters
appropriately chosen.
However, it is very important to understand the role of a
nonzero cosmological constant in the astrophysically most
relevant, rotating, Kerr backgrounds. Equatorial motion of
photons has been studied extensively for Kerr–Newman–
共anti-兲de Sitter spacetimes describing both black holes and
naked singularities and some unusual effects have been
found 共for details, see 关35,39兴兲. Here, attention will be focused on the circular equatorial motion of test particles in the
Kerr–de Sitter backgrounds.
In Sec. II, the Kerr–de Sitter backgrounds are separated in
the parameter space into black-hole and naked-singularity
spacetimes. In Sec. III, the equations of the equatorial motion of test particles are presented. In Sec. IV, the constants
of motion of the circular orbits are determined, and their
properties are discussed. As in Kerr spacetimes, there exist
two different sequences of the equatorial circular geodesics.
We call them plus 共minus-兲 family orbits. All minus-family
orbits are counterrotating relative to the locally nonrotating
frames 共LNRF; for a definition of these frames see 关13兴兲,
while the plus-family orbits are mostly corotating, but in
some regions are counterrotating relative to the LNRF. Only
outside the outer horizon of the Kerr black holes are all the
plus-family orbits corotating relative to the LNRF. On the
other hand, in vicinity of the ring singularity of the Kerr
naked singularities with rotational parameter low enough, the
plus-family orbits become counterrotating relative to the
LNRF 关29兴. We shall see that in all Kerr–de Sitter spacetimes this happens nearby the so-called static radius, where
the sequences of plus-family and minus-family orbits coalesce. 关Note that in asymptotically flat Kerr spacetimes, the
orbits corotating 共counterrotating兲 relative to the LNRF also
corotate 共counterrotate兲 from the point of view of stationary
observers at infinity. However, the last criterion cannot be
used in the asymptotically de Sitter spacetimes under consideration.兴 In Sec. V, the properties of the circular orbits are
discussed with attention focused on their relevance for thin
accretion disks. Regions where the orbits of plus-family are
counterrotating relative to the LNRF are determined; further,
it is established where these orbits could have a negative
energy parameter. The efficiency of the accretion process in
geometrically thin disks is determined. In Sec. VI, concluding remarks are presented.
II. KERR–de SITTER BLACK-HOLE
AND NAKED-SINGULARITY SPACETIMES
In the standard Boyer-Lindquist coordinates (t,r, ␪ , ␸ )
and geometric units (c⫽G⫽1), the Kerr–共anti-兲de Sitter geometry is given by the line element
ds 2 ⫽⫺
⫹
⌬r
I 2␳ 2
共 dt⫺a sin2 ␪ d␸ 兲 2
⌬ ␪ sin2 ␪
I 2␳ 2
关 adt⫺ 共 r 2 ⫹a 2 兲 d␸ 兴 2 ⫹
␳2 2 ␳2 2
dr ⫹ d␪ ,
⌬r
⌬␪
共1兲
where
064001-2
1
⌬ r ⫽⫺ ⌳r 2 共 r 2 ⫹a 2 兲 ⫹r 2 ⫺2M r⫹a 2 ,
3
共2兲
1
⌬ ␪ ⫽1⫹ ⌳a 2 cos2 ␪ ,
3
共3兲
1
I⫽1⫹ ⌳a 2 ,
3
共4兲
␳ 2 ⫽r 2 ⫹a 2 cos2 ␪ .
共5兲
PHYSICAL REVIEW D 69, 064001 共2004兲
EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
The parameters of the spacetime are mass (M ), a specific
angular momentum (a), and cosmological constant (⌳). It
is convenient to introduce a dimensionless cosmological parameter
1
y⫽ ⌳M 2 .
3
共6兲
For simplicity, we put M ⫽1 hereafter. Equivalently, also the
coordinates t,r, the line element ds, and the parameter of the
spacetime a are expressed in units of M and become dimensionless.
For y⬍0 corresponding to the attractive cosmological
constant, the line element 共1兲 describes a Kerr–anti–de Sitter
geometry. Here we focus our attention on the case y⬎0 corresponding to the repulsive cosmological constant, when Eq.
共1兲 describes a Kerr–de Sitter spacetime.
The event horizons of the spacetime are given by the
pseudosingularities of the line element 共1兲, determined by the
condition ⌬ r ⫽0. The loci of the event horizons are determined by the relation
a 2 ⫽a 2h 共 r;y 兲 ⬅
r 2 ⫺2r⫺yr 4
yr 2 ⫺1
共7兲
.
The asymptotic behavior of the function a 2h (r;y) is given by
a 2h (r→0,y)→0, a 2h (r→⬁,y)→⫺⬁. For y⫽0, the function
a 2h (r)⫽2r⫺r 2 determines loci of the horizons of Kerr black
holes. The divergent points of a 2h (r;y) are determined by
y⫽y d(h) 共 r 兲 ⬅
1
r2
r⫺2
r3
which is the limiting value for the existence of
Schwarzschild–de Sitter black holes 关36兴. In Reissner–
Nordström–de Sitter spacetimes, the critical value of the cosmological parameter limiting the existence of black-hole
spacetimes is 关37兴
共8兲
,
its zero points are given by
y⫽y z(h) 共 r 兲 ⬅
FIG. 1. Functions y e(h)⫺ (r) 共solid curve兲, y z(h) (r) 共dashed
curve兲, y d(h) (r) 共dash-dotted curve兲, and y e(h)⫹ (r) 共dotted curve兲
characterizing the properties of the function a 2h (r;y) determining
the horizons of Kerr–de Sitter spacetimes. The maximum of
y e(h)⫺ (r) corresponds to the critical value of the cosmological parameter y c(KdS) ⬟0.05924 for which the function a 2h (r;y) has an
inflex point 共and no extrema兲. Any Kerr–de Sitter spacetime with
y⬎y c(KdS) is of naked-singularity type containing only a cosmological horizon. The function y z(h) (r) determines the horizons of
the Schwarzschild–de Sitter spacetimes and its maximum y c(SdS)
⫽1/27 corresponds to the limiting value for the existence of
Schwarzschild–de Sitter black holes.
共9兲
,
y c(RNdS) ⫽
If y⫽y c(KdS) , the function a 2h (r;y) has an inflex point at
r⫽r crit , corresponding to a critical value of the rotation parameter of Kerr–de Sitter spacetimes
2
⫽
a crit
and its local extrema are determined by the relation
y⫽y e(h)⫾ 共 r 兲 ⬅
2r⫹1⫾ 冑8r⫹1
2r
3
.
共10兲
The functions y d(h) (r), y z(h) (r), and y e(h)⫾ (r) are illustrated
in Fig. 1. The function y e(h)⫺ (r) has its maximum at r crit
⫽(3⫹2 冑3)/4, where the value of the cosmological parameter takes a critical value
y c(KdS) ⫽
16
共 3⫹2 冑3 兲 3
⬟0,05924;
共11兲
for y⬎y c(KdS) , only naked-singularity backgrounds exist for
a 2 ⬎0. A common point of the functions y z(h) (r) and
y e(h)⫺ (r) is located at r⫽3, where is the maximum of
y z(h) (r) taking a critical value
y c(SdS) ⫽
1
⬟0.03704,
27
共12兲
2
⬟0.07407.
27
3
共 3⫹2 冑3 兲 ⬟1,21202.
16
共13兲
2
only, while
Kerr–de Sitter black holes can exist for a 2 ⬍a crit
Kerr–de Sitter naked singularities can exist for both a 2
2
2
⬍a crit
and a 2 ⬎a crit
.
For y⬎0, the function y e(h)⫺ (r) determines two local
2
(r 1 ,y),
extrema of a 2h (r;y) at y⬍y c(KdS) , denoted as a max(h)
2
2
a min(h) (r 2 ,y), with r 1 ⬍r 2 . If y⬍y c(SdS) , a min(h) (r 2 ,y)⬍0,
and the minimum is unphysical. The function a 2h (r) diverges
at r d ⫽1/冑y, and it is discontinuous there. The function
y e(h)⫹ (r) determines a maximum of a 2h (r;y) at a negative
value of a 2 which is, therefore, physically irrelevant 关see Fig.
2 giving typical behavior of a 2h (r;y)].
If 0⬍y⬍y c(SdS) , black-hole spacetimes exist for a 2
2
⭐a max(h)
(y), and naked-singularity spacetimes exist for a 2
2
⬎a max(h) (y). If y c(SdS) ⬍y⭐y c(KdS) , black-hole spacetimes
2
2
exist for a min(h)
(y)⭐a 2 ⭐a max(h)
(y), while naked-singularity
2
2
2
(y). The
spacetimes exist for a ⬍a min(h) (y) and a 2 ⬎a max(h)
2
2
functions a min(h) (y),a max(h) (y) are implicitly given by Eqs.
064001-3
PHYSICAL REVIEW D 69, 064001 共2004兲
Z. STUCHLÍK AND P. SLANÝ
FIG. 2. Horizons of Kerr–de Sitter spacetimes. They are given
for five typical values of the cosmological parameter y by the function a 2h (r;y). For y⬎y c(KdS) ⬟0.05924 (y⫽0.08) the function has
no local extrema and only naked-singularity spacetimes are allowed
共the only horizon is the cosmological horizon兲. For y⫽y c(KdS) , the
function has an inflex point where the black-hole and the cosmological horizons coincide. For y c(SdS) ⫽1/27⬍y⬍y c(KdS) (y
⫽0.045) the function has two local extrema in positive values and
the black-hole spacetimes exist for a 2 between those extrema. For
y⫽y c(SdS) the local minimum resides on the axis a 2 ⫽0. The critical value y c(SdS) represents the limiting value of cosmological parameter for which the Schwarzschild–de Sitter black holes can exist; the Kerr–de Sitter black holes again exist for a 2 between those
extrema. For 0⬍y⬍y c(SdS) (y⫽0.03) the local minimum resides in
the nonphysical region a 2 ⬍0 and the black holes exist for a 2 up to
the local maximum. For completeness, we present the gray curve
determining horizons of the Kerr (y⫽0) black holes. In all cases,
the local extrema correspond to the extreme black holes.
共7兲 and 共10兲; the separation of Kerr–de Sitter black-hole and
naked-singularity spacetimes in the parameter space y –a 2 is
shown in Fig. 3. In black-hole spacetimes, there are two
black-hole horizons and the cosmological horizon, with r h⫺
⬍r h⫹ ⬍r c . In naked-singularity spacetimes, there is the cosmological horizon r c only.
The extreme cases, when two 共or all three兲 horizons coalesce, were discussed in detail for the case of Reissner–
Nordström–de Sitter spacetimes 关40,41兴. In Kerr–de Sitter
spacetimes, the situation is analogical. If r h⫺ ⫽r h⫹ ⬍r c , the
extreme black-hole case occurs; if r h⫺ ⬍r h⫹ ⫽r c , the marginal naked-singularity case occurs; if r h⫺ ⫽r h⫹ ⫽r c , the
‘‘ultra-extreme’’ case occurs which corresponds to the nakedsingularity case.
FIG. 3. Classification of Kerr–de Sitter spacetimes. The space
of parameters a 2 and y is separated into six regions. Dashed curves
separate regions of black holes and naked singularities. Solid curves
divide the parametric space into spacetimes differing by properties
of the stable circular orbits relevant for Keplerian accretion disks.
For large values of a 2 both the solid lines tend to the a 2 axis. 共I兲
Black-hole spacetimes with both corotating and counterrotating
stable or bound circular orbits, 共II兲 black-hole spacetimes with no
counterrotating stable or bound circular orbits, 共III兲 black-hole
spacetimes with no corotating and counterrotating stable or bound
circular orbits, 共IV兲 naked-singularity spacetimes with no corotating
and counterrotating stable or bound circular orbits, 共V兲 nakedsingularity spacetimes with both corotating and counterrotating
stable or bound circular orbits, and 共VI兲 naked-singularity spacetimes with no counterrotating stable or bound circular orbits of the
minus family. The dash-dotted curve defines the subregion of
naked-singularity spacetimes, where the plus-family circular orbits
could be stable and counterrotating 共from the point of view of a
locally nonrotating observer兲; shaded is the subregion allowing
stable circular orbits with E ⫹ ⬍0.
the equations were obtained by Carter 关6兴. For the motion
restricted to the equatorial plane (d␪ /d␭⫽0, ␪ ⫽ ␲ /2) the
Carter equations take the form
r2
dr
⫽⫾R 1/2共 r 兲 ,
d␭
共14兲
r2
aI P r
d␸
,
⫽⫺I P ␪ ⫹
d␭
⌬r
共15兲
r2
dt
共 r 2 ⫹a 2 兲 I P r
⫽⫺aI P ␪ ⫹
,
d␭
⌬r
共16兲
III. EQUATORIAL MOTION
where
In order to understand basic properties of thin accretion
disks in the field of rotating black holes or naked singularities, it is necessary to study equatorial geodetical motion,
especially circular motion, of test particles, as it can be
shown that due to the dragging of the inertial frames any
tilted disk has to be driven to the equatorial plane of the
rotating spacetimes 关42兴.
R 共 r 兲 ⫽ P r2 ⫺⌬ r 共 m 2 r 2 ⫹K 兲 ,
共17兲
P r ⫽IE共 r 2 ⫹a 2 兲 ⫺Ia⌽,
共18兲
P ␪ ⫽I 共 aE⫺⌽ 兲 ,
共19兲
K⫽I 2 共 aE⫺⌽ 兲 2 .
A. Carter equations
The motion of a test particle with rest mass m is given by
the geodesic equations. In a separated and integrated form,
共20兲
The proper time of the particle, ␶ , is related to the affine
parameter ␭ by ␶ ⫽m␭. The constants of the motion are
064001-4
PHYSICAL REVIEW D 69, 064001 共2004兲
EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
energy (E), related to the stationarity of the geometry; axial
angular momentum (⌽), related to the axial symmetry of the
geometry; ‘‘total’’ angular momentum (K), related to the
hidden symmetry of the geometry. For the equatorial motion,
K is restricted through Eq. 共20兲 following from the conditions on the latitudinal motion 关43兴. Notice that E and ⌽
cannot be interpreted as energy and axial angular momentum
at infinity, since the spacetime is not asymptotically flat.
by an ‘‘effective potential’’ given by the condition R(r)⫽0
for turning points of the radial motion. It is useful to define
specific energy and specific angular momentum by the relations
E⬅
关共 1⫹ya 2 兲 r 共 r 2 ⫹a 2 兲 ⫹2a 2 兴
E (⫹) 共 r;X,a,y 兲 ⬅
or
共25兲
E⭐E (⫺) 共 r;L,a,y 兲 .
Conditions E⫽E (⫹) (r,L,a,y) 关or E⫽E (⫺) (r;L,a,y)] give
the turning points of the radial motion; at the dynamic regions (⌬ r ⬍0), the turning points are not allowed. In the
region between the outer black-hole horizon and the cosmological horizon, the motion of particles in the positive-root
states—i.e., particles with positive energy as measured by
local observers—being future directed (dt/d␭⬎0) and having a direct ‘‘classical’’ physical meaning, is determined by
the effective potential E (⫹) (r;L,a,y). The character of the
motion in the whole Kerr–de Sitter background and the relevance of the effective potential E (⫺) (r;L,a,y), determining
the motion of particles in the negative-root states between
the black-hole and cosmological horizons, is qualitatively the
same as discussed in 关16兴. In the following we restrict our
attention to the positive-root states determined by the effective potential E (⫹) (r;L,a,y). Trajectories of the equatorial
motion are then determined by the equation
r
d␸
⫽⫾
dr
⌬r
.
关 aX⫹⌬ r1/2共 r 2 ⫹X 2 兲 1/2兴
共28兲
共 r⫹2 兲 X⫹aEr
冑共 r
2
E⫺aX 兲 2 ⫺⌬ r 共 r 2 ⫹X 2 兲
.
共29兲
The equatorial circular orbits can most easily be determined by solving simultaneously the equations
R 共 r 兲 ⫽r 4 E 2 ⫺2ar 2 EX⫹ 共 a 2 ⫺⌬ r 兲 X 2 ⫺r 2 ⌬ r ⫽0,
共30兲
dR
⫽4r 3 E 2 ⫺4arEX⫺⌬ r⬘ X 2 ⫺⌬ r⬘ r 2 ⫺2r⌬ r ⫽0,
dr
共31兲
where ⌬ r⬘ ⬅d⌬ r /dr. Combining Eqs. 共30兲 and 共31兲, we arrive at a quadratic equation
冉冊
X
E
2
⫹B 共 r 兲
冉冊
X
⫹C 共 r 兲 ⫽0,
E
共32兲
with
共26兲
Nevertheless, it is convenient to redefine the axial angular
momentum by the relation
X⬅L⫺aE;
r
2
IV. EQUATORIAL CIRCULAR ORBITS
2
冑关 E 共 r 2 ⫹a 2 兲 ⫺aL 兴 2 ⫺⌬ r 关 r 2 ⫹ 共 aE⫺L 兲 2 兴
1
and the equation of trajectories 共26兲 transforms to the form
A共 r 兲
aE 共 r ⫹a ⫺⌬ r 兲 ⫹ 共 ⌬ r ⫺a 兲 L
共23兲
.
for an analogous redefinition in the case of equatorial photon
motion see 关35兴. With the constant of motion, X, instead of L,
the effective potential takes the simple form
共24兲
E⭓E (⫹) 共 r;L,a,y 兲
r
d␸
⫽⫾
dr
⌬r
共21兲
we find the effective potential in the form
a 关 yr 共 r 2 ⫹a 2 兲 ⫹2 兴 L⫾⌬ r1/2兵 r 2 L 2 ⫹r 关共 1⫹ya 2 兲 r 共 r 2 ⫹a 2 兲 ⫹2a 2 兴 其 1/2
In the stationary regions (⌬ r ⭓0), the motion is allowed
where
2
I⌽
.
m
R 共 r 兲 ⬅ 关 E 共 r 2 ⫹a 2 兲 ⫺aL 兴 2 ⫺⌬ r 关 r 2 ⫹ 共 aE⫺L 兲 2 兴 ⫽0, 共22兲
The equatorial motion is governed by the constants of
motion E,⌽. Its properties can be conveniently determined
2
L⬅
Solving the equation
B. Effective potential
E (⫾) 共 r;L,a,y 兲 ⬅
IE
,
m
共27兲
064001-5
A 共 r 兲 ⫽2⌬ r 共 a 2 ⫺⌬ r 兲 ⫹a 2 ⌬ r⬘ r,
共33兲
B 共 r 兲 ⫽⫺2a⌬ r⬘ r 3 ,
共34兲
C 共 r 兲 ⫽r 4 共 ⌬ r⬘ r⫺2⌬ r 兲 .
共35兲
PHYSICAL REVIEW D 69, 064001 共2004兲
Z. STUCHLÍK AND P. SLANÝ
Its solution can be expressed in the relatively simple form
冉冊
X
E
共 r;a,y 兲 ⫽
⫾
r 2 共 r⫺a 2 ⫺yr 4 兲
.
ar 关 r⫺1⫺yr 共 2r 2 ⫹a 2 兲兴 ⫾⌬ r 关 r 共 1⫺yr 3 兲兴 1/2
共36兲
Assuming now
X ⫹ ⫽E ⫹
冉冊
X
E
X ⫺ ⫽E ⫺
,
⫹
冉冊
X
E
共37兲
,
⫺
substituting into Eq. 共30兲, and solving for the specific energy
of the orbit, we obtain
2
E ⫾ 共 r;a,y 兲 ⫽
冉 冊
冉 冊册
1
1/2
1⫺ ⫺ 共 r ⫹a 兲 y⫾a
⫺y
r
r3
冋
2
2
3
1/2 1/2
1
. 共38兲
1⫺ ⫺a 2 y⫾2a
⫺y
r
r3
The related constant of motion, X, of the orbit is then given
by the expression
⫺a⫾r 2
X ⫾ 共 r;a,y 兲 ⫽
冋
冉 冊
冉 冊册
1
r3
1/2
⫺y
1/2 1/2
1
3
, 共39兲
1⫺ ⫺a 2 y⫾2a
⫺y
r
r3
while the specific angular momentum of the circular orbits is
determined by the relation
冉 冊
冉 冊册
2a⫹ar 共 r ⫹a 兲 y⫿r 共 r ⫹a 兲
2
L ⫾ 共 r;a,y 兲 ⫽⫺
冋
2
2
3
2
1
r3
r 1⫺ ⫺a 2 y⫾2a
⫺y
r
r3
a2
1/2
1⫹
⫺y
1/2 1/2
1
E ⫾ 共 r;a 兲 ⫽
冋
3
2a
1⫺ ⫾
r r 3/2
册
1/2
,
L ⫾ 共 r;a 兲 ⫽⫾r
.
共40兲
Relations 共38兲–共40兲 determine two families of the circular
orbits. We call them plus-family orbits and minus-family orbits according to the ⫾ sign in relations 共38兲–共40兲. The typical behavior of the functions E ⫾ (r;a,y) and L ⫾ (r;a,y) giving the specific energy and specific angular momentum is
illustrated in Figs. 4 and 5, respectively, for Kerr–de
Sitter black-hole spacetimes with appropriately taken parameters. Figure 6 shows the typical behavior of these functions for some Kerr–de Sitter naked-singularity spacetimes.
In the limit of y→0, relations 共38兲 and 共40兲 reduce to the
expression given by Chandrasekhar 共in units of M ) 关24兴 for
circular orbits in the Kerr backgrounds:
a
2
1⫺ ⫾
r r 3/2
FIG. 4. Specific energy of the equatorial circular orbits in
Kerr–de Sitter black-hole spacetimes. The spacetimes are specified
by the cosmological parameter y and the rotational parameter a (a 2
varies from 0.0 to 1.0 in steps of 0.2兲. The left column corresponds
to the plus-family orbits; the right column corresponds to the
minus-family orbits. The local extrema of the curves correspond to
the marginally stable orbits, the rising parts correspond to stable
orbits, and the descending parts correspond to unstable ones. The
behavior of the curves for the spacetimes with y⬍10⫺5 is similar to
the case of y⫽10⫺5 .
共41兲
1/2
冋
r2
⫿
3
2a
r 3/2
2a
1⫺ ⫾
r r 3/2
册
1/2
.
共42兲
In the limit of a→0 we arrive at the formulas determining
the specific energy and the specific angular momentum of
circular orbits in the field of Schwarzschild–de Sitter black
holes 关36兴:
E 共 r;y 兲 ⫽
L 共 r;y 兲 ⫽
r⫺2⫺yr 3
关 r 共 r⫺3 兲兴 1/2
r 共 1⫺yr 3 兲 1/2
共 r⫺3 兲 1/2
,
共43兲
;
共44兲
here, we do not give L for the minus-family orbits as these
are equivalent to the plus-family orbits in spherically symmetric spacetimes.
The formulas for the specific energy and angular momentum of the equatorial circular orbits hold equally for both
Kerr–de Sitter (y⬎0) and Kerr–anti-de Sitter (y⬍0) spacetimes. Here, we shall concentrate our discussion on the circular motion in Kerr–de Sitter spacetimes. We shall deter-
064001-6
PHYSICAL REVIEW D 69, 064001 共2004兲
EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
A. Existence of circular orbits
Inspecting expressions 共38兲 and 共40兲, we find two reality
conditions on the circular orbits. The first restriction on the
existence of circular orbits is given by the relation
y⭐y s ⬅
1
r3
共45兲
,
which introduces the notion of the ‘‘static radius,’’ given by
the formula r s ⫽y ⫺1/3 independently of the rotational parameter a. It can be compared with the formally identical result
in Schwarzschild–de Sitter spacetimes 关36兴. A ‘‘free’’ or
‘‘geodetical’’ observer on the static radius has only U t component of four-velocity nonzero. The position on the static
radius is unstable relative to radial perturbations, as follows
from the discussion on stability of the circular orbits performed below.
The second restriction on existence of circular orbits is
given by the condition
FIG. 5. Specific angular momentum of the equatorial circular
orbits in Kerr–de Sitter black-hole spacetimes. The spacetimes are
specified by the cosmological parameter y and the rotational parameter a (a 2 varies from 0.0 to 1.0 in steps of 0.2兲. The left column
corresponds to the plus-family orbits; the right column corresponds
to the minus-family orbits. The local extrema of the curves correspond to the marginally stable orbits, the rising parts of L ⫹ and the
descending parts of L ⫺ correspond to the stable orbits, the descending parts of L ⫹ , and the rising parts of L ⫺ correspond to the unstable ones. The behavior of the curves for the spacetimes with y
⬍10⫺5 is similar to the case of y⫽10⫺5 .
mine radii where the existence of circular orbits is allowed,
the orientation of the circular motion relative to the locally
nonrotating frames, stability of the circular motion relative to
radial perturbations. Finally, we shall introduce the notion of
marginally bound orbits.
冉 冊
1
3
1⫺ ⫺a 2 y⫾2a 3 ⫺y
r
r
1/2
⭓0;
共46兲
the equality determines the radii of photon circular orbits,
where both E→⬁ and L→⫾⬁.
The photon circular orbits of the plus-family are given by
the relation
a⫽a (⫹)
ph(1,2) 共 r;y 兲 ⬅
共 1⫺yr 3 兲 1/2⫾ 共 1⫺3yr 2 兲 1/2
yr 3/2
,
共47兲
while for the minus-family orbits they are given by the relation
a⫽a (⫺)
ph(1,2) 共 r;y 兲 ⬅
⫺ 共 1⫺yr 3 兲 1/2⫾ 共 1⫺3yr 2 兲 1/2
yr 3/2
. 共48兲
FIG. 6. Specific energy and
specific angular momentum of the
equatorial circular orbits in
Kerr–de Sitter naked-singularity
spacetimes.
The
plus-family
curves are plotted for the rotational parameter a 2 ⫽10, 20, 30,
50, 100, 300; the minus-family
curves are plotted for a 2 ⫽2, 5,
10, 20, 30. The meaning of particular parts of the curves is the
same as in black-hole spacetimes.
064001-7
PHYSICAL REVIEW D 69, 064001 共2004兲
Z. STUCHLÍK AND P. SLANÝ
The photon circular orbits can be determined by a ‘‘common’’ formula related to a 2 :
a 2 ⫽a 2ph(1,2) 共 r;y 兲
⬅
共 1⫺yr 3 兲 ⫹ 共 1⫺3yr 2 兲 ⫾2 冑共 1⫺yr 3 兲共 1⫺3yr 2 兲
y 2r 3
,
共49兲
where the notation a 2ph(1) and a 2ph(2) is used for the ⫾ parts
of Eq. 共49兲 because these functions can define photon orbits
for both plus- 共minus-兲 family circular orbits. A detailed discussion of the equatorial photon motion is presented in 关35兴,
where more general, Kerr–Newman–共anti-兲de Sitter spacetimes are studied. In the Kerr–de Sitter spacetimes the situation is much simpler. Since
⳵ a 2ph(1,2) 关 1⫺yr 2 共 r⫹3 兲 ⫹3y 2 r 5 兴 1/2共 yr 2 ⫺2 兲 ⫾ 关 ⫺2⫹yr 2 共 r⫹4 兲 ⫺y 2 r 5 兴
⫽
,
⳵r
y 2 r 4 关 1⫺yr 2 共 r⫹3 兲 ⫹3y 2 r 5 兴 1/2
we find that the local extrema of a 2ph(1,2) (r;y) are located at
radii determined by the relation
y⫽y e(ph)⫾ 共 r 兲 ⬅
2r⫹1⫾ 冑8r⫹1
a 2ph(1,2) (r;y)
2r 3
⫽y e(h)⫾ 共 r 兲 .
Therefore,
and
have common points at
their local extrema. Nevertheless, in order to obtain directly
limits on the existence of the plus- 共minus-兲 family circular
orbits, it is convenient to consider the plus- 共minus-兲 photon
circular orbits determined by relations 共47兲 and 共48兲, respectively, under the assumption a⭓0. We have to introduce a
critical value of the rotational parameter corresponding to the
(⫺)
situation where a (⫹)
ph(1) (r;y)⫽a ph(1) (r;y)—i.e., where these
functions reach the static radius r s ⫽y ⫺1/3:
2
a c(s)
共 y 兲⬅
1⫺3y 1/3
.
y
共52兲
Further, it is necessary to determine 共by a numerical procedure兲 the related critical value of the cosmological parameter
2
(y) correy c(s) such that for y⬍y c(s) the critical value a c(s)
sponds to a naked-singularity spacetime. The numerical procedure implies
y c(s) ⬟0.033185.
共53兲
The results can be summarized in dependence on the cosmological parameter and are illustrated in Fig. 7.
1. yÏy c„s… (Figs. 7a,7b)
In black-hole spacetimes there are three photon circular
orbits. Their loci satisfy the conditions
r ph(1) ⬍r h(⫺) ⬍r h(⫹) ⬍r ph(2) ⬍r ph(3) ⬍r s .
共54兲
The orbits r ph(1) and r ph(2) belong to the plus-family orbits,
while r ph(3) belongs to the minus-family orbits. We can conclude that in the black-hole backgrounds, the plus-family circular orbits are located at radii satisfying the relations
0⬍r⬍r ph(1)
and
r ph(2) ⬍r⬍r s ,
while the minus-family orbits are located at radii satisfying
the relation
r ph(3) ⬍r⬍r s .
共51兲
a 2h (r;y)
共55兲
共50兲
共56兲
In the naked-singularity spacetimes, we have to distinguish
two qualitatively different cases.
2
(y), there is one photon circular orbit belongIf a 2 ⭐a c(s)
ing to the minus-family orbits. In such spacetimes, the plusfamily orbits are located in the region
0⬍r⬍r s .
共57兲
2
If a 2 ⬎a c(s)
(y), the situation changes dramatically as the
minus-family orbits 共and the notion of the static radius兲 cease
to exist. There is only one plus-family photon circular orbit.
Therefore, the plus-family circular orbits are located in the
region
0⬍r⬍r ph .
共58兲
2. y c„s… ËyÏy c„SdS… (Figs. 7c,7d)
Now, we have to distinguish two cases in black-hole
spacetimes.
2
(y), the loci of the photon circular orbits are
If a 2 ⭐a c(s)
again related by relation 共54兲, and the limits on the existence
of plus-family and minus-family circular orbits are the same
as in the case of y⬍y c(s) —see relations 共55兲 and 共56兲, respectively.
2
(y), black-hole spacetimes admit only the
If a 2 ⬎a c(s)
plus-family circular orbits and all of the three photon circular
orbits limit them by the relation
0⬍r⬍r ph(1) and r ph(2) ⬍r⬍r ph(3) .
共59兲
In naked-singularity spacetimes, only one plus-family photon
circular orbit exists and the plus-family circular orbits are
limited by relation 共58兲.
3. y c„SdS… ËyÏy c„KdS… (Fig. 7e)
Only the plus-family circular orbits exist that are limited
by three photon circular orbits through relation 共59兲 in black-
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PHYSICAL REVIEW D 69, 064001 共2004兲
EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
FIG. 7. Existence of the circular orbits. Locations of the horizons and the photon circular orbits are given as functions of the rotational
parameter for typical values of the cosmological parameter of Kerr–de Sitter spacetimes and the regions of the existence of the circular orbits
are shown. Dash-dotted curves depict the horizons, solid curves depict the plus-family photon circular orbits, dashed curves depict the
minus-family photon circular orbits, and the gray line gives the static radius (r s ⫽y ⫺1/3). Notice that the static radius is relevant only in the
spacetimes admitting the minus-family circular orbits. The plus-family circular orbits exist in the whole shaded region, while the minusfamily circular orbits exist in the dark-gray regions only—i.e., in the spacetimes with y⬍y c(SdS) . In the naked-singularity spacetimes only
one photon circular orbit exists and the plus-family orbits always approach the ring singularity at r⫽0.
064001-9
PHYSICAL REVIEW D 69, 064001 共2004兲
Z. STUCHLÍK AND P. SLANÝ
hole spacetimes and by one photon circular orbit through
relation 共58兲 in naked-singularity spacetimes.
4. y c„KdS… Ëy (Fig. 7f)
Naked-singularity spacetimes exist for any a⬎0. The
spacetimes admit only the plus-family circular orbits limited
by one photon circular orbit through relation 共58兲.
Locally measured components of the four-momentum are
given by the projection of a particle’s four-momentum onto
the tetrad:
p ( ␣ ) ⫽ p ␮ ␻ ␮( ␣ ) ,
where
p ␮ ⫽m
B. Orientation of the circular orbits
The behavior of the circular orbits in the field of Kerr
black holes (y⫽0) suggests that the plus-family orbits correspond to the corotating orbits, while the minus-family circular orbits correspond to the counterrotating ones. However,
this statement is not generally correct even in some of the
Kerr naked-singularity spacetimes—namely, in the spacetimes with the rotational parameter low enough, where counterrotating plus-family orbits could exist nearby the ring singularity 关29兴. In Kerr–de Sitter spacetimes, the situation is
even more complicated and we cannot identify the plusfamily circular orbits with purely corotating orbits even in
black-hole spacetimes. Moreover, in rotating spacetimes with
a nonzero cosmological constant it is not possible to define
the corotating 共counterrotating兲 orbits in relation to stationary observers at infinity, as can be done in Kerr spacetimes,
since these spacetimes are not asymptotically flat.
The natural way of defining the orientation of the circular
orbits in Kerr–de Sitter spacetimes is to use the point of view
of locally nonrotating frames that is used in asymptotically
flat Kerr spacetimes too. The tetrad of one-forms corresponding to these frames in the Kerr–de Sitter backgrounds is
given by 关35兴
␻ ⬅
(t)
␻ (␸)⬅
冉 冊
冉 冊
⌬ r⌬ ␪% 2
A sin ␪
2
冉 冊
冉 冊
1/2
%2
⌬␪
1/2
␻ (␪)⬅
p (␸)⫽
mA 1/2
共 ␸˙ ⫺⍀ ṫ 兲 ,
Ir
dr,
d␪ ,
ṫ ⫽
␸˙ ⫽
I
r2
I
r
2
冋
冋
aX⫹
X⫹
册
共 r 2 ⫹a 2 兲共 r 2 E⫺aX 兲
,
⌬r
⌬ ␪ ⬅1⫹ya 2 cos2 ␪ ,
共61兲
共71兲
共64兲
共65兲
共66兲
A 1/2
p (␸)⫽
共62兲
共63兲
mr
共 aE⫹X 兲
共72兲
and using Eq. 共27兲 we obtain intuitively anticipated relation
mr
A 1/2
L.
共73兲
We can see that the sign of the azimuthal component of the
four-momentum measured in the locally nonrotating frames
is given by the sign of the specific angular momentum of a
particle on the orbit of interest. Therefore, the circular orbits
with p ( ␸ ) ⬎0 (L⬎0) we call corotating, and the circular orbits with p ( ␸ ) ⬍0 (L⬍0) we call counterrotating, in agreement with the approach used in the case of asymptotically
flat Kerr spacetimes.
C. Stability of the circular orbits
and the angular velocity of the locally nonrotating frames,
d␸ a
⫽ 关 ⫺⌬ r ⫹ 共 r 2 ⫹a 2 兲 ⌬ ␪ 兴 .
⍀⬅
dt A
共70兲
册
a 2
共 r E⫺aX 兲 .
⌬r
共60兲
where
A⬅ 共 r 2 ⫹a 2 兲 2 ⫺a 2 ⌬ r
共69兲
where the temporal and azimuthal components of the fourmomentum, determined by the geodesic equations, can be
expressed in the form containing the specific constants of
motion E,X:
p (␸)⫽
共 d␸ ⫺⍀dt 兲 ,
共68兲
are the coordinate components of particle’s four-momentum,
the affine parameter ␭⫽ ␶ /m, m denotes the rest mass of the
particle, and ␶ is its proper time.
In the equatorial plane, ␪ ⫽ ␲ /2, the azimuthal component
of the four-momentum measured in the locally nonrotating
frames is given by the relation
1/2
I 2% 2
%2
␻ ⬅
⌬r
(r)
dt,
dx ␮
dx ␮
⬅mẋ ␮ ⫽
d␶
d␭
A simple calculation reveals
1/2
I 2A
共67兲
The loci of the stable circular orbits are given by the
condition
d2 R
Note that ⌬ ␪ ⫽1 in the equatorial plane.
dr 2
064001-10
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共74兲
PHYSICAL REVIEW D 69, 064001 共2004兲
EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
which has to be satisfied simultaneously with the conditions
R(r)⫽0 and dR/dr⫽0 determining the specific energy and
the specific angular momentum of the circular orbits. Using
relations 共38兲 and 共39兲, we find that the radii of the stable
orbits of both families are restricted by the condition
r 关 6⫺r⫹r 3 共 4r⫺15兲 y 兴 ⫿8a 关 r 共 1⫺yr 3 兲 3 兴 1/2⫹a 2 关 3⫹r 2 y 共 1
⫺4yr 3 兲兴 ⭓0.
共75兲
The marginally stable orbits of both families can be described together by the relation
2
a 2 ⫽a ms(1,2)
共 r;y 兲
⬅ 关 3⫹r 2 y 共 1⫺4yr 3 兲兴 ⫺2 r„关 r⫺6⫺r 3 共 4r⫺15兲 y 兴
⫻ 关 3⫹r 2 y 共 1⫺4yr 3 兲兴 ⫹32共 1⫺yr 3 兲 3
⫾8 共 1⫺yr 3 兲 3/2共 1⫺4yr 3 兲 1/2
⫻ 兵 r 关 3⫺ry 共 6⫹10r⫺15yr 3 兲兴 ⫺2 其 1/2….
共76兲
The (⫾) sign in Eq. 共76兲 is not directly related to the plus2
, correfamily and minus-family orbits. The function a ms(1)
sponding to the ⫹ sign in Eq. 共76兲, determines marginally
stable orbits of the plus-family orbits, while the function
2
, corresponding to the ⫺ sign in Eq. 共76兲, is relevant
a ms(2)
for both the plus-family and minus-family orbits. The reality
2
(r;y) are directly given
conditions for the functions a ms(1,2)
by Eq. 共76兲. The standard condition y⭐y s (r)⬅1/r 3 is guaranteed by the first relevant condition
y⭐y ms 共 r 兲 ⬅
1
4r 3
FIG. 8. Reality conditions for the existence of the stable circular
orbits. Black and gray solid curves correspond to the functions
y ms (r) and y s (r), respectively; dash-dotted and dashed curves correspond to the functions y ms⫹ (r) and y ms⫺ (r), respectively. Stable
orbits can exist only in the shaded region, where the local maximum
corresponds to the critical value of the cosmological parameter
y crit(ms⫹) ⬟0.06886.
The plus-family stable circular orbits are allowed for y
⬍y ms (r), if y⬍y i , and for y⬍y ms(⫺) (r), if y i ⬍y
⬍y crit(ms⫹) .
The condition determining the local extrema of
2
(r;y),
a ms(1,2)
2
⳵ a ms(1,2)
共 r;y 兲
⫽0,
⳵r
implies very complicated relations; however, they lead to one
simple relevant relation
共77兲
.
y⫽y e(ms) 共 r 兲 ⬅
The other two conditions can be given in the form
y⭐y ms⫺ 共 r 兲
or
共78兲
y⭓y ms⫹ 共 r 兲 ,
共82兲
1
共83兲
10r 3
2
(r;y)
determining important local extrema of both a ms(1,2)
simultaneously, both located on the radius
where the functions y ms(⫾) (r) are given by the relation
y ms⫾ 共 r 兲 ⫽
3⫹5r⫾ 共 60r⫺20r 2 ⫹9 兲 1/2
15r
3
.
共79兲
The behavior of the functions y s (r), y ms (r), and y ms⫾ (r) is
illustrated in Fig. 8. The function y ms(⫹) (r) is irrelevant; the
relevant function y ms(⫺) (r) intersects the function y s (r) at
r⫽3, where y⫽y c(SdS) ⫽1/27, and the function y ms (r) at r
⫽(3⫹2 冑3)/4, where y⫽y i ⫽16/(3⫹2 冑3) 3 . The critical
value of the cosmological parameter for the existence of the
stable 共plus-family兲 orbits, corresponding to the local maximum of y ms(⫺) (r), is given by
y crit(ms⫹) ⫽
100
共 5⫹2 冑10兲 3
⬟0.06886.
共80兲
The related critical value of the rotational parameter is
2
⫽
a crit(ms⫹)
955⫹424冑10
⬟1.41716.
1620
r⫽r e(ms) 共 y 兲 ⬅
共81兲
1
共 10y 兲 1/3
.
共84兲
The critical value of the cosmological parameter for the
existence of the minus-family stable circular orbits, deter2
(r e(ms) ;y)⫽0, is given by
mined by the condition a ms(2)
y crit(ms⫺) ⫽
12
154
.
共85兲
It coincides with the limit on the existence of the stable circular orbits in Schwarzschild–de Sitter spacetimes 关36兴.
2
(r;y) can be summaProperties of the functions a ms(1,2)
rized in the following way.
共i兲 y⬎y crit(ms⫹) . No stable circular orbits are allowed
for any value of the rotational parameter.
共ii兲 y crit(ms⫹) ⬎y⬎y crit(ms⫺) . At r⫽r e(ms) , the func2
2
(r;y) has a local maximum (a ms(max)
), and the
tion a ms(1)
2
2
function a ms(2) (r;y) has a local minimum (a ms(min)
). For
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Z. STUCHLÍK AND P. SLANÝ
FIG. 9. Marginally stable circular orbits in Kerr–de Sitter spacetimes. The relevant functions are given for some typical values of
the cosmological parameter y. 共a兲 The black-hole region of Kerr–de
Sitter spacetimes. For y⬍12/154 there exist spacetimes containing
four marginally stable 共ms兲 orbits. For a given spacetime, the innermost and the outermost ms orbits belong to the plus-family orbits;
the two orbits in between belong to the minus-family orbits. 共b兲 In
the naked-singularity region there exist spacetimes with no stable
orbits for a fixed value of y 关spacetimes with a 2 greater than the
2
(r;y) for a given y]. Stable counglobal maximum of function a ms
terrotating 共minus-family兲 orbits exist only in shaded regions of the
presented spacetimes. But some naked-singularity spacetimes contain counterrotating plus-family orbits; for more details, see the
text. The dashed line corresponds to the radius (10y) ⫺1/3 where
2
both maxima of a ms
(r;y) are located.
2
2
2
a ms(min)
⬍a 2 ⬍a ms(max)
, the equation a 2 ⫽a ms(1,2)
(r;y) determines two marginally stable plus-family circular orbits 共an
2
inner one and an outer one兲. For 0⬍a 2 ⬍a ms(min)
and a 2
2
⬎a ms(max) , no stable circular orbits are allowed.
共iii兲 y⬍y crit(ms⫺) . There are two zero points of the
2
(r;y) corresponding to its local minima,
function a ms(2)
2
while it has a local maximum a ms(max2)
at r⫽r e(ms) , where
2
the maximum of the function a ms(1) (r;y) is located too. For
2
a 2 ⬎a ms(max)
, there is no stable circular orbit. For
2
2
2
, there are two marginally stable
a ms(max2) ⬍a ⬍a ms(max)
2
plus-family circular orbits. For a 2 ⬍a ms(max2)
, there are four
marginally stable orbits. The innermost and the outermost
orbits belong to the plus-family orbits; the two orbits in between belong to the minus-family orbits.
2
The functions a ms(1,2)
are illustrated for typical values of the
cosmological parameter in Fig. 9. In the parameter space
y-a 2 , separation of Kerr–de Sitter spacetimes according to
the existence of stable circular orbits, determined by the
2
(r;y) and y e(ms) (r), is given in Fig. 3.
functions a ms(1,2)
D. Marginally bound circular orbits
The behavior of the effective potential 共28兲 enables us to
introduce the notion of the marginally bound orbits—i.e.,
unstable circular orbits where a small radial perturbation
causes infall of a particle from the orbit to the center or its
escape to the cosmological horizon. For some special value
of the axial parameter X, denoted as X mb , the effective potential has two local maxima related by the condition
E (⫹) 共 r mb(i) ;X mb ,a,y 兲 ⫽E (⫹) 共 r mb(o) ;X mb ,a,y 兲 , 共86兲
and corresponding to both the inner and outer marginally
bound orbits; see Fig. 10 共and Fig. 15, below兲. For completeness, the figures include the effective potentials defining both
FIG. 10. Effective potential of the equatorial radial motion of
test particles in an appropriatelly chosen Kerr–de Sitter black-hole
spacetime (y⫽10⫺4 ,a 2 ⫽0.36) allowing stable circular orbits for
corotating particles. Marginally bound 共mb兲 orbits are given by the
solid curve corresponding to the angular momentum parameter X
⫽X mb⫹ ⬟2.38445. The curve has two local maxima of the same
value, E mb ⬟0.93856, corresponding to the inner 关mb共i兲兴 and the
outer 关mb共o兲兴 marginally bound orbits. The dashed effective potential defines the inner marginally stable orbit 关ms共i兲兴 by coalescing
the local minimum and the 共inner兲 local maximum. It corresponds
to the parameter X⫽X ms(i)⫹ ⬟2.20307 with specific energy
E ms(i)⫹ ⬟0.90654. In an analogous manner, the dash-dotted potential defines the outer marginally stable orbit 关ms共o兲兴 with specific
energy E ms(o)⫹ ⬟0.94451 corresponding to the parameter X
⫽X ms(o)⫹ ⬟2.90538.
the inner and outer marginally stable orbits 共corresponding to
special values of the parameter X: X ms(i) ,X ms(o) ). The search
for the marginally bound orbits in a concrete Kerr–de Sitterspacetime must be realized in a numerical way and can be
successful only in the spacetimes admitting stable circular
orbits. Clearly, in the spacetimes with y⭓12/154 , the minusfamily marginally bound orbits do not exist. Figure 3 offers
insight into the possibility of the existence of both stable and
bound circular orbits of both families. The limiting 共solid兲
curves are obtained from the conditions 共76兲,共83兲 that have
to be solved simultaneously.
The location of the astrophysically important circular orbits 共photon orbits, marginally stable and marginally bound
orbits兲 in dependence on the rotational parameter a is given
in Fig. 11 for three appropriately chosen values of the cosmological parameter y. The values of y reflect the dependence of the existence of stable minus-family orbits on y.
Stable plus-family orbits exist for all chosen values of y in
the relevant range of the parameter a. Spacetimes without
stable circular orbits or without any circular orbits are inferred from Figs. 7,8.
V. DISCUSSION
In comparison with asymptotically flat Kerr spacetimes,
where the effect of the rotational parameter vanishes for asymptotically large values of the radius, in Kerr–de Sitter
spacetimes the properties of the circular orbits must be
treated more carefully, because the rotational effect is rel-
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EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
FIG. 11. Mutual positions of the astrophysically important circular orbits in Kerr–de Sitter spacetimes. The figures are constructed for
three representative values of y. The radii of the special equatorial circular orbits are plotted as functions of the rotational parameter a. The
wide dashed line is given by the value of rotational parameter corresponding to the extreme black hole and it splits up Kerr–de Sitter
spacetimes into black-hole 共BH兲 and naked-singularity 共NS兲 regions. Thin curves are used for the plus-family orbits 共in most cases they
correspond to the corotating orbits from the point of view of the locally nonrotating observers, but there are exceptions described in the text兲.
Bold curves are used for the minus-family orbits 共in all spacetimes under consideration: counterrotating orbits兲. Solid curves determine the
inner and outer black-hole horizons. Dotted curves determine the photon circular orbits; dashed curves determine the marginally bound 共mb兲
circular orbits. There is a disconnection between BH and NS regions for the plus-family orbits. Lower gray dashed curves determine the
marginally bound orbits hidden under the inner black-hole horizon; the upper one, approaching the static radius for small a, is its outer
analogy. Dash-dotted curves determine the marginally stable 共ms兲 orbits. For y⭓12/154 there are no minus-family mb and ms orbits.
evant in whole the region where the circular orbits are allowed and it survives even at the cosmological horizon.
The minus-family orbits have specific angular momentum
negative, L ⫺ ⬍0, in every Kerr–de Sitter spacetime and such
orbits are counterrotating from the point of view of locally
nonrotating frames.
In black-hole spacetimes, the plus-family orbits are corotating in almost all radii where the circular orbits are allowed
except some region in vicinity of the static radius, where
they become counterrotating, as their specific angular momentum L is slightly negative there. In naked-singularity
spacetimes, the plus-family orbits behave in a more complex
way; nevertheless, they are always counterrotating in vicinity
of the static radius.
The specific angular momentum of particles located on
the static radius, where the plus-family orbits and the minus-
family orbits coalesce, is given by the relation
L 共 r s ;y,a 兲 ⫽L s ⬅⫺a
3y 1/3⫹a 2 y
共 1⫺3y 1/3⫺a 2 y 兲 1/2
,
共87兲
and their specific energy is
E 共 r s ;y,a 兲 ⫽E s ⬅ 共 1⫺3y 1/3⫺a 2 y 兲 1/2.
共88兲
A. Circular orbits with zero angular momentum
Separation of the corotating and counterrotating orbits as
defined by their azimuthal angular momentum relative to the
locally nonrotating frames is determined by the orbits with
L⫽0. The orbits with zero angular momentum are defined
by the relation
y⫽y (L⫽0) 共 r;a 兲
⬅
⫺r 关 r 共 r 2 ⫹a 2 兲 ⫹4a 2 兴 ⫹r 1/2共 r 2 ⫹a 2 兲 1/2关共 r 2 ⫹a 2 兲共 r 3 ⫹4a 2 兲 ⫹8a 2 r 2 兴 1/2
2a 2 r 2 共 r 2 ⫹a 2 兲
064001-13
.
共89兲
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Z. STUCHLÍK AND P. SLANÝ
FIG. 12. Circular orbits with zero angular momentum. The function y (L⫽0) (r;a) is plotted for a 2 ⫽ 兵 1,1.8,2.4406,5其 . For a 2 ⫽1,
when the black-hole spacetimes are allowed, the first two orbits are
hidden under the black-hole event horizon. The dashed curve determines horizons of Kerr–de Sitter spacetimes with a 2 ⫽1. In the
other cases only naked-singularity spacetimes are allowed and even
three L⫽0 orbits are possible for a 2 ⬍2.4406. The case a 2 ⬟2.4406
共the curve with an inflex point at y⬟0.03998) corresponds to the
maximum value of the rotational parameter admitting the stable
counterrotating plus-family orbits.
At these orbits, the locally nonrotating observers follow circular geodesics at the equatorial plane.
The physically relevant zero points of the function y (L⫽0)
are given by the function
a z(L⫽0) 共 r 兲 ⬅r 1/2关 1⫹ 共 1⫺r 兲 1/2兴
共90兲
determining circular orbits with L⫽0 in the asymptotically
flat Kerr backgrounds. A detailed study reveals that such orbits exist only in Kerr naked-singularity spacetimes with 1
⬍a/M ⭐ 34 冑3. 共In Kerr black-hole spacetimes orbits of this
kind are hidden under the event horizon.兲 The typical behavior of the function y (L⫽0) (r;a) is presented in Fig. 12 for
some appropriately chosen values of the rotational parameter
2
a 2 . The function has a local extremum for a 2 ⬍a cL
2
⬅2.4406 共where the critical value a cL is obtained by a nu-
y⫽y (E⫽0) 共 r;a 兲 ⬅
merical procedure兲—we can conclude that up to three zeroangular-momentum orbits can exist in the corresponding
Kerr–de Sitter spacetimes. In the naked-singularity spacetimes all three orbits with zero angular momentum are relevant and the middle orbit is stable. In black-hole spacetimes, however, two of these orbits are hidden under the
black-hole horizons and only the unstable one, located
nearby the static radius, is physically important. The plusfamily orbits between the zero-angular-momentum orbit and
the static radius are counterrotating. Comparison of the functions y (L⫽0) (r;a) and y ms (r;a) 共determining the position of
the outermost marginally stable orbit for a given cosmological parameter y) reveals that the discussed orbits are unstable.
In naked-singularity spacetimes, the behavior of the plusfamily orbits is more intriguing. Except for the unstable
counterrotating orbits located nearby the static radius 共discussed above兲, some stable counterrotating plus-family circular orbits exist in the vicinity of the ring singularity of
naked-singularity spacetimes with rotational parameter low
enough. Spacetimes admitting such orbits belong to the dashdotted naked-singularity region of the parametric space
(y,a 2 ) presented in Fig. 3. The limiting 共dash-dotted兲 curve
was obtained by solving simultaneously the conditions for
the marginally stable orbits given by Eq. 共76兲 and the condition for the orbits with zero angular momentum given by Eq.
共89兲.
In a given Kerr–de Sitter naked-singularity spacetime relations 共76兲,共89兲 determine the innermost and outermost
stable counterrotating plus-family orbits, respectively.
B. Circular orbits with negative energy
In the rotating naked-singularity spacetimes the potential
well can be deep enough nearby the ring singularity to allow
the existence of stable 共plus-family兲 counterrotating circular
orbits with negative specific energy, indicating an extremely
high efficiency of conversion of the rest mass into heat energy during accretion in a corotating 共or, more precisely, a
plus-family兲 thin disk. The plus-family circular orbits with
zero energy are given by the relation
r 关 2 共 r 2 ⫹a 2 兲共 r⫺2 兲 ⫺a 2 r 兴 ⫹ar 1/2兵 4 共 r 2 ⫹a 2 兲 2 ⫺r 2 关 4 共 r 2 ⫹a 2 兲共 r⫺2 兲 ⫺a 2 r 兴 其 1/2
2r 2 共 r 2 ⫹a 2 兲 2
The reality conditions of the function y (E⫽0) (r;a) are given
by the relations
r⭓0,
4 共 r 2 ⫹a 2 兲 2 ⫺r 2 关 4 共 r 2 ⫹a 2 兲共 r⫺2 兲 ⫺a 2 r 兴 ⭓0.
2
a 2 ⭓a min(E⫽0)
共 r 兲⬅
.
共91兲
r2
关 3r⫺16⫹ 共 9r 2 ⫺32r⫹64兲 1/2兴 ,
8
共94兲
共92兲
共93兲
which is relevant for r⬎3. For 0⬍r⬍3 the function
2
(r) is negative. The zero points of the function
a min(E⫽0)
y (E⫽0) (r;a) are given by the function
The condition 共93兲 can be transferred into the relation
a z(E⫽0) 共 r 兲 ⬅r 1/2共 2⫺r 兲 ,
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which determines the circular orbits with zero specific energy in Kerr spacetimes. For y⫽0, such orbits exist only in
Kerr naked-singularity spacetimes with 1⬍a/M ⭐ 34 冑 23 ,
which are a subset of spacetimes with zero-angularmomentum orbits; in fact, orbits with E⫽0 have L⬍0 共for
details see 关29兴兲. The behavior of the function y (E⫽0) (r;a) is
presented for some typical values of the rotational parameter
a in Fig. 13. In Kerr–de Sitter spacetimes with 0⬍a 2
⬍1.47, the function y (E⫽0) has two local extrema leading up
to three circular orbits with zero energy. The ending points of
the curves are given by the condition 共94兲 and are represented by the function
y min(E⫽0) 共 r 兲 ⬅y (E⫽0) 共 r;a⫽a min(E⫽0) 兲
⫽
3r 2 ⫺10r⫹16⫺ 共 r⫺2 兲共 9r 2 ⫺32r⫹64兲 1/2
2r 4
.
共96兲
Details of the properties of the plus-family orbits with E
⫽0 can be inferred from Fig. 13. Here, we give a short
overview of them.
In the black-hole spacetimes, there is always one orbit
with E⫽0 located under the inner black-hole horizon, and
there can exist, for properly chosen parameters a and y, one
orbit with E⫽0 located above the outer black-hole horizon.
Both the orbits must be unstable relative to radial perturbations.
In
the
naked-singularity
spacetimes,
if
a2
2
⬍a c(E⫽0) ⬟1.18518, there can exist one orbit with E⫽0
共unstable兲, two such orbits 共the inner one unstable, the outer
one stable兲, or three such orbits 共the inner and outer being
2
unstable, the intermediate being stable兲. If a 2 ⬎a c(E⫽0)
and y
is properly chosen, there can be an additional possibility of
the nonexistence of the circular orbit with E⫽0. If a 2
2
⬎a s(E⫽0)
⬟1.47000, there can exist no stable zero-energy
orbits for any y 共cf. Fig. 3兲.
Examples of naked-singularity spacetimes admitting
stable counterrotating plus-family circular orbits with negative energy are presented in Fig. 14. The efficiency of conversion of the rest mass into heat energy during accretion,
given by the relation
␩ ⬅E ms(o) ⫺E ms(i) ,
共97兲
is limited by the specific energy of the outermost stable circular plus-family orbit, E ms(o) ⬍1, which can be directly inferred from Fig. 15. Correspondingly, extraction of the rotational energy from a naked singularity with rotational
parameter low enough is possible with subsequent conversion of the naked singularity into a black hole 共see, e.g.,
关44兴兲. Spacetimes allowing such processes are contained in
the shaded naked-singularity region of the parametric space
(y,a 2 ) in Fig. 3. The limiting 共dotted兲 curve was obtained by
solving simultaneously conditions for the marginally stable
orbits 共76兲 and the circular orbits with E⫽0 共91兲.
VI. CONCLUDING REMARKS
Many properties of Kerr–de Sitter spacetimes and circular
orbits of both families can be clearly viewed from figures
which are presented in the paper. Table I contains a certain
classification of the figures which could be helpful for quick
orientation in the topic.
Both black-hole and naked-singularity Kerr–de Sitter
spacetimes can be separated into three classes according to
the existence of stable 共and, equivalently, marginally bound兲
circular orbits 共see Fig. 3兲. Stable orbits of both the plus
family and minus family exist in the spacetimes of class I
共black holes兲 and class V 共naked singularities兲. Solely stable
orbits of the plus family exist in the spacetimes of classes II
共black holes兲 and VI 共naked singularities兲. No stable orbits
exist in the spacetimes of classes III and IV. In dependence
on the cosmological parameter, there are three qualitatively
different types of the behavior of the loci of the marginally
stable, marginally bound, and photon circular orbits as functions of the rotational parameter. These functions are illustrated for three representative values of y in Fig. 11, enabling
us to make in a straightforward way separation of Kerr–de
Sitter spacetimes into classes I–VI. In the special case of
Kerr spacetimes (y⫽0), these functions can be found in
关13,29兴.
The marginally stable circular orbits are crucial in the
context of Keplerian 共geometrically thin兲 accretion disks as
these orbits determine the efficiency of conversion of rest
mass into heat energy of any element of matter transversing
the disks from their outer edge located on the outer marginally stable orbit to their inner edge located on the inner marginally stable orbit.
Clearly, accretion disks constituted from minus-family orbits are everywhere counterrotating relative to the locally
nonrotating frames. For the minus-family disks, the specific
energy of both the outer and inner marginally stable circular
orbits and the efficiency parameter ␩ ⫺ ⫽E ms(o)⫺ ⫺E ms(i)⫺
are given for three typical values of y as functions of a in
Fig. 16. In the limit of a→0 with y being fixed, we obtain
the known values of the specific energy E ms(o) ,E ms(i) and
the efficiency parameter of the accretion process ␩ for the
Schwarzschild–de Sitter black holes 关36兴. Both the specific
energy parameters E ms(o)⫺ (a), E ms(i)⫺ (a) and the efficiency
␩ ⫺ (a) vary smoothly at values of the rotational parameter
corresponding to the extreme black holes.
The Keplerian accretion disks constituted from the plusfamily orbits behave in much more complex way in comparison with those of the minus-family orbits. First, usually these
disks could be considered as corotating relative to the locally
nonrotating frames; recall that in asymptotically flat Kerr
black-hole spacetimes the plus-family disks are corotating at
all radii down to the marginally stable orbit, while in the field
of naked singularities with a/M ⬍ 43 冑3 the stable circular
orbits corotating at large distances are transformed into counterrotating orbits in vicinity of the marginally stable orbit
关29兴. A similar behavior occurs in Kerr–de Sitter spacetimes;
however, in the spacetimes with y→y c(KdS) , the stable plusfamily orbits can be counterrotating even at all allowed radii
共see, e.g., Fig. 14兲. 共Moreover, there are always counterrotat-
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Z. STUCHLÍK AND P. SLANÝ
FIG. 13. Location of the plus-family orbits with E⫽0. The dashed curves determine the horizons of Kerr–de Sitter spacetimes. The solid
curves determine the orbits with E⫽0. The rising 共descending兲 part of y (E⫽0) (r;a) corresponds to the stable 共unstable兲 orbits. The intervals
of the rotational parameter giving different behavior of the curves y (E⫽0) (r;a),y h (r;a) in cases 共a兲–共g兲 are 共a兲 0⬍a 2 ⬍1, 共b兲 1⬍a 2
2
2
2
2
⬍1.06992, 共c兲 1.06992⬍a 2 ⬍a c(E⫽0)
⬟1.18518, 共d兲 a c(E⫽0)
⬍a 2 ⬍a crit
⬟1.21202, 共e兲 a crit
⬍a 2 ⬍1.25976, 共f兲 1.25976⬍a 2
2
2
2
⬍a s(E⫽0) ⬟1.47000, 共g兲 a ⬎a s(E⫽0) .
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EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
FIG. 14. Naked-singularity spacetimes admitting stable circular orbits with negative specific energy. We can see that stable circular orbits
in vicinity of the innermost stable orbit 共the left and middle columns兲 or even all the stable orbits 共the right column兲 can have negative
energy. The behavior of the specific angular momentum reveals counterrotation of such orbits.
ing plus-family orbits in the vicinity of the static radius,
where the plus-family orbits and the minus-family orbits
coalesce; these orbits are, however, unstable relative to radial
perturbations and cannot be related to accretion disks.兲
Second, the specific energy E ms(i)⫹ (y,a) of the inner
marginally stable plus-family orbit can be negative. Recall
that E ms(i)⫹ ⬍0 in asymptotically flat Kerr naked-singularity
spacetimes with the rotational parameter a/M ⬍ 34 冑 23 , indicating the efficienty of the accretion process ␩ ⫹ ⫽E ms(o)⫹
⫺E ms(i)⫹ ⬎1, because in asymptotically flat Kerr spacetimes the outer edge of the accretion disks can be at arbi-
FIG. 15. Effective potential of the equatorial radial motion of
test particles in an appropriately chosen Kerr–de Sitter nakedsingularity spacetime (y⫽10⫺4 ,a 2 ⫽1.05) allowing the stable circular orbits with negative energy. Marginally bound 共mb兲 orbits are
given by the solid curve corresponding to X⫽X mb⫹ ⬟⫺0.18444.
The curve has two local maxima of the same value, E mb ⬟0.92857,
corresponding to the inner 关mb共i兲兴 and the outer 关mb共o兲兴 marginally
bound orbits. The dashed effective potential defines the inner marginally stable orbit 关ms共i兲兴 resulting from coalescence of the local
minimum and 共inner兲 local maximum, with X⫽X ms(i)⫹
⬟⫺0.40713 and the specific energy E ms(i)⫹ ⬟⫺0.23321. In an
analogous manner, the dash-dotted potential defines the outer marginally stable orbit 关ms共o兲兴 with the specific energy
E ms(o)⫹ ⬟0.94405 and X⫽X ms(o)⫹ ⬟2.40379.
trarily large radii, implying thus E ms(o)⫹ ⫽1. In Kerr–de Sitter spacetimes allowing E ms(i)⫹ ⬍0, the efficiency of the
accretion process can be both ␩ ⬎1 and ␩ ⬍1, as it depends
strongly on E ms(o)⫹ , which for y⬃y c(KdS) can be even negative 共see Fig. 14兲. For three typical values of y, the functions
E ms(o)⫹ (a),E ms(i)⫹ (a), ␩ ⫹ (a) are illustrated in Fig. 17. The
specific energy function E ms(i)⫹ (a) falls for a growing in the
black-hole region and for a descending in the nakedsingularity region. The specific energy function E ms(o)⫹ (a)
has a local minimum at some value of the rotational parameter a strongly dependent on the cosmological parameter y.
For y being fixed, the accretion efficiency ␩ ⫹ (a) grows for a
growing in the black-hole sector up to the critical value corresponding to the extreme black-hole spacetime, and it also
grows for a descending in the naked-singularity sector down
to the critical value.
Third, there is a strong discontinuity of the specific energy
function E ms(i)⫹ (a) for spacetimes approaching the extreme
black hole state from the black-hole and the nakedsingularity sectors. For extreme Kerr black holes (y
⫽0,a/M ⫽1), there is the limiting value of the specific energy E ms(bh) ⫽1/冑3, while for naked singularities approaching the extreme hole states (a/M →1 from above兲, there is
E ms(ns) ⫽⫺1/冑3. For extreme Kerr–de Sitter spacetimes,
the dependence of the specific energy of the inner marginally
stable orbit on the cosmological parameter is shown in Fig.
18a. Clearly, there is E ms(ns) (y)⫽⫺E ms(bh) (y), where for a
TABLE I. Classification of the figures.
Spacetime properties
Properties
of circular orbits
General
Specific energy
Spec. ang. mom.
Accretion efficiency
064001-17
Figs. 1–3
(⫹) family
(⫺) family
Figs. 7–11, 15
Figs. 4, 6, 13, 14, 17–19
Figs. 5, 6, 12, 19
Figs. 17, 18
Figs. 7–9, 11
Figs. 4, 6, 16
Figs. 5, 6, 12
Fig. 16
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Z. STUCHLÍK AND P. SLANÝ
FIG. 16. Specific energy of the marginally stable minus-family orbits 关共a兲 Inner, 共b兲 Outer兴 and 共c兲 the accretion efficiency ␩ ⫺
⬅E ms(o)⫺ ⫺E ms(i)⫺ given as a function of the rotational parameter for three representative values of the cosmological parameter. For Kerr
spacetimes, y⫽0, we assume E ms(o)⫺ ⫽1.
given cosmological parameter y the rotational parameter a of
the corresponding extreme black hole is determined by the
upper branch of the limiting line separating black-hole and
naked-singularity states in Fig. 3. For y→y c(KdS) , there is
E ms(bh) (y)→0. For the specific energy function
E ms(o)⫹ (y,a) of the outer marginally stable orbits there is no
discontinuity at the states corresponding to extreme blackhole spacetimes 共see Fig. 18b兲. The accretion efficiency
␩ ⫹ (y) in the field of extreme black holes 关 ␩ bh (y) 兴 and in
the field of the naked singularities infinitesimally close to the
extreme hole states 关 ␩ ns (y) 兴 is shown in Fig. 18c. For y
⫽0 their difference takes the maximum ( ␩ ns ⫽1⫹1/冑3,
␩ bh ⫽1⫺1/冑3), while at y⫽y c(KdS) the difference vanishes
( ␩ ns ⫽0, ␩ bh ⫽0).
As a result of accretion in a plus-family or a minus-family
Keplerian disk, a hypothetical naked singularity can be converted into an extreme black hole. In the case of Kerr naked
singularities their evolution into an extreme hole state was
discussed in 关44 – 46兴. Such a conversion can be a rather
dramatic process in the case of the plus-family accretion
disks, because of the discontinuity of the plus-family orbits
at the extreme black-hole state. We can understand this process if we show how the stable circular orbits are distributed
in naked-singularity spacetimes approaching the extreme
black-hole state 共Fig. 19兲. We can see that all the orbits with
the specific energy ranging from E ms(ns) (y) up to E ms(bh) (y)
are distributed at an infinitesimally small range of the radial
coordinate in the vicinity of the radius corresponding to the
event horizon of the extreme black hole. Of course, it is well
known that at these radii the physically relevant proper radial
length, along which the accretion disk is distributed, becomes very 共almost infinitely兲 long 共see 关13兴兲. If the conversion of a hypothetical naked singularity into an extreme
black hole is realized, the part of the accretion disk located
under the marginally stable circular orbit of the created black
hole becomes unstable relative to radial perturbations and
will be immediately swallowed by the black hole. It can be
expected that the collapse of the unstable internal part of the
disk with the specific energy ranging from E ms(ns) (y) up to
E ms(bh) (y) could be observationally important, leading to an
abrupt fall down of observable luminosity of the accretion
disk.
Finally, we shall give to our results proper astrophysical
relevance by presenting numerical estimates for the observationally established value of the current value of the cosmological constant. A wide range of recent cosmological observations give a strong ‘‘concordance’’ indication 关47兴 that the
observed value of the vacuum energy density is
% v ac(0) ⬇0.66% crit(0) ,
共98兲
with present values of the critical energy density % crit(0) and
the Hubble parameter H 0 given by
064001-18
PHYSICAL REVIEW D 69, 064001 共2004兲
EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
FIG. 17. Specific energy of the marginally stable plus-family orbits 关共a兲 inner, 共b兲 outer兴 and 共c兲 the accretion efficiency ␩ ⫹ ⬅E ms(o)⫹
⫺E ms(i)⫹ 共right column兲 as a function of the rotational parameter for three representative values of the cosmological parameter. The gray
line corresponds to the extreme black hole. We can see a strong discontinuity of the specific energy of the inner marginally stable orbits and
the accretion efficiency when black holes and naked singularities approach the extreme black-hole state. For Kerr spacetimes, y⫽0, we
assume E ms(o)⫹ ⫽1.
FIG. 18. Specific energy of the marginally stable orbits and accretion efficiency near the extreme black-hole states. 共a兲 Specific energy
of the inner marginally stable plus-family orbit in extreme black-hole and related limiting naked-singularity spacetimes approaching the
extreme hole states as a function of the cosmological parameter y. The solid curve corresponds to the extreme black holes; the dashed curve
corresponds to the limiting naked singularities. The curves are symmetric around the zero-energy axis and tend to zero for y⫽y c(KdS) . In
extreme Kerr spacetimes (y⫽0), the specific energies in the black-hole and naked-singularity cases are 1/冑3 and ⫺1/冑3, respectively. 共b兲
Specific energy of the outer marginally stable plus-family orbit in extreme Kerr–de Sitter black-hole spacetimes is the same as for
naked-singularity spacetimes approaching the extreme hole state; i.e., there is no discontinuity in this case. The specific energy tends to zero
for y→y c(KdS) . 共c兲 Accretion efficiency for the extreme black holes ␩ bh 共solid curve兲 and for the limiting naked singularities ␩ ns 共dashed
curve兲. For y⫽0 共pure Kerr spacetimes兲 we obtain the maximum value 0.42 for black holes and 1.58 for naked singularities. For y
→y c(KdS) the efficiency tends to zero for both black holes and naked singularities.
064001-19
PHYSICAL REVIEW D 69, 064001 共2004兲
Z. STUCHLÍK AND P. SLANÝ
FIG. 19. Distribution of the specific energy and the specific angular momentum of the equatorial circular orbits in naked-singularity
spacetimes approaching the extreme black-hole state. The orbits with the specific energy in the interval E ms(ns) ⬍E⬍E ms(bh) are located in
an extremely small interval of the radial coordinate having, however, an extremely long proper length 关13兴. After conversion of a hypothetical naked singularity into an extreme black hole all these circular orbits become unstable relative to radial perturbations and will be
immediately swallowed by the black hole. The figures are drawn for a⫽a 0 (1⫹ ␦ ) and y⫽y 0 (1⫺ ␦ ), where y 0 ⫽10⫺4 and a 0 ⫽1.0001 are
chosen to correspond to an extreme black hole and, subsequently, ␦ ⫽10⫺3 ,10⫺4 ,10⫺6 .
% crit(0) ⫽
3H 20
8␲
,
H 0 ⫽100h km s⫺1 Mpc⫺1 .
共99兲
Taking the value of the dimensionless parameter h⬇0.7, we
obtain the ‘‘relict’’ repulsive cosmological constant to be
⌳ 0 ⫽8 ␲ % v ac(0) ⬇1.1⫻10⫺56 cm⫺2 .
共100兲
Having this value of ⌳ 0 , we can determine the mass parameter of the spacetime corresponding to any value of y, parameters of the equatorial circular geodesics, and basic characteristics of the thin accretion disks. For extreme black holes
TABLE II. Mass parameter, static radius, and radius of the outer
marginally stable circular orbit determining the outer edge of corotating Keplerian disks in extreme Kerr–de Sitter black-hole spacetimes are given for the relict repulsive cosmological constant indicated
by
recent
cosmological
observations:
⌳ 0 ⬇1.1
⫻10⫺56 cm⫺2 . Note that accretion efficiency ␩ ⫹ is, in principle,
smaller than the value for the pure Kerr case (y⫽0) but, in practice, for small values of cosmological parameter y contained in the
table, ␩ ⫹ is undistinguishable from the Kerr limit ␩ ⬇0.42.
y
M
关 M 䉺兴
rs
关kpc兴
r ms(o)⫹
关kpc兴
10⫺46
10⫺44
10⫺42
10⫺40
10⫺34
10⫺32
10⫺30
10⫺28
10⫺26
10⫺24
10⫺22
1.1
11.1
111.4
1.1⫻103
1.1⫻106
1.1⫻107
1.1⫻108
1.1⫻109
1.1⫻1010
1.1⫻1011
1.1⫻1012
0.1
0.2
0.5
1.1
11.4
24.5
52.8
113.8
245.2
528.3
1138.4
0.07
0.15
0.3
0.7
7.2
15.5
33.3
71.7
154.5
332.9
717.1
共we have chosen some typical values of the black-hole
mass兲, the dimensions of the static radius and the outer marginally stable circular orbit of the plus-family accretion disk
are given in Table II. For more detailed information in the
case of thick disks around Schwarzschild–de Sitter black
holes see 关38兴, where the estimates for primordial black holes
in the early Universe with a repulsive cosmological constant
related to a hypothetical vacuum energy density connected
with the electroweak symmetry breaking or the quark confinement are presented.
It is well known 共see, e.g., 关48兴兲 that dimensions of accretion disks around stellar-mass black holes (M ⬃10M 䉺 ) in
binary systems are typically 10⫺3 pc, dimensions of large
galaxies with central black-hole mass M ⬃108 M 䉺 , of both
spiral and elliptical type, are in the interval 50–100 kpc, and
extremely large elliptical galaxies of cD type with central
black-hole mass M ⬃3⫻109 M 䉺 extend up to 1 Mpc. Therefore, we can conclude that the influence of the relict cosmological constant is quite negligible in the accretion disks in
binary systems of stellar-mass black holes as the static radius
exceeds in many orders dimension of the binary systems. But
it can be relevant for accretion disks in galaxies with large
active nuclei as the static radius puts limit on the extension
of the disks well inside of the galaxies. Moreover, the agreement 共up to one order兲 of the dimension of the static radius
related to the mass parameter of central black holes at nuclei
of large or extremely large galaxies with extension of such
galaxies suggests that the relict cosmological constant could
play an important role in the formation and evolution of such
galaxies. Of course, the first step in confirming such a suggestion is modeling of the influence of the repulsive cosmological constant on self-gravitating accretion disks. Some
hints this way could be given by recent results of Rezzolla
et al. 关49兴, based on sophisticated numerical hydrodynamic
methods developed by Font 关50,51兴, who showed that mass
outflow from the outer edge of thick accretion disks, induced
by the relict cosmological constant, could efficiently stabilize
the accretion disks against the runaway dynamical instability.
064001-20
PHYSICAL REVIEW D 69, 064001 共2004兲
EQUATORIAL CIRCULAR ORBITS IN THE KERR-de . . .
ACKNOWLEDGMENTS
The present work was supported by GAČR grant No. 205/
03/1147 and by the Bergen Computational Physics Laboratory project, a EU Research Infrastructure at the University
of Bergen, Norway, supported by the European Community
Access to Research Infrastructure Action of the Improving
Human Potential Program. The authors would like to express
their gratitude to Professor L. P. Csernai for hospitality at the
University of Bergen. Z.S. and P.S. would like to acknowledge the excellent working conditions at the CERN’s Theory
Division and SISSA’s Astrophysic Sector, respectively.
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064001-21
INSTITUTE OF PHYSICS PUBLISHING
Class. Quantum Grav. 22 (2005) 3623–3651
CLASSICAL AND QUANTUM GRAVITY
doi:10.1088/0264-9381/22/17/019
Relativistic thick discs in the Kerr–de Sitter
backgrounds
Petr Slaný and Zdeněk Stuchlı́k
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
E-mail: [email protected] and [email protected]
Received 13 May 2005, in final form 15 July 2005
Published 15 August 2005
Online at stacks.iop.org/CQG/22/3623
Abstract
Perfect fluid tori with a uniform distribution of the specific angular momentum,
(r, θ ) = const, orbiting the Kerr–de Sitter black holes or naked singularities
are studied. It is well known that the structure of equipotential surfaces of such
marginally stable tori reflects the basic properties of any tori with a general
distribution of the specific angular momentum. Closed equipotential surfaces
corresponding to stationary thick discs are allowed only in the spacetimes
admitting stable circular geodesics. The last closed surface crosses itself in
the cusp(s) enabling the outflow of matter from the torus due to the violation
of hydrostatic equilibrium. The inner cusp enables an accretion onto the
central object. The influence of the repulsive cosmological constant, > 0,
on the equipotential surfaces lies in the existence of the outer cusp (with a
stabilizing effect on the thick discs) and in the strong collimation of open
equipotential surfaces along the rotational axis. Both the effects take place
near a so-called static radius where the gravitational attraction is just balanced
by the cosmic repulsion. The outer cusp enables excretion, i.e., the outflow of
matter from the torus into the outer space. The plus-family discs (which are
always co-rotating in the black-hole backgrounds but can be counter-rotating,
even with negative energy of the fluid elements, in some naked-singularity
backgrounds) are thicker and more extended than the minus-family ones (which
are always counter-rotating in all backgrounds). For co-rotating discs in
the naked-singularity spacetimes, the potential well between the centre of
the disc and its edges at the cusps is usually much higher than in the black-hole
spacetimes. If the parameters of naked-singularity spacetimes are very close to
the parameters of extreme black-hole spacetimes, the family of possible disclike configurations includes members with two isolated discs where the inner
one is always a counter-rotating accretion disc.
PACS numbers: 04.70.Bw, 04.20.Dw, 95.30.Sf, 95.30.Lz, 98.62.Mw, 98.80.Es
0264-9381/05/173623+29$30.00 © 2005 IOP Publishing Ltd Printed in the UK
3623
3624
P Slaný and Z Stuchlı́k
1. Introduction
The presence of a relic repulsive cosmological constant ∼ 10−56 cm−2 , or, equivalently,
a relic vacuum energy, indicated by a wide range of independent cosmological tests, see,
e.g., [1–3], leads to very important consequences for the properties of both the thin and thick
accretion discs around black holes. It was shown that thin discs orbiting the Schwarzschild–
de Sitter black holes have, besides the standard inner edge at the inner marginally stable
circular orbit, an outer edge at the outer marginally stable circular orbit located slightly under
the static radius [4]. Similarly, thick discs have, besides the standard inner cusp enabling
the accretion inflow of matter from the disc onto the black hole, an outer cusp located near
the static radius and enabling the outflow of matter from the system of the black hole and its
accretion disc into the outer space [5].
Rezzolla et al [6] suggested that the repulsive vacuum energy can have a stabilizing
effect on dynamics of thick discs. Using the high-resolution shock-capturing technique,
see [7, 8], they considered the influence of the outflow of matter through the outer cusp on
the development of the so-called runaway instability of thick discs. The runaway instability,
see [9], compare with [10, 11], appears when the mass increase of the black hole induced by
the accretion process results in such modifications of the equipotential surfaces that cause the
inner cusp to move deeper inside the disc more rapidly than the inner edge of the disc. Then
additional matter of the disc will be captured by the hole in an accelerated manner and the
matter of the disc is swallowed by the black hole in an exponentially fast process. Rezzolla
et al [6] performed time-dependent axisymmetric hydrodynamical simulations of adiabatic
perfect fluid equilibrium configurations in a sequence of Schwarzschild–de Sitter black holes
with increasing masses due to accretion. The calculation shows that for a wide range of initial
conditions the mass outflow through the outer cusp has a strong stabilizing effect leading to
an efficient suppression of the runaway instability.
Studies of rotating, Kerr black-hole backgrounds are crucial to understand astrophysically
more realistic situations. Of course, it is very important to study the combined effect of a
repulsive vacuum energy and rotation of black holes on the properties of accretion discs. We
perform this in the framework of Kerr–de Sitter spacetimes which were first analysed by Carter
or Demianski [12, 13]. Derivation of the timelike geodesic equations in Kerr–(anti)-de Sitter
spacetimes, following Carter’s approach of integration of the Hamilton–Jacobi equation by
a separation of variables, and their exact solutions in the case of spherical polar orbits and
equatorial circular orbits, can be found in the work of Kraniotis [14]. The equatorial circular
geodesics of the Kerr–de Sitter spacetimes and their relevance for the geometrically thin
accretion discs were discussed in detail by Stuchlı́k and Slaný [15]. As in the Schwarzschild–
de Sitter spacetimes, the outer marginally stable orbit always exists in those Kerr–de Sitter
spacetimes admitting any stable circular orbit. Since the existence of naked singularities is not
excluded at the present state of knowledge [16], despite the cosmic censorship hypothesis [17],
it is useful to consider both the Kerr–de Sitter black-hole and naked-singularity spacetimes
and focus attention on those effects that could evidently distinguish a hypothetical Kerr–
de Sitter naked singularity from black holes. Because the Kerr–de Sitter spacetimes are
asymptotically de Sitter, not flat, the notion of co-rotating or counter-rotating motion cannot
be related to the observers at infinity but only to the locally non-rotating observers/frames
(LNRF). As in the Kerr spacetimes, the circular geodesics of the Kerr–de Sitter spacetimes
can be separated into two families, see [15]. The minus-family orbits are all counter-rotating,
while the plus-family orbits are usually co-rotating relative to the LNRF. However, the plusfamily orbits become counter-rotating in the vicinity of the static radius in all Kerr–de Sitter
spacetimes (these orbits are unstable), and also near the ring singularity in Kerr–de Sitter
Relativistic thick discs in the Kerr–de Sitter backgrounds
3625
naked-singularity spacetimes with the rotational parameter low enough (these orbits can be
stable including the inner marginally stable orbit). In such spacetimes, the efficiency of the
conversion of rest energy into heat energy in the geometrically thin plus-family accretion
discs, given by the difference of energies of a particle at the outer and the inner marginally
stable orbits (η = Ems(o) − Ems(i) ), can reach extremely high values exceeding the efficiency
of the annihilation process. It should be noted, however, that in all Kerr–de Sitter spacetimes
containing stable circular orbits, the accretion efficiency η is smaller in comparison with the
one for pure Kerr case (y = 0). Moreover, it was shown that transformation of a Kerr–de
Sitter naked singularity into an extreme black hole, caused by the accretion process, leads to
an abrupt instability of the innermost parts of the plus-family accretion discs that can have
strong observational consequences [15].
Here, we shall study the structure of the equipotential surfaces in barotropic perfect
fluid tori with a uniform distribution of the specific angular momentum, (r, θ ) = const,
rotating in the Kerr–de Sitter black-hole and naked-singularity backgrounds. In section 2,
the general theory of equilibrium configurations of a barotropic perfect fluid orbiting in a
stationary and axisymmetric background is outlined. In section 3, Kerr–de Sitter spacetimes
admitting stable equatorial circular orbits of test particles and the properties of such orbits
are briefly discussed. The structure of equipotential surfaces determining the equilibrium
configurations of a barotropic fluid is studied in detail for both the black-hole and nakedsingularity backgrounds in section 4. In section 5, the properties of the tori around black holes
and naked singularities are summarized and compared, and dimensions of the configurations
in astrophysical units are presented for the current, observationally established value of the
relic cosmological constant and typical masses of central black holes, covering the range from
stellar ones up to supermassive ones, existing in galactic nuclei.
2. Equilibrium configurations of barotropic perfect fluid
The analytic theory of equilibrium configurations of rotating perfect fluid bodies was developed
by Boyer [18] and then studied by many authors. The main result of the theory, known as
‘Boyer’s condition’, states that the boundary of any stationary, barotropic, perfect fluid body
has to be an equipotential surface. In this section, we briefly discuss its application to the
relativistic test perfect fluid orbiting in a stationary and axisymmetric way in a stationary,
axisymmetric background. Such a problem has already been studied by Abramowicz and
co-workers [19, 20] in the case of Schwarzschild and Kerr black holes, and by Stuchlı́k et al
[5] in the case of Schwarzschild–de Sitter black holes.
In the standard Boyer–Lindquist coordinates the spacetime is described by the line element
ds 2 = gtt dt 2 + 2gtϕ dt dϕ + gϕϕ dϕ 2 + grr dr 2 + gθθ dθ 2
(1)
satisfying the properties of stationarity and axial symmetry, i.e., ∂t gµν = ∂ϕ gµν = 0. Further,
we shall consider test perfect fluid moving in the azimuthal direction only and forming
the toroidal configurations (discs). Its 4-velocity vector field U µ has only two non-zero
components U t , U ϕ which can be the functions of coordinates r, θ , and the stress–energy
tensor field has the well-known form
T µν = ( + p)U µ U ν + pg µν
(2)
where and p are the total energy density and the pressure measured in the frame comoving
with the element of the fluid. The angular velocity and the specific angular momentum of the
rotating fluid are defined in terms of the 4-velocity field as
Uϕ
Uϕ
=− .
(3)
= t,
U
Ut
3626
P Slaný and Z Stuchlı́k
These definitions lead to the relation between and in the form
=−
gtt + gtϕ
.
gtϕ + gϕϕ
(4)
The equation of motion of the fluid, i.e., the relativistic Euler equation, obtained by the
projection of the conservation law ∇µ T µν = 0 onto the hypersurface orthogonal to the
4-velocity U µ , has the axially symmetric form
∂i ∂i p
= −∂i (ln Ut ) +
,
+p
1 − (5)
where i = r, θ and
(Ut )2 =
2
− gtt gϕϕ
gtϕ
gtt 2 + 2gtϕ + gϕϕ
.
(6)
For a barotropic fluid, i.e., for a body with an equation of state p = p(), the surfaces
of constant pressure are given, in accordance with Boyer’s approach, by the equipotential
surfaces of the potential W (r, θ ) defined by the relations [20]
p
dp
d
= ln(Ut )in − ln(Ut ) +
≡ Win − W,
(7)
0 +p
in 1 − where the subscript ‘in’ refers to the inner edge of the disc. The explicit form of the potential,
W = W (r, θ ), is given by equation (7), if one specifies the metric tensor of the background
and the ‘rotational law’, i.e., the function = (l). The simplest but also astrophysically
very important is the case of a uniform distribution of the specific angular momentum
(r, θ ) = const
(8)
through the disc. It has been known for a long time that the tori with (r, θ ) = const are
marginally stable [21] and capable of producing maximal luminosity at all [22]. Moreover,
topological properties of the equipotential surfaces seem to be rather independent of the
distribution of the specific angular momentum (r, θ ), see [5, 19, 20, 22–24]. In this special
case, the potential is given by the simple formula
1/2
2
− gtt gϕϕ
gtϕ
(9)
W (r, θ ) = ln
gtt 2 + 2gtϕ + gϕϕ
and is fully determined by the geometry of the background. Note that the points where ∂i W = 0
correspond to free-particle (geodesic) motion due to the vanishing of the pressure-gradient
forces there.
3. Kerr–de Sitter spacetimes admitting stable circular orbits
Stationary toroidal configurations corresponding to thick discs can exist only in the spacetimes
allowing the motion along stable circular geodetical orbits. The reason for such a claim lies in
the fact that at the centre of any perfect fluid torus the pressure gradient vanishes and matter
must follow a stable geodesic there. An analysis of equatorial circular geodesics in Kerr–de
Sitter spacetimes has been done in our recent work [15] where their relevance for the thin
(Keplerian) discs was also discussed. In this section, we describe those characteristics of the
circular geodesics which are useful for further discussion on thick discs.
Relativistic thick discs in the Kerr–de Sitter backgrounds
3627
The geometry of Kerr–de Sitter spacetimes is given by the line element
ds 2 = −
r
θ sin2 θ
ρ2 2 ρ2 2
2
2
2
2
2
(dt
−
a
sin
θ
dϕ)
+
[a
dt
−
(r
+
a
)
dϕ]
+
dr +
dθ
I 2ρ2
I 2ρ2
r
θ
(10)
where
r = r 2 − 2Mr + a 2 − 13 r 2 (r 2 + a 2 ),
(11)
θ = 1 + 13 a cos θ,
(12)
I =1+
(13)
2
2
1
a 2 ,
3
ρ 2 = r 2 + a 2 cos2 θ
(14)
and geometric units (c = G = 1) are used. The spacetime is specified by three parameters:
central mass (M), rotational parameter (a) corresponding to the specific angular momentum
of the central object, and positive cosmological constant (). It is convenient to introduce the
dimensionless ‘cosmological parameter’
y = 13 M 2
(15)
and reformulate relations (10)–(14) into the completely dimensionless form by putting M = 1
hereafter. The spacetime is stationary, axially symmetric and asymptotically de Sitter.
The spacetime horizons are determined by the condition r = 0 giving the relation
r 2 − 2r + a 2
(16)
r 2 (r 2 + a 2 )
determining implicitly the radii of the horizons. Local extrema of the function yh (r; a) are
given by the relation
r
2
(r) ≡ [1 − 2r + (8r + 1)1/2 ].
(17)
a 2 = aex(h)
2
2
Function aex(h)
(r) has one extreme (maximum)
√ .
2
3
= 16
(3 + 2 3) = 1.212 02
(18)
a 2 = acrit
√
2
the function yh (r; a)
at r = rcrit = (3 + 2 3)/4. We can conclude that for 0 < a 2 < acrit
2
these extrema coincide. Blackhas two local extrema, ymin (a) and ymax (a); for a 2 = acrit
hole spacetimes exist for ymin (a) y ymax (a). In general, three horizons (the inner
and the outer of the black hole, rh− and rh+ , and the cosmological one, rc ) exist. If y =
ymin (a), rh− = rh+ < rc , which corresponds to the extreme black hole. If y = ymax (a),
rh− < rh+ = rc , which corresponds to the marginal naked singularity, as two dynamical
regions are joined together. If y < ymin (a) or y > ymax (a), naked-singularity spacetimes
2
, the ‘ultra-extreme’ case occurs
occur. Note that for 0 < a < 1, ymin (a) < 0. If a 2 = acrit
(rh− = rh+ = rc ), corresponding to a naked-singularity case, and we obtain the maximal value
of the cosmological parameter enabling the existence of black holes to be
y = yh (r; a) ≡
y = ycrit =
16
.
= 0.059 24.
√
3
(3 + 2 3))
(19)
2
For a 2 > acrit
, only the cosmological horizon exists, and the Kerr–de Sitter geometry describes
a naked-singularity spacetime. The behaviour of the function yh (r; a) for some values of the
rotational parameter a is presented in figure 1.
3628
P Slaný and Z Stuchlı́k
0.1
yh (r;a)
0.08
1.3
0.06
1.1
0.04
1
0.02
0
0.5
0
-0.5
0.5
0
0.5
log r
1
1.5
Figure 1. Behaviour of the function yh (r; a) for some values of the rotational parameter squared,
a 2 . The grey curve corresponds to the Schwarzschild–de Sitter case with the maximal value of
.
the cosmological parameter enabling the existence of black holes being y = ycrit(SdS) = 1/27 =
0.037 04. Horizons of the Kerr–de Sitter spacetime are determined by the relation y = yh (r; a).
.
2
If a 2 < acrit
= 1.212 02, the black-hole spacetimes can exist for appropriate values of the
cosmological parameter y containing up to three horizons—two black-hole horizons and one
cosmological horizon—and also naked-singularity spacetimes exist containing the cosmological
2 , only the naked-singularity spacetimes exist.
horizon. If a 2 > acrit
As in the Kerr spacetimes, in the Kerr–de Sitter spacetimes also we can distinguish two
families of equatorial circular geodesics denoted by + (plus) or − (minus), if the spacetime
admits circular geodesics1 . The angular velocity on such (Keplerian) orbits is given by the
simple formula
K± =
1
a±
r 3/2 /(1
− yr 3 )1/2
(20)
revealing that no circular orbits can exist for r > rs , where rs is the ‘static radius’ defined by
rs ≡ y −1/3 .
(21)
Note that at r = rs , K = 0. Relation (21) for the static radius is independent of the rotation
parameter a and coincides with the formula for the static radius at the Schwarzschild–de Sitter
spacetimes. In both backgrounds, this is the only radius where the static geodesic observer
exists. The direction of the circular geodesics can be determined from the point of view of
LNRF moving in the equatorial plane with the angular velocity
LNRF =
a(r 2 + a 2 − r )
.
(r 2 + a 2 )2 − a 2 r
(22)
Orbits with > LNRF or < LNRF are called co-rotating or counter-rotating, respectively.
In the black-hole spacetimes, all stable plus/minus-family orbits are co-rotating/
counter-rotating relative to the LNRF, whereas in the naked-singularity spacetimes with a
rotational parameter low enough, in accordance with the Kerr case [25], stable plus-family
orbits lying near the ring singularity become counter-rotating relative to the LNRF. More
precisely, such orbits can exist in specifically chosen naked-singularity spacetimes with
.
.
2
= 2.4406 and 0 < y < yc(ms+) = 0.068 86 (the shaded region
parameters 1 < a 2 < acL
in figure 2). Moreover, a constant of motion E+ , connected with the stationarity of the
geometry and possessing the physical meaning of the specific energy of a particle on such
an orbit, can be negative for the counter-rotating plus-family orbits in the naked-singularity
2 .
= 1.47 (the subregion
spacetimes with specifically chosen rotational parameter 1 < a 2 < acE
1 For a more detailed view of the problem of existence and properties of the equatorial circular geodesics in the
Kerr–de Sitter spacetimes, see [15].
Relativistic thick discs in the Kerr–de Sitter backgrounds
NS(0)
-1
-2
log y
3629
BH(0)
E+
-3
BH(+)
NS(+)
<
0
-4
BH(±)
-5
0.5
NS(±)
1
1.5
a2
2
2.5
Figure 2. Classification of the Kerr–de Sitter spacetimes according to the existence of stable
circular orbits of test particles in the equatorial plane. The dashed curve separates regions of black
holes (BH) and naked singularities (NS), full curves separate spacetimes admitting either both
families (±) of stable circular orbits or the plus-family (+) only or even no (0) stable circular
orbits. For large values of a 2 both the full lines tend to the a 2 -axis. The shaded region corresponds
to the naked-singularity spacetimes admitting counter-rotating stable plus-family orbits. The
dashed-dotted curve forms the boundary of the subregion where these counter-rotating stable plusfamily orbits possess negative total energy (the constant of motion connected with the existence of
a timelike Killing vector field in the Kerr–de Sitter spacetime).
with the dashed-dotted boundary in figure 2), see [15]. All stable minus-family circular orbits
are counter-rotating from the point of view of LNRF in all naked-singularity spacetimes.
Discussion of the stability of circular orbits against radial perturbations enables us to
divide the parametric Kerr–de Sitter space (a 2 , y) into six regions according to the existence
of stable circular orbits of both families [15]. The result is presented in figure 2 where the
abbreviations BH and NS denote the black-hole and naked-singularity regions, respectively,
the signs (+) and (−) correspond to the family of circular orbits which can be stable in a given
region, and (0) states that no stable circular orbits are possible in the region.
4. Equipotential surfaces
Relevant metric coefficients of Kerr–de Sitter geometry occurring in the expression (9) for the
potential W (r, θ ) have the form
gtt = −
1
(
r − θ a 2 sin2 θ ),
I 2ρ2
gtϕ = −
gϕϕ =
1
(23)
[
θ (r 2 + a 2 ) − r ]a sin2 θ,
(24)
1
[
θ (r 2 + a 2 )2 − r a 2 sin2 θ ] sin2 θ
I 2ρ2
(25)
I 2ρ2
and the equipotential surfaces are given by the formulae
1/2
2
r θ sin2 θ
ρ
.
(26)
W (r, θ ) = ln 2
I θ (r 2 + a 2 − a)2 sin2 θ − r ( − a sin2 θ )2
As shown by Kozłowski et al [19], all the relevant properties of the equipotential surfaces
are described by the behaviour of the potential in the equatorial plane, i.e., by the formula
1/2
2
r
r
.
(27)
W (r, θ = π/2) = ln 2 2
I (r + a 2 − a)2 − r ( − a)2
3630
P Slaný and Z Stuchlı́k
Therefore, we have to discuss its properties.
function (27), namely
There are two reality conditions of the
r 0,
(28)
(r + a − a) − r ( − a) > 0.
2
2
2
2
(29)
Condition (28) can be rewritten into the relation y yh (r; a), corresponding to the stationary
parts of the backgrounds, and compared with figure 1. For further discussion, we do not
consider the stationary regions located under the inner black-hole horizons. The second
reality condition (29) is connected with a limit given by the photon motion, since it implies
the inequality ph− < < ph+ where
ph± (r; a, y) = a +
r2
√
a ± r
(30)
correspond to the effective potentials of the photon geodesic motion; see [26] for an alternative
definition.
The local extrema of the function W (r, θ = π/2) lie at those radii where the specific
angular momentum coincides with the specific angular momentum of test particles moving in
the geodetical (Keplerian) circular orbits, i.e., where
= K± (r; a, y) ≡ ±
(r 2 + a 2 )(1 − yr 3 )1/2 ∓ ar 1/2 [2 + r(r 2 + a 2 )y]
.
r 3/2 [1 − (r 2 + a 2 )y] − 2r 1/2 ± a(1 − yr 3 )1/2
(31)
Those extrema are the only local extrema of the function W (r, θ ).
Further, we will discuss black-hole and naked-singularity spacetimes separately.
4.1. Black-hole backgrounds
Typical behaviour of the Keplerian specific angular momentum K± (r; a, y) together with
the photon potential ph± (r; a, y) is presented in figure 3, where we chose spacetimes of the
BH (±) class, i.e., the black-hole spacetimes admitting both the co-rotating and the counterrotating stable circular orbits. If the spacetime contains stable circular orbits of any family,
the appropriate function K (r; a, y) has two local extrema, ms(i) and ms(o) , corresponding to
the inner and the outer marginally stable orbits, respectively. The local extreme of the photon
potential corresponds to the impact parameter of the circular null geodesic, ph(c) . If the specific
angular-momentum distribution (r, θ ) through the disc is constant and ∈ (ms(i) , ms(o) ) for
the given family of orbits, the stationary disc-like configurations exist. More precisely, if
ms(i)+ < < ph(c)+ < ms(o)+
or
ms(i)+ < < ms(o)+ < ph(c)+
(32)
for the plus-family (co-rotating) discs, and
ms(i)− > > ph(c)− > ms(o)−
or
ms(i)− > > ms(o)− > ph(c)−
(33)
for the minus-family (counter-rotating) ones, then = const intersects K (r; a, y) in three
points corresponding to the three free-particle orbits in the marginally stable configurations
of the perfect fluid. The innermost and the outermost ones are unstable and determine
the inner and the outer local maxima—cusps—of the potential, respectively; the middle
one is stable determining the local minimum of the potential, i.e., the ‘centre’ of the
equilibrium configuration. In general, any stable angular-momentum distribution through
the disc intersects the curve of Keplerian angular momentum in three points leading to the
existence of two cusps and one centre in the structure of equipotential surfaces.
Relativistic thick discs in the Kerr–de Sitter backgrounds
3631
8
-2
lK- , lph-
0
lK+ , lph+
10
6
4
-4
-6
-8
2
0
0.5
1
log r
1.5
2
0
0.5
1.5
2
(b)
7
0
6
-2
5
lK- , lph-
lK+ , lph+
(a)
1
log r
4
3
-4
-6
-8
2
0.4
0.6
0.8
log r
1
1.2
0
0.25 0.5 0.75 1
log r
(c)
1.25 1.5
(d)
Figure 3. Keplerian specific angular momentum (the solid curve) and the effective potential of
the photon geodesic motion (the dashed curve) in some Kerr–de Sitter black-hole spacetimes. The
grey line determines the boundary of a black hole. The rising (descending) part(s) of K+ and the
descending (rising) part(s) of K− correspond to the stable (unstable) orbits. The local extrema of
the functions K± determine the specific angular momentum of marginally stable orbits—the inner
and the outer, ms(i) and ms(o) , respectively. The local extreme of the photon potential determines
the impact parameter of the photon circular geodesic, ph(c) . (a) Plus-family (co-rotating)
orbits in the Kerr–de Sitter spacetime (y = 10−6 , a 2 = 0.6) where ms(i) < ph(c) < ms(o) .
(b) Minus-family (counter-rotating) orbits in the Kerr–de Sitter spacetime (y = 10−6 , a 2 = 0.6)
where ms(i) > ph(c) > ms(o) . (c) Plus-family (co-rotating) orbits in the Kerr–de Sitter spacetime
(y = 10−4 , a 2 = 0.3) where ms(i) < ms(o) < ph(c) . (d ) Minus-family (counter-rotating) orbits
in the Kerr–de Sitter spacetime (y = 10−5 , a 2 = 0.4) where ms(i) > ms(o) > ph(c) .
In some cases, a constant specific angular momentum through the disc can satisfy the
relations
ph(c)+ < < ms(o)+
or
ph(c)− > > ms(o)− .
(34)
As a consequence, only two free-particle orbits exist in the marginally stable configurations
of the perfect fluid corresponding to the centre and the outer cusp of the configuration, while
the existence of the inner cusp is not allowed and the inflow of matter onto the black hole is
forbidden.
Behaviour of the function (27) for the parameter between ms(i) and ms(o) (figure 4)
reveals the existence of two kinds of stationary toroidal configurations, namely the one
containing the inner cusp and corresponding to the accretion discs, and the other with the
outer cusp where the accretion onto the central black hole is impossible but the outflows from
the torus through the outer cusp are still possible. We call them ‘excretion discs’. The value of
separating these two kinds of discs is given by the specific angular momentum of a particle
on the marginally bound circular orbit, mb . For
ms(i)+ < < mb+
or
ms(i)− > > mb− ,
(35)
P Slaný and Z Stuchlı́k
0
0
0.02
0.02
W(r, /2)
W(r, /2)
3632
0.04
0.06
0.04
0.06
0.08
0.08
0.1
0.1
0
0.5
1
1.5
log r
2
2.5
0
0.5
1
(a)
W(r, /2)
W(r, /2)
2.5
2
2.5
2
2.5
2
2.5
0.05
0.025
0.05
0
0.05
0
0.025
0.05
0.075
0.1
0.1
0
0.5
1
1.5
log r
2
2.5
0
0.5
1
W(r, /2)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0
0.5
1
1.5
log r
(d)
(c)
W(r, /2)
2
(b)
0.1
1.5
log r
2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
2.5
0
0.5
1
(e)
1.5
log r
(f )
0.01
0.01
0.011
0
W(r, /2)
W(r, /2)
1.5
log r
0.01
0.02
0.012
0.013
0.014
0.03
0
0.5
1
1.5
log r
2
2.5
0
0.5
1
1.5
log r
(h)
(g)
Figure 4. Behaviour of the potential W (r, θ = π/2) in the Kerr–de Sitter black-hole background
(y = 10−6 , a 2 = 0.6) for the specific angular momentum between the values of Keplerian
specific angular momentum on the inner (ms(i) ) and the outer (ms(o) ) marginally stable orbits:
.
(a) ms(i)+ < < mb+ , = 2.9, (b) = mb+ = 2.933 78, (c) mb+ < < ph(c)+ < ms(o)+ , =
.
3.03, (d) ph(c)+ < < ms(o)+ , = 3.5, (e) ms(i)− > > mb− , = −4.47, (f ) = mb− =
−4.554 51, (g) mb− > > ph(c)− > ms(o)− , = −4.7, (h) ph(c)− > > ms(o)− , = −6.8.
The dashed lines give asymptotes of W (r, θ = π/2) and no equipotential surface exists in between
them.
the configurations correspond to the accretion discs, for
mb+ < < ms(o)+
or
mb− > > ms(o)− ,
(36)
3633
2
2
1
1
(log r) cos
(log r) cos
Relativistic thick discs in the Kerr–de Sitter backgrounds
0
1
0
1
2
2
0
0.5
1
1.5
(log r) sin
2
0
0.5
(a)
1
1.5
(log r) sin
2
(b)
2
2
1
1
(log r) cos
(log r) cos
3
0
1
0
1
2
2
0
0.5
1
1.5
(log r) sin
2
3
0
(c)
0.5
1
1.5
(log r) sin
2
(d)
Figure 5. Equipotential surfaces (meridional sections) for the cases described in figure 4. Shaded
regions containing closed equipotential surfaces correspond to the possible disc-like configurations;
the last closed surface (the critical surface) is self-crossing in the cusp(s). (a)–(d ) Plus-family
(co-rotating) fluid: (a) accretion disc, (b) marginally bound accretion disc, (c) excretion disc,
(d ) excretion disc; this case differs from the previous one in the behaviour of the open equipotential
surfaces since the one with the inner cusp does not exist and the remainder do not form an open throat
connected with the black hole. (e)–(h) The same situation as in the (a)–(d ) cases, respectively,
now for the minus-family (counter-rotating) fluid.
the configurations correspond to the excretion discs. The case
= mb+
or
= mb−
(37)
corresponds to the marginally bound accretion disc of a given family.
Figure 4 implies a one-dimensional (1D) insight into the structure of equipotential surfaces
of a perfect fluid orbiting the Kerr–de Sitter black holes. A two-dimensional (2D) insight is
given by the meridional sections through the equipotential surfaces which are presented in
figure 5. The shaded regions correspond to the possible disc-like configurations with a
boundary given by any closed equipotential surface, as the equilibrium tori can exist inside the
closed equipotential surfaces only. Accretion (excretion) is possible through the inner (outer)
cusp due to the violation of mechanical equilibrium when matter overfills the closed critical
surface with the cusp (or cusps in the special case of = mb ). Thus, the inner (outer) cusp
naturally defines the inner (outer) edge of the accretion (excretion) disc.
P Slaný and Z Stuchlı́k
2
2
1
1
(log r) cos
(log r) cos
3634
0
1
0
1
2
2
0
0.5
1
1.5
(log r) sin
2
0
0.5
3
3
2
2
1
1
0
1
0
1
2
2
3
2
(f )
(log r) cos
(log r) cos
(e)
1
1.5
(log r) sin
3
0
0.5
1
1.5
(log r) sin
(g)
2
0
0.5
1
1.5
(log r) sin
2
(h)
Figure 5. (Continued.)
4.2. Naked-singularity backgrounds
We start a discussion on the equilibrium configurations of a perfect fluid in the nakedsingularity backgrounds with spacetimes of class NS (±) containing stable circular orbits
of both families. As in the black-hole backgrounds, the curves of Keplerian angular
momentum K± (r; a, y) possess two local extrema corresponding to the inner and the outer
marginally stable orbits ms(i) and ms(o) , respectively, and the rising parts of K+ (r; a, y)
and the descending part of K− (r; a, y) determine stable circular orbits. In naked-singularity
spacetimes with the rotational parameter low enough to admit stable negative-energy orbits,
the curve of K+ (r; a, y) contains two points of discontinuity corresponding to the zero-energy
orbits (figures 6(a) and (b)). In naked-singularity spacetimes where the innermost stable plusfamily orbits are still counter-rotating but correspond to the states with E > 0, the local
minimum of the function K+ (r; a, y) lies in negative values of (figures 6(c) and (d )). In
the remaining naked-singularity spacetimes admitting stable plus-family orbits, the behaviour
of the function K+ (r; a, y) is similar to its behaviour above the outer black-hole horizon
in the black-hole spacetimes (figures 6(e) and ( f )). Keplerian angular momentum for the
minus-family orbits K− (r; a, y) behaves in a similar manner as in the black-hole spacetimes.
3635
15
10
7.5
5
2.5
0
-2.5
-5
-7.5
0.1
10
lK , lph
lK , lph
Relativistic thick discs in the Kerr–de Sitter backgrounds
5
0
-5
-10
0.5 1
5 10
50 100
0.1
0.5 1
5 10
r
(b)
10
10
5
5
lK , lph
lK , lph
(a)
0
-5
0
-5
-10
0.1
50 100
r
-10
0.5 1
5 10
r
50 100
0.1
0.5 1
(c)
5 10
r
50 100
(d)
20
5
lK , lph
lK , lph
10
0
-5
-10
0.1
0
-10
-20
0.5 1
5 10
50 100
0.1
0.5 1
5 10
r
(e)
50 100
r
(f)
Figure 6. Keplerian specific angular momentum and the effective potential of the photon
geodesic motion in some appropriately chosen Kerr–de Sitter naked-singularity spacetimes
admitting stable circular geodesics of both families.
The behaviour of the functions
K+ (r; a, y), K− (r; a, y), ph+ (r; a, y) and ph− (r; a, y) is described by the solid, dashed, dasheddotted and dotted curves, respectively. The vertical solid and dotted straight lines correspond to
the asymptotes of K+ (r; a, y) and ph− (r; a, y), respectively. The rising (descending) part(s) of
K+ and the descending (rising) part(s) of K− correspond to the stable (unstable) orbits. The
local extrema of the functions K± determine the specific angular momentum of marginally stable
orbits—the inner and the outer, ms(i) and ms(o) , respectively. The local maximum of ph−
determines the impact parameter of the photon circular geodesic, ph(c) . (a) y = 10−6 , a 2 =
1.05, (b) y = 10−6 , a 2 = 1.15, (c) y = 10−7 , a 2 = 1.22, (d) y = 10−7 , a 2 = 1.3, (e) y =
10−6 , a 2 = 5, (f ) y = 10−6 , a 2 = 64.
In the naked-singularity spacetimes of class NS (+) containing stable circular orbits of the
plus-family only, a special subset of spacetimes with sufficiently small values of the rotational
parameter a and sufficiently large values of the cosmological parameter y exists, in which
all stable orbits in the equatorial plane are counter-rotating with the negative specific energy
being located between the ring singularity and the photon orbit. The corresponding behaviour
of the functions K± (r; a, y) and ph± (r; a, y) is presented in figure 7.
3636
P Slaný and Z Stuchlı́k
6
lK , lph
5
4
3
2
0.6
1
r
1.5
Figure 7. Keplerian specific angular momentum and the effective potential of the photon geodesic
motion in the Kerr–de Sitter naked-singularity spacetime (y = 0.061, a 2 = 1.24) admitting stable
circular geodesics of the plus-family with negative specific energy (E+ < 0) only. The behaviour
of the functions K+ (r; a, y), K− (r; a, y), ph+ (r; a, y) and ph− (r; a, y) is described by the solid,
dashed, dashed-dotted and dotted curves, respectively. The rising part of K+ corresponds to the
only stable circular orbits. The local extrema of the function K+ determine the specific angular
momentum of the inner and the outer marginally stable orbits, ms(i)+ and ms(o)+ , respectively. The
local maximum of ph− determines the impact parameter of the photon circular geodesic, ph(c) .
In all of the naked-singularity spacetimes, only minus-family photon circular orbits
exist2 , corresponding to the local maximum ph(c) of the function ph− (r; a, y); the function
ph+ (r; a, y) has no local extrema.
In most of the naked-singularity spacetimes, the necessary condition for the existence of
stationary tori is the same as in the black-hole spacetimes, i.e., the specific angular momentum
has to be chosen between the values of Keplerian angular momentum on the inner and the
outer marginally stable orbits, ∈ (ms(i) , ms(o) ), of a given family, however, the exceptions
exist concerning the plus-family discs in naked-singularity backgrounds with the rotational
parameter low enough to admit counterrotating stable plus-family circular geodesics.
The interplay between the functions K± (r; a, y) and ph± (r; a, y) reveals a varied set
of possible stationary disc-like configurations. First, we shall consider naked-singularity
spacetimes of class NS(±); this enables us to cover almost all possible toroidal equilibrium
configurations of a perfect fluid. Next we shall consider the spacetime of class NS(+), in
which all stable orbits are negative-energy counterrotating ones:
(i) Naked-singularity spacetimes with a rotational parameter low enough to admit plusfamily stable circular orbits with E < 0, where ms(i)+ < ms(o)+ (figure 6(a)):
(a) > ms(o)+ . Two free-particle circular orbits exist in a given disc; the inner
one is unstable and the outer one is stable corresponding to the inner cusp of the
equipotential surfaces and to the centre of the disc, respectively. Both orbits are
negative-energy counter-rotating ones.
(b) ms(i)+ < < ms(o)+ . Two pairs of circular geodesics exist, separated by a forbidden
region for the occurrence of particles and fluid; the boundary of the forbidden region
is given by the functions ph± (r; a, y). The inner pair is identical to that described
in case (i)a. The outer pair contains the inner-stable orbit and the outer-unstable
one corresponding to the centre of the second disc and to the outer cusp of the
equipotential surfaces, respectively. Both orbits are co-rotating.
2 A comprehensive analysis of the photon geodesic motion in a general Kerr–Newman–de Sitter background can be
found in [26].
Relativistic thick discs in the Kerr–de Sitter backgrounds
3637
(c) a < < ms(i)+ . Two co-rotating circular geodesics with properties identical to the
second pair of the previous case (i)b exist.
(d) ms(i)− < < a. In a given disc, three circular geodesics exist. The innermost and
the outermost ones are unstable corresponding to the inner and outer cusps of the
equipotential surfaces, respectively, and the middle one is stable determining the
centre of the disc. Depending on the sign of these orbits are co-rotating ( > 0)
or counter-rotating ( < 0).
(e) ms(o)− < ph(c) < < ms(i)− . Five circular geodesics exist for the given angularmomentum distribution. Counted in the direction from the singularity, the second
one and the fourth one are stable corresponding to two centres of the configuration,
the remaining ones are unstable corresponding to the cusps. All circular geodesics
counter-rotate the ring singularity. In fact, the configuration consists of two counterrotating stationary tori. If ph(c) < ms(o)− , such a configuration occurs for the whole
range ms(o)− < < ms(i)− .
(f) ms(o)− < < ph(c) . In such a configuration, two pairs of counter-rotating circular
geodesics exist separated by a forbidden region for the occurrence of particles and
fluid; now, the boundary of the forbidden region is given by the function ph− (r; a, y)
only. This situation is similar to case (i)b, however, the orbits with E > 0 constitute
the inner pair now. If ph(c) < ms(o)− , such a configuration does not exist.
(g) < ms(o)− . This situation is identical to case (i)a with one exception: the circular
geodesics correspond to states with E > 0.
(ii) Naked-singularity spacetimes with a rotational parameter low enough to admit plusfamily stable circular orbits with E < 0, where ms(i)+ > ms(o)+ (figure 6(b)). Except for
case (i)b, all the previously mentioned cases (i)a, (i)c–(i)g are possible.
(iii) Naked-singularity spacetimes admitting stable counter-rotating plus-family orbits with
E > 0, where ms(o)− < ms(i)+ < ms(i)− (figure 6(c)).
From the cases mentioned above, only (i)c–(i)f are possible here and, in addition, two
new ones arise:
(a) ms(o)− < < ms(i)+ < ph(c) . Two circular geodesics exist in a given disc; the
inner one is stable and the outer one is unstable corresponding to the centre of
the disc and to the outer cusp of the equipotential surfaces, respectively. Both the
orbits are counter-rotating and the disc is separated from the ring singularity by the
forbidden region for the occurrence of matter determined by the photon potential.
(b) ms(o)− < ph(c) < < ms(i)+ . In a given disc, three circular geodesics exist.
The innermost and the outermost ones are unstable corresponding to the inner and
the outer cusps of the equipotential surfaces, respectively, and the middle one is
stable determining the centre of the disc. All the orbits are counter-rotating. The
same situation occurs when ph(c) < ms(o)− < < ms(i)+ .
(iv) Naked-singularity spacetimes admitting stable counter-rotating plus-family orbits with
E > 0, where ms(i)− < ms(i)+ < 0 (figure 6(d )).
Cases (i)c, (i)d, (iii)a, (iii)b could occur in such a spacetime. Moreover, if ph(c) <
ms(o)− , case (iii)a is impossible.
(v) Naked-singularity spacetimes admitting co-rotating stable plus-family orbits3 where
ms(o)+ > a (figure 6(e)).
The situation is the same as in (iv) with one exception: case (i)d can be co-rotating only.
(vi) Naked-singularity spacetimes where ms(o)+ < a (figure 6( f )).
3
The counter-rotating ones belong to the minus-family only.
3638
P Slaný and Z Stuchlı́k
Just two possibilities, the co-rotating one (i)d and its counter-rotating analogy (iii)b,
could occur in such a spacetime.
(vii) Naked-singularity spacetimes of class NS(+), in which all stable orbits are counterrotating with E < 0 (figure 7):
(a) ms(i)+ < ph(c) < < ms(o)+ . The situation is similar to case (i)d but all three
geodetical orbits are counter-rotating only.
(b) ms(i)+ < < ph(c) . The situation is similar to case (i)a. If ph(c) < ms(i)+ , such a
configuration does not exist.
The presented analysis enables us to anticipate new features in the behaviour of the
potential W (r, θ ) in the naked-singularity backgrounds. In an effort to cover all the possible
toroidal configurations, we present in figure 8, where the Kerr–Schild coordinate x instead
of the Boyer–Lindquist coordinate r is used4 , the behaviour of the function W (r, θ = π/2)
in appropriately chosen spacetimes (each of them has already been discussed in terms of the
behaviour of the Keplerian angular momentum, see figures 6 and 7) and only for the values
of specific angular momentum enabling the existence of stationary discs. The local maxima
correspond to the cusps of the equipotential surfaces and the local minima correspond to the
centres of the discs. Again, we can see that both the accretion discs (figures 8(a), (d ), (e),
( j), (k), (m), ( p), (q) and (s)) and the excretion ones (figures 8(c), ( f ), (l), (o) and (r)) could
exist. Moreover, some naked-singularity backgrounds admit the configurations with two discs
(figures 8(b), (g)–(i), (n)). The inner disc is always a counter-rotating accretion disc but the
outer accretion or excretion disc is co-rotating in some cases, and counter-rotating in the other
cases. The region between the discs either contains the forbidden region for the occurrence of
the fluid with a given specific angular momentum, determined by the photon potential through
the relation = ph± (figures 8(b) and (n)), or can be filled by an accretion flow from the
outer accretion disc (figures 8(g) and (h)), or corresponds to a non-stationary outer part of
the inner accretion disc (figure 8(i)). In the cases where both the accretion and the excretion
discs are possible, a limiting value of the parameter separating the accretion discs from the
excretion ones corresponds to the specific angular momentum of a particle on the marginally
bound circular geodesic, mb .
Meridional sections through the equipotential surfaces of the equilibrium configurations
just mentioned are depicted in figure 9. As in the black-hole backgrounds, the boundary of
the shaded regions (corresponding to the stationary discs) is formed by the critical closed
equipotential surface self-crossing in the cusp(s). The presented configurations correspond to
and possess the subsequent properties5 :
(a) The counter-rotating accretion disc.
The equipotential surface with the outer cusp does not exist. The specific energy of the
fluid elements in the centre and on the inner edge (where the fluid follows the geodesic
motion) is negative and we can expect that every fluid element in the disc has energy
E < 0. Moreover, no open equipotential surface going out from the singularity is
connected with such a configuration.
(b) The counter-rotating negative-energy accretion disc (the inner one) and the co-rotating
excretion disc (the outer one).
The region between the discs is the ‘forbidden region’, hence no open equipotential
surfaces leave the inner disc.
4 The relation between the Boyer–Lindquist coordinates r, θ and the Kerr–Schild coordinates x, y, z is given through
the expressions x 2 + y 2 = (r 2 + a 2 ) sin2 θ, z = r cos θ indicating that in the equatorial plane (θ = π/2) and for y = 0
the Kerr–Schild coordinate x = ±(r 2 + a 2 )1/2 .
5 Only the properties which differ between naked singularities and black holes are mentioned.
Relativistic thick discs in the Kerr–de Sitter backgrounds
3639
0
0
W(r, /2)
W(r, /2)
0.1
−1
−2
0
0.5
1
1.5
2
2.5
0.05
−1
-0.05
0
y = 10−6
a2 = 1.05
= 3.5
-0.1
− 1.5
y = 10−6
a2 = 1.05
=8
−3
− 0.5
-0.15
−2
0.5
3
0
0.5
1
1.5
1
1.5
log (r2+a2)1/2
log (r +a )
2
2 1/2
(a)
2
2.5
2
2.5
3
(b)
0.2
y = 10−6
a2 = 1.05
= 2.5
0
W(r, /2)
W(r, /2)
0.1
0
− 0.1
− 0.2
− 0.5
−1
− 1.5
y = 10−6
a2 = 1.05
= 0.5
−2
0
0.5
1
1.5
2
2.5
0
3
0.5
log (r2+a2)1/2
1
0
0
− 0.5
− 0.5
−1
− 1.5
y = 10−6
a2 .= 1.05
= 0.84193
−2
0.5
1
1.5
2
2.5
− 1.5
−2
3
0
0.5
1
y = 10−6
a2 = 1.05
= −4.65
−2
-0.025
-0.03
−3
-0.035
− 3.5
0
0
0.5
0.5
1
1
1.5
1.5
2
log (r +a )
2
(g)
-0.01
−1
-0.02
− 2.5
2.5
3
y = 10−6
a2 .= 1.05
= −4.7194
− 0.5
W(r, /2)
W(r, /2)
-0.015
2
(f)
0
− 1.5
1.5
log (r2+a2)1/2
0
−1
3
y = 10−6
a2 = 1.05
= 0.9
(e)
-0.01
2.5
−1
log (r2+a2)1/2
− 0.5
2
(d )
W(r, /2)
W(r, /2)
(c)
0
1.5
log (r2+a2)1/2
2 1/2
2
-0.02
−2
-0.025
− 2.5
-0.03
−3
-0.035
− 3.5
2.5
2.5
-0.015
− 1.5
3
0
0
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
3
log (r +a )
2
2 1/2
(h)
Figure 8. Behaviour of the potential W (r, θ = π/2) in the appropriately chosen Kerr–de Sitter
naked-singularity backgrounds. The values of constant specific angular momentum were chosen
to cover all the possible disc-like configurations in naked-singularity backgrounds and are referred
to in the enumeration and discussion presented in the text. A region in between the non-solid vertical
lines, determined by the conditions = ph+ (r; a, y) (dashed-dotted lines) and = ph− (r; a, y)
(dotted lines), is the ‘forbidden region’. (a) Case (i)a. (b) Case (i)b. (c) Case (i)c. (d ) Case (i)d,
< mb+ . (e) Case (i)d, = mb+ . ( f ) Case (i)d, > mb+ . (g) Case (i)e, > mb− . (h) Case
(i)e, = mb− . (i) Case (i)e, < mb− . ( j) Case (iii)b, > mb− . (k) Case (iii)b, = mb− . (l)
Case (iii)b, < mb− . (m) Case (i)g. (n) Case (i)f. (o) Case (iii)a. ( p) Case (vii)a, < mb+ . (q)
Case (vii)a, = mb+ . (r) Case (vii)a, > mb+ . (s) Case (vii)b.
(c) The co-rotating excretion disc with the ‘forbidden region’ between the disc and the ring
singularity.
3640
P Slaný and Z Stuchlı́k
0
-0.01
y = 10−6
a2 = 1.05
= −4.85
-0.5
0.01
-1
-0.015
-1.5
-0.01
-2
-2.5
-0.02
-3
-0.03
y = 10−6
a2 = 5
= −5.3
-0.0175
W(r, /2)
W(r, /2)
0
-0.0125
-0.02
-0.0225
-0.025
-0.0275
-3.5
0
0
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
0.5
3
log (r2+a2)1/2
1
1.5
(i)
2.5
3
(j)
0
-0.01
-0.0125
-0.015
y = 10−6
a2 .= 5
= −5.3855
y = 10−6
a2 = 5
= −5.5
-0.005
-0.01
-0.0175
W(r, /2)
W(r, /2)
2
log (r2+a2)1/2
-0.02
-0.0225
-0.025
-0.015
-0.02
-0.025
-0.0275
0.5
1
1.5
2
2.5
3
0.5
log (r2+a2)1/2
1
1.5
2
2.5
3
log (r2+a2)1/2
(k)
(l)
1
1
0
0
y = 10−7
a2 = 1.22
= −7.5
W(r, /2)
W(r, /2)
-3.108
-1
-2
-3
−6
y = 10
a2 = 1.05
= −8
-4
0
0.5
1
1.5
2
2.5
-1
-3.11
0.02
-3.112
0.01
-2
-3.114
0
-0.01
-3.116
-0.02
-3
3
-3.118
-0.03
-0.04
0.08 0.11 0.14
1
0
0.5
1
log (r2+a2)1/2
2
2
2.5
2.5
3
3
log (r2+a2)1/2
(m)
(n)
-1.6
0.01
y = 10−7
a2 = 1.22
= −9
0.005
-1.7
y = 0.061
a2 = 1.24
= 4.1
-1.8
W(r, /2)
0
W(r, /2)
1.5
1.5
-0.005
-0.01
-1.9
-2
-2.1
-0.015
0
0.5
1
1.5
log (r2+a2)1/2
(o)
2
2.5
3
0.1
0.2
0.3
0.4
log (r2+a2)1/2
(p)
Figure 8. (Continued.)
(d)
(e)
(f)
(g)
The co-rotating accretion disc.
The co-rotating marginally bound accretion disc.
The co-rotating excretion disc.
Two counter-rotating accretion discs.
The matter from the outer disc as well as from the region between the critical surfaces
(belonging to the outer configuration) can flow through the inner cusp and a throat formed
by open equipotential surfaces onto the singularity. If some sufficiently strong dissipative
processes are present during such in-fall, the matter could fill the inner accretion disc.
Relativistic thick discs in the Kerr–de Sitter backgrounds
3641
-1.9
-1.95
y = 0.061
a2 = 1.24
= 4.35
-2.05
-2.1
-2.05
W(r, /2)
W(r, /2)
-2
-2
y = 0.061
a2 .= 1.24
=
4.2556
-2.1
-2.15
-2.2
-2.15
-2.2
-2.25
-2.25
0.1
0.2
0.3
0.4
log (r2+a2)1/2
0.1
0.2
0.3
0.4
log (r2+a2)1/2
(q)
(r)
- 1.6
W(r, /2)
- 1.65
y = 0.061
a2 = 1.24
= 3.7
- 1.7
- 1.75
- 1.8
0.1
0.2
0.3
0.4
log (r2+a2)1/2
(s)
Figure 8. (Continued.)
(h) The counter-rotating accretion disc (the inner one) and the counter-rotating marginally
bound accretion disc (the outer one).
Again, if the dissipative processes, present in the accretion flow from the outer disc,
efficiently drain the energy from the accreting matter, it could feed the inner accretion
disc instead of a direct in-fall through the throat onto the singularity.
(i) The counter-rotating accretion disc (the inner one) and the counter-rotating excretion disc
(the outer one).
The outflow from the outer disc is possible through the outer cusp of the last closed
(critical) surface only, since the critical equipotential surface with the inner cusp is open
and the region between the critical surfaces contains open (cylindrical) surfaces.
(j) The counter-rotating accretion disc.
(k) The counter-rotating marginally bound accretion disc.
(l) The counter-rotating excretion disc.
(m) The counter-rotating accretion disc.
The plus-energy analogy of case (a).
(n) The counter-rotating negative-energy accretion disc (the inner one) and the counterrotating excretion disc (the outer one).
The fully counter-rotating analogy of case (b).
(o) The counter-rotating excretion disc with the ‘forbidden region’ between the disc and the
ring singularity.
(p) The counter-rotating negative-energy accretion disc.
(q) The counter-rotating negative-energy marginally bound accretion disc.
(r) The counter-rotating negative-energy excretion disc.
(s) The counter-rotating negative-energy accretion disc.
Cases (p)–(s) have some common properties with case (a), especially that the matter
following geodetical motion at the centre and in the cusp(s) of the potential possesses
negative energy (and we can expect that every fluid element in the disc has energy E < 0)
3642
P Slaný and Z Stuchlı́k
0.75
0.25
0
sinh
1
r cos
0.5
0.25
0.5
0.75
0.6 0.7 0.8 0.9
sinh
1
r2
1
1.1 1.2
a2 sin
(a)
0.6
6
0.4
r cos
2
sinh
sinh
0.2
0
1
0
1
r cos
4
2
4
0.2
0.4
6
0.6
0.9 0.95
1
2
sinh
3
1
r2
4
a2
sin
5
sinh
1
1
r2
1.05 1.1 1.15
a2 sin
(b)
Figure 9. Equipotential surfaces (meridional sections) for the cases described in figure 8.
The shaded regions containing closed equipotential surfaces correspond to the possible toroidal
configurations; the last closed surface (the critical surface) is self-crossing in the cusp(s) which
naturally determines the edge(s) of a disc. The Kerr–Schild coordinates together with the scale
of axes were chosen to show clearly the whole range of a disc including the ring singularity (the
grey point on the left in the equatorial plane) as well as the region near the static radius. In cases
(b), (d ), (g)–(i) and (n), the inner region near the ring singularity is enlarged in the right figure. A
detailed discussion of the depicted stationary configurations is presented in the text.
and that the configuration is imprisoned by a photon shell around the singularity. However,
in contrast to (a), configurations (p)–(r) contain two critical equipotential surfaces.
5. Concluding remarks
The influence of a repulsive cosmological constant, or equivalently a vacuum/dark energy, on
barotropic perfect fluid tori orbiting Kerr–de Sitter black holes and naked singularities can be
summarized as follows6 :
.
(i) Black-hole backgrounds (0 < y < ycrit = 0.059 24).
(a) Stationary tori exist for the range of specific angular momentum between the values
corresponding to the specific angular momenta of the inner and the outer marginally
stable circular orbits, ∈ (ms(i) , ms(o) ), as such values of enable the existence
6
We are only interested in the Kerr–de Sitter backgrounds admitting, at least, stable circular orbits of the plus-family.
Relativistic thick discs in the Kerr–de Sitter backgrounds
3643
6
4
sinh
1
r cos
2
0
2
4
6
1
2
3
sinh
1
4
r2
5
6
a2 sin
1.5
4
1
2
0.5
r cos
6
2
sinh
sinh
0
1
0
1
r cos
(c)
0.5
1
4
1.5
6
0
1
2
sinh
1
3
4
r2
a2 sin
0
5
0.25 0.5 0.75
sinh
1
r2
1
1.25 1.5
a2 sin
(d)
Figure 9. (Continued.)
of closed equipotential surfaces. Moreover, the equipotential surface with the outer
cusp always exists.
(b) For ∈ (ms(i) , mb ), where mb denotes the specific angular momentum of a
marginally bound circular orbit, the last closed surface is self-crossing in the inner
cusp, enabling the outflow of matter from the disc into the black hole caused by
overfilling of this critical surface by matter which further cannot remain in hydrostatic
equilibrium. Such a configuration corresponds to the accretion disc. An equipotential
surface with the outer cusp is open and matter from the region between the
critical surfaces contributes to the accretion flow along the inner cusp. When
the critical surface with the outer cusp is overfilled, an outflow of matter through
the outer cusp begins to complement the accretion inflow, having the capability to
regulate the accretion.
(c) For ∈ (mb , ms(o) ), the last closed surface is self-crossing in the outer cusp enabling
the outflow of matter from the disc into the outer space by a violation of hydrostatic
equilibrium. Such a configuration is called an excretion disc. The equipotential
surface with the inner cusp, if such a surface exists, is open (cylindrical) and separated
from the critical surface with the outer cusp by additional cylindrical surfaces which,
in fact, disable accretion onto the black hole.
P Slaný and Z Stuchlı́k
6
4
4
2
2
r cos
6
sinh
sinh
0
1
0
1
r cos
3644
2
2
4
4
6
6
1
2
sinh
1
3
4
r2
a2 sin
5
1
2
sinh
1
(e)
3
4
r2
a2 sin
5
(f)
6
1
4
r cos
sinh
sinh
0
1
0
1
r cos
0.5
2
2
0.5
4
1
6
1
2
sinh
1
3
4
r2
a2 sin
0.2
5
0.4
0.6
sinh
1
0.8
r2
1
1.2
a2 sin
(g)
Figure 9. (Continued.)
(d) For = mb , the last closed surface is self-crossing in both cusps and an overfilling
of the critical surface causes the accretion inflow through the inner cusp as well as
the excretion outflow through the outer cusp. Such a configuration corresponds to
the marginally bound accretion disc.
.
(ii) Naked-singularity backgrounds (0 < y < yc(ms+) = 0.068 86).
All the previously mentioned conclusions concerning the black-hole backgrounds are
2 .
=
also relevant for naked-singularity backgrounds with the rotational parameter a 2 > acL
.
2.4406 (y < yc(ms+) ), or even for a 2 > 27/16 = 1.6875 (corresponding to the Kerr
limit [25]), if the cosmological parameter y is not very large, typically y < 10−4 . In the
naked-singularity backgrounds with the rotational parameter low enough, especially for
.
2 .
= 1.47, and typically for a 2 < 32/27 = 1.1852 (corresponding to the Kerr limit
a 2 < acE
[25] which holds sufficiently well for y < 10−4 ), exceptions and additional possibilities
exist:
(a) If ms(i)+ < ms(o)+ , stationary discs exist for an arbitrary value of the specific angular
momentum . Spacetimes where ms(i)+ > ms(o)+ admit no stationary discs for the
specific angular momentum satisfying the relation ms(i)+ > > ms(o)+ . For the
remaining values of , the stationary configurations always exist.
(b) Moreover, for > ms(o)+ > ms(i)+ or > ms(i)+ > ms(o)+ , the configuration
corresponds to the counterrotating accretion disc with matter in states with E < 0.
The disc is isolated from the outer space by the region without any equipotential
Relativistic thick discs in the Kerr–de Sitter backgrounds
3645
6
1
4
r cos
sinh
sinh
0
1
0
1
r cos
0.5
2
2
0.5
4
1
6
1
2
sinh
3
1
r2
4
0.4
5
(h)
a2 sin
6
0.6
sinh
1
0.8
r2
1
a2
1.2
sin
1
4
r cos
sinh
sinh
0
1
0
1
r cos
0.5
2
2
0.5
4
1
6
1
2
sinh
3
1
r2
4
a2
sin
0.4
5
(i)
0.6
sinh
1
0.8
r2
1
1.2
a2 sin
Figure 9. (Continued.)
surfaces. The inner parts including the ring singularity are screened by the disc itself.
For < ms(o)− < ph(c) or < ph(c) < ms(o)− , the configuration corresponds to a
counterrotating accretion disc with matter in states with E > 0.
(c) In the part of naked-singularity spacetimes of class NS(+) with sufficiently large
values of the cosmological parameter y (very close to yc(ms+) ), in which all stable plusfamily circular orbits are counter-rotating with negative specific energy, the stationary
tori exist for ∈ (ms(i)+ , ms(o)+ ). Together with closed equipotential surfaces the
equipotential surface with the inner cusp always exists. We can expect that all fluid
elements in the torus have negative energy (E < 0). For ms(i)+ < < mb+ the
configuration corresponds to the counter-rotating negative-energy accretion disc, for
mb+ < < ms(o)+ the configuration corresponds to the counter-rotating negativeenergy excretion disc, and for = mb+ the configuration corresponds to the counterrotating negative-energy marginally bound accretion disc with both accretion and
excretion outflows from the torus. Since the outer marginally bound circular orbit in
the equatorial plane is located under the photon circular orbit, rmb(o)+ < rph(c) , the
whole torus is cut off from the outer space by the photon shell.
(d) Special values of the specific angular momentum = const can lead to stationary
configurations with two discs. The inner one is always the counter-rotating accretion
disc (for ms(o)+ > > ms(i)+ > a matter in the disc is in the states with E < 0;
otherwise the matter is in the states with E > 0), but the outer disc can be, depending
on the value of , the co-rotating or counter-rotating excretion disc, as well as
P Slaný and Z Stuchlı́k
6
4
4
2
2
r cos
6
sinh
sinh
0
1
0
1
r cos
3646
2
2
4
4
6
6
2
3
sinh
1
4
r2
5
2
a2 sin
3
sinh
1
(j)
4
r2
5
a2 sin
(k)
6
0.75
0.5
r cos
2
sinh
sinh
0.25
0
1
0
1
r cos
4
2
4
0.25
0.5
0.75
6
0.4
2
3
sinh
1
4
r2
(l)
a2 sin
0.6
5
sinh
1
0.8
r2
1
1.2
a2 sin
(m)
Figure 9. (Continued.)
the counter-rotating accretion disc. The region between the discs can be a region
forbidden for matter with prescribed specific angular momentum, if = const has
common points with any of the functions ph± (r; a, y). However, in the case of two
counter-rotating accretion discs, the region between the discs is filled by the matter
falling from the outer disc through its inner cusp onto the ring singularity. If, in
addition, some efficient dissipative processes are present in the accretion flow from
the outer disc, the matter could fill the inner accretion disc, rather than be directly
falling onto the ring singularity. If the inner accretion disc has already been created,
it could partly shield the ring singularity from the direct in-fall of accreting material
coming from the outer disc.
We conclude that in the structure and properties of equipotential surfaces, there are two
qualitatively new features connected with a cosmic repulsion, both of which take place near
the static radius of a given spacetime. The first one is the outer cusp which enables the outflow
of matter from the new type of stationary tori—the excretion discs. The outflow through the
outer cusp is driven by the same mechanism as the outflow through the inner cusp in the case of
accretion discs, i.e., by a violation of hydrostatic equilibrium. However, the outer cusp plays
an important role also for the accretion discs, since the outflow through the outer cusp (in the
case of large overfilling of the critical surface with the inner cusp when the critical surface
with the outer cusp is also overfilled) could stabilize the disc against dynamical instabilities,
Relativistic thick discs in the Kerr–de Sitter backgrounds
3647
0.075
0.05
2.5
0.025
r cos
5
2.5
5
0
0.025
1
0
sinh
sinh
1
r cos
7.5
0.05
0.075
7.5
1
2
sinh
3
1
r2
4
5
0.975
6
sinh
a2 sin
1.025
1.075
1
a2 sin
r2
1.125
(n)
7.5
1
5
0.5
r cos
0
1
0
2.5
sinh
sinh
1
r cos
2.5
5
0.5
1
7.5
1
2
sinh
1
3
r2
(o)
4
5
a2 sin
6
0
0.2 0.4 0.6 0.8
sinh
1
r2
1
1.2 1.4
a2 sin
(p)
Figure 9. (Continued.)
as shown by Rezzolla et al [6] in the case of perfect fluid tori orbiting Schwarzschild–de
Sitter black holes. On the other hand, the excretion outflow from the excretion disc cannot be
complemented by the accretion outflow into the black hole/naked singularity. As the inner
cusp determines the inner edge of the accretion disc, the outer cusp naturally determines the
outer edge of the excretion disc; however, it also determines the maximal extension of any
accretion disc. Similarly, in the case of marginally bound accretion discs, the inner (outer) cusp
determines the inner (outer) edge of the disc. The maximal potential difference W (in units
of c2 ) between the boundary and the centre of the torus, corresponding to the co-rotating (plusfamily) marginally bound accretion disc, is presented for various Kerr–de Sitter backgrounds
in table 1, where the particular rotational parameter a of the background is determined as the
α-multiple of the rotational parameter of the extreme black hole for a given y, a = αaex (y).
Note that the line y = 0 describes the Kerr backgrounds, whereas the column α = 0 describes
the Schwarzschild–de Sitter ones.
The second feature connected with the cosmic repulsion consists in strong collimation of
open equipotential surfaces near the axis of rotation, being evident near and behind the static
radius7 , when compared with the Kerr case, see figure 10 (left).
7 The location of the ‘static radius’ out of the equatorial plane is roughly given by its location in the Schwarzschild–de
Sitter spacetime corresponding to the same cosmological parameter y (relation (21)), since far away from the rotating
black hole these two backgrounds almost coincide.
3648
P Slaný and Z Stuchlı́k
0.5
0.5
r cos
1
r cos
1
0
1
sinh
sinh
1
0
0.5
1
0.5
1
0
0.2 0.4 0.6 0.8
sinh
1
r2
1
1.2 1.4
0
0.2 0.4 0.6 0.8
a2 sin
sinh
1
(q)
r2
1
1.2 1.4
a2 sin
(r)
1
r cos
0.5
sinh
1
0
0.5
1
0.9
1
sinh
1.1
1
r2
1.2
1.3
1.4
a2 sin
(s)
Figure 9. (Continued.)
Table 1. Potential differences, W = Wcusp − Wcentre , between the boundary (with both the inner
and the outer cusp) and the centre of plus-family accretion discs with (r, θ ) = mb+ = const
along the disc for various Kerr–de Sitter backgrounds. The rotational parameter a of the particular
background is determined by a = αaex (y), where aex (y) is the rotational parameter of the extreme
black hole for a given cosmological parameter y. Maximal values of W correspond to the Kerr
cases (y = 0), where for the extreme Kerr black hole W ≈ 0.549 whereas for the limiting
(a → 1) Kerr naked singularity W → ∞.
α:
y
0
0.5
0.9
0.999
0.999 99
W/c2
1.000 01
1.001
1.1
1.5
0
10−10
10−8
10−6
10−4
10−3
10−2
0.043
0.043
0.040
0.031
0.003
–
–
0.063
0.063
0.061
0.051
0.015
–
–
0.129
0.129
0.126
0.116
0.072
0.023
–
0.352
0.352
0.349
0.338
0.287
0.213
0.071
0.479
0.478
0.475
0.464
0.410
0.331
0.164
5.568
5.568
5.566
5.556
5.510
5.436
5.232
3.266
3.266
3.264
3.254
3.208
3.134
2.931
1.022
1.022
1.020
1.010
0.964
0.891
0.696
0.407
0.407
0.405
0.395
0.350
0.280
0.124
The rotation of the background manifested by the dragging of inertial frames, see, e.g.,
[27, 28], influences the shape of tori: the co-rotating discs are thicker and more extended than
the counter-rotating ones, generating a narrower funnel where highly collimated relativistic
streams of particle-jets are most probably created [29]. In figure 10 (right) the shapes
Relativistic thick discs in the Kerr–de Sitter backgrounds
3649
collimated
open surfaces
rotation axis
static radius
open surfaces
thick disc corotating
the KdS black hole
thick disc
orbiting
the SdS b. h.
accretion disc
(closed surfaces)
outer edge
(outer cusp)
outer cusp
thick disc counterrotating
the KdS black hole
static radius
Figure 10. (Left) Influence of a repulsive cosmological constant on the structure of equipotential
surfaces. Configurations corresponding to accretion discs corotating the Kerr (y = 0, a 2 = 0.99)
and the Kerr–de Sitter (y = 10−6 , a 2 = 0.99) black hole are compared. (Right) Influence of
the spacetime’s rotation on the shape of marginally bound accretion discs ( = mb ) orbiting the
Kerr–de Sitter black hole (y = 10−6 , a 2 = 0.99). For comparison, the thick marginally bound
accretion disc orbiting the Schwarzschild–de Sitter black hole (y = 10−6 , a = 0) is presented.
Table 2. The mass parameter, the static radius and radii of the outer marginally stable and outer
marginally bound circular orbits corresponding to the extreme Kerr–de Sitter black-hole spacetimes
with the current value of the ‘relic’ cosmological constant 0 ≈ 1.3 × 10−56 cm−2 . Note that
rms(o)− < rms(o)+ , rmb(o)− < rmb(o)+ as well as rmb(o)+ < rs in general, but for the presented values
of the cosmological parameter y these pairs of radii are indistinguishable.
y
M (M
)
rs (kpc)
rms(o)± (kpc)
rmb(o)± (kpc)
10−46
1
10
1 × 102
1 × 103
1 × 106
1 × 107
1 × 108
1 × 109
1 × 1010
1 × 1011
1 × 1012
0.11
0.23
0.50
1.1
11
23
50
1.1 × 102
2.3 × 102
5.0 × 102
1.1 × 103
0.067
0.15
0.31
0.67
6.7
15
31
67
1.5 × 102
3.1 × 102
6.7 × 102
0.11
0.23
0.50
1.1
11
23
50
1.1 × 102
2.3 × 102
5.0 × 102
1.1 × 103
10−44
10−42
10−40
10−34
10−32
10−30
10−28
10−26
10−24
10−22
of marginally bound accretion discs co-rotating/counter-rotating with the Kerr–de Sitter
black hole are compared with the marginally bound accretion disc orbiting the non-rotating
Schwarzschild–de Sitter black hole (corresponding to the same value of the cosmological
parameter y = 10−6 ).
Finally, we shall give an idea on scales at which the discussed effects take place, by
expressing basic characteristics of the tori in astrophysical units. For this purpose, we use
the current value of the cosmological constant8 , = 0 . Various cosmological observations
8 Analogical estimates for primordial Schwarzschild–de Sitter black holes and effective cosmological constants
related to vacuum energy density connected with the electroweak symmetry breaking or the quark confinement are
presented in [5].
3650
P Slaný and Z Stuchlı́k
indicate the present value of the vacuum/dark energy density [1]
vac(0) ≈ 0.73crit(0) ,
(38)
where the present value of the critical energy density crit(0) is related to the Hubble parameter
H0 by
3H02
,
H0 = 100h km s−1 Mpc−1 .
(39)
8π
Taking the value of the dimensionless parameter h ≈ 0.7, we obtain the ‘relic’ repulsive
cosmological constant to be
crit(0) =
0 = 8π vac(0) ≈ 1.3 × 10−56 cm−2 .
(40)
Having this value of 0 , we can determine the mass parameter of the spacetime, corresponding
to any value of the cosmological parameter y, using relation (15), however, the most interesting
are those values of y which correspond to typical masses of black holes. For extreme black
holes, the dimensions of the static radius and the outer marginally stable and marginally bound
circular orbits of both families are presented in table 2. As the outer edge of tori is located
between the outer marginally stable orbit and the static radius, rms(o) < rout < rs , the repulsive
cosmological constant puts a limit on the maximal extension of disc-like structures in a given
background. Note that, e.g., for supermassive black holes (106 M
–1010 M
), the dimensions
of test tori are roughly comparable with the dimensions of galaxies [30]. In the case of
such large structures, to get a more realistic picture, the influence of the cosmic repulsion on
self-gravitating tori has to be studied and we plan it for the future.
Acknowledgments
The authors are supported by Czech grants GAČR 205/03/1147 and MSM4781305903.
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Relativistic thick discs in the Kerr–de Sitter backgrounds
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PHYSICAL REVIEW D 71, 024037 (2005)
Aschenbach effect: Unexpected topology changes in the motion of particles and fluids orbiting
rapidly rotating Kerr black holes
Zdeněk Stuchlı́k,1,2,* Petr Slaný,1,2,† Gabriel Török,1,2,‡ and Marek A. Abramowicz1,2,3,x
1
Institute of Physics, Silesian University at Opava, Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
2
NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
3
Theoretical Physics, Göteborg & Chalmers Universities, S-412 96 Göteborg, Sweden
(Received 12 November 2004; published 28 January 2005)
Newtonian theory predicts that the velocity V of free test particles on circular orbits around a spherical
gravity center is a decreasing function of the orbital radius r, dV =dr < 0. Only very recently, Aschenbach
[B. Aschenbach, Astronomy and Astrophysics, 425, 1075 (2004)] has shown that, unexpectedly, the same
is not true for particles orbiting black holes: for Kerr black holes with the spin parameter a > 0:9953, the
velocity has a positive radial gradient for geodesic, stable, circular orbits in a small radial range close to
the black-hole horizon. We show here that the Aschenbach effect occurs also for nongeodesic circular
orbits with constant specific angular momentum ‘ ‘0 const. In Newtonian theory it is V ‘0 =R,
with R being the cylindrical radius. The equivelocity surfaces coincide with the R const surfaces
which, of course, are just coaxial cylinders. It was previously known that in the black-hole case this simple
topology changes because one of the ‘‘cylinders’’ self-crosses. The results indicate that the Aschenbach
effect is connected to a second topology change that for the ‘ const tori occurs only for very highly
spinning black holes, a > 0:99979.
DOI: 10.1103/PhysRevD.71.024037
PACS numbers: 04.20.Gz, 04.70.–s, 95.30.Sf
I. INTRODUCTION
Aschenbach [1] found a very interesting and rather
surprising fact about the circular orbits of free particles
around the Kerr black holes with high spin. Contrary to
what is true for Kerr black holes with a small spin, for
orbits around Kerr black holes with a > 0:9953 the
Keplerian orbital velocity V LNRF
measured in locally nonrotating frames (LNRF) is a nonmonotonic function of
radius.
In this article we show that there is a corresponding
change of behavior of orbital velocity in the case of nonKeplerian orbits with constant specific angular momentum1, ‘r; const: for low spin black holes the radial
gradient of the orbital velocity, @V LNRF =@r, changes its
sign once, but for a sufficiently rapidly rotating black
holes, it changes the sign 3 times.
We discuss the geometrical reason for this puzzling
behavior of orbital velocity in terms of the von Zeipel
surfaces, defined as the surfaces of constant Rr; ‘=V LNRF
. In Newtonian physics (Euclidean geometry),
the von Zeipel surfaces have topology of coaxial cylinders
R r sin
const. In the black-hole geometry, the topology of the von Zeipel surfaces is remarkably different.
*Electronic address: [email protected]
†
Electronic address: [email protected]
‡
Electronic address: [email protected]
x
Electronic address: [email protected]
1
Here the specific angular momentum means the axial (conserved) angular momentum per total (conserved) energy and we
use this term for both particles and fluid elements in the whole
article.
1550-7998= 2005=71(2)=024037(9)$23.00
It was known for a long time that for a nonrotating blackhole one of the von Zeipel surfaces self-crosses at the
location of the photon orbit [2]. We found that the
Aschenbach effect is due to a second topology change, as
the another surface with a cusp together with toroidal
surfaces appear. The second change is strictly connected
to the first one, but it occurs only for very rapidly rotating
black holes.
In Sec. II, we summarize basic relations characterizing
the constant specific angular momentum tori. In Sec. III,
the orbital velocity relative to the LNRF is given and its
properties are determined. In Sec. IV, the notion of von
Zeipel radius is introduced and properties of the von Zeipel
surfaces are analyzed. In Sec. V, we present discussion and
some concluding remarks.
II. CONSTANT SPECIFIC ANGULAR
MOMENTUM TORI
In general, stationary and axially symmetric spacetimes
with the line element
d s2 gtt dt2 2gt dtd g d2 grr dr2 g
d
2 ;
(1)
the stationary and axisymmetric fluid tori with the stressenergy tensor T % pU U pg are characterized by 4-velocity field
U Ut ; 0; 0; U (2)
with Ut Ut r; ; U U r; , and by the distribution
of specific angular momentum
024037-1
 2005 The American Physical Society
STUCHLÍK, SLANÝ, TÖRÖK, AND ABRAMOWICZ
‘
U
:
Ut
PHYSICAL REVIEW D 71, 024037 (2005)
The angular velocity of orbiting matter, U =Ut , is
then related to ‘ by the formula
III. THE ORBITAL VELOCITY IN LNRF
(3)
‘gtt gt
:
‘gt g
(4)
The locally nonrotating frames are given by the tetrad of
1-forms [5,6]
1=2
e t dt;
(16)
A
e
The tori considered here are assumed to have constant
specific angular momentum,
‘ ‘r; const:
Ut
Ut in
(6)
1=2
dr;
e 1=2 d
;
(5)
Their structure is determined by equipotential surfaces
W Wr; defined by [3,4]
W Win ln
r
e 1=2
A
sin
d !dt;
2
Ut !
gtt g
gtt ‘ 2gt ‘ g
;
(7)
gt g
(8)
2arsin2 ;
V
V
where
2
2
(11)
2
r a cos ;
(12)
A r2 a2 2 a2 sin2 :
(13)
We make our formulas dimensionless by using the standard
c G M 1 convention. The relation (4) for the angular velocity of matter orbiting the black hole acquires the
form
r; ; a; ‘ a2 sin2 ‘
2arsin2 A 2‘arsin2 LNRF
U e
U et
LNRF
A sin
p !:
A a2 sin2 4a2 r2 sin2 p
‘:
A 2a‘r sin
K r; a (21)
(22)
1
r a
3=2
(24)
together with Eq. (21) to get, in the equatorial plane,
V
K r; a
r2 a2 2 a2 2arr3=2 a
p
: (25)
r2 r3=2 a Obviously, the velocity formally diverges at r r . The
Keplerian specific angular momentum is given by
W Wr; ; a; ‘ 1
sin2 ln 2
:
2
2
2 r a a‘ sin2 ‘ asin2 2
The velocity vanishes
at infinity, r ! 1, and at the horip
zon, r ! r 1 1 a2 . Thus, a change of the sign of
the radial gradient of velocity occurs for any pair a, ‘. This
is not true for Keplerian orbits. To see this, one may use the
formula for Keplerian angular velocity [5]
(14)
and the potential W, defined in Eq. (6), has the explicit
form
(20)
In the equatorial plane, =2, Eq. (22) reduces to
p
r ‘:
V r; =2; a; ‘ 2
rr a2 2a‘ a
(23)
(10)
r2 2r a2 ;
2ar
:
A
Substituting for the angular velocities and ! from the
relations (14) and (20), respectively, we arrive at the formula
(9)
Asin2 ;
(19)
The azimuthal component of 3-velocity in LNRF reads
the subscript ‘‘in’’ refers to the inner edge of the torus.
The metric components of the Kerr spacetime (with a >
0) in the Boyer-Lindquist coordinates are:
a2 sin2 ;
gtt (18)
where the angular velocity of LNRF, ! gt =g , is
given by the relation
with
g2t
2
(17)
‘K r; a (15)
r2 2ar1=2 a2
:
r3=2 2r1=2 a
(26)
The minimum of ‘K r; a corresponds to the marginally
024037-2
ASCHENBACH EFFECT: UNEXPECTED TOPOLOGY. . .
PHYSICAL REVIEW D 71, 024037 (2005)
stable circular geodesic at r rms , which is the innermost
possible circular, stable geodesic. The innermost possible
circular fluid orbit in the constant specific angular momentum tori is given by the condition ‘ ‘K rin . It is known
that rmb < rin < rms . Here rmb denotes the radius of the
circular marginally bound geodesic.
The radial gradient of the equatorial orbital velocity of
tori reads
2
2
r3r2 a2 @V r 1rrr a 2a‘ ap
‘;
rr2 a2 2a‘ a2 @r
and it changes its orientation at radii determined by the
condition
‘ ‘ex r; a a r2 r2 a2 r 1 2r
: (28)
2a rr 1
We have to discuss properties of ‘ex r; a above the
event horizon r taking into account the limits on the inner
boundary of the tori, ‘ 2 ‘ms ; ‘mb where ‘ms ‘mb denotes specific angular momentum of the marginally stable
(marginally bound) circular geodesic. The local extrema of
‘ex r; a are given by the relation
p
3 18r 7r2 Dr
2
2
;
(29)
a aex r r
23r 2
(27)
a2ex r is given in Fig. 2. Clearly, only a2ex r is relevant
for black holes. The minimum of a2ex r, denoted I, is
located at radius rmin 1:19466
_
and the critical value of the
rotational parameter is
p
_
(31)
acbh _ 0:99928 0:99964:
Note that both the functions a2ex r are relevant for Kerr
naked singularities. The maximum of a2ex r, denoted II,
a2ex− (r)
1.0015
1.001
with
1.0005
Dr 9 108r 150r2 12r3 23r4
23r r1 r r2 r r3 r r4 ; (30)
0.9
where
1.2
1.3
r
0.9995
r1 _ 3:11363;
r3 0:74939;
_
r2 0:09602;
_
a2ex+ (r)
0.8
-2
1.2
400
1.8
300
1.6
200
1.4
✠
-1
-100
II
❄
2.2
500
100
I
(a)
D(r)
r1
✻
0.999
r4 1:74648:
_
The situation is illustrated in Fig. 1 which implies that only
the interval r 2 r3 ; r4 is relevant for the region outside of
the black-hole event horizon. Behavior of the functions
✠
-3
1.1
r2
1.4
1.6
r
1.2
✻1
r3
✻2
r
(b)
r4
FIG. 1. Reality condition for the existence of local extrema of
the function ‘ex r; a. The extrema are allowed, if Dr > 0.
Clearly, the physically relevant extrema, located above the outer
horizon, can exist in the interval r 2 1; r4 .
FIG. 2. Loci of local extrema of the function ‘ex r; a. They
are determined by the functions a2ex r. (a) The function a2ex r
is relevant for both black holes and naked singularities; its local
minimum is denoted I. (b) The function a2ex r is relevant for
naked singularities only; its local maximum is denoted II.
024037-3
STUCHLÍK, SLANÝ, TÖRÖK, AND ABRAMOWICZ
PHYSICAL REVIEW D 71, 024037 (2005)
is located at radius rmax 1:43787
_
and the critical value of
the rotational parameter is
p
acns _ 2:26289 1:50429:
_
(32)
Therefore, the possibility to have three changes of the sign
of @V =@r in constant specific angular momentum tori is
limited from below for black holes, and from above for
naked singularities. Here we restrict our attention to the
Kerr black holes.
Now, we have to compare the local extrema of the
function ‘ex r; a, determined by the condition (29), with
the functions characterizing the marginally stable, ‘ms a,
and the marginally bound, ‘mb a, circular geodesics as
these determine the limits of allowance of stationary toroidal structures in the Kerr spacetimes [3]. For each given
value of a, location of both the marginally stable and the
marginally bound circular geodesics is uniquely given by
the functions rms rms a; rmb rmb a [5], and
‘ms a; ‘mb a can then be determined using the formulas
for ‘K r; a and rms a; rmb a, respectively. In Fig. 3, behavior of the local extrema ‘exmin a; ‘exmax a and the
functions ‘ms a; ‘mb a is illustrated. It is clear immediately that the sign’s change of @V =@r is relevant only
for tori orbiting the Kerr black holes with the rotational
parameter
a > actori 0:99979;
_
(33)
which is much closer to the extreme case a 1 than the
critical value acK 0:9953
_
determined by Aschenbach for
ms
2.015
I
mb
A
B
2.01
I = (0.99964, 2.01471)
A = (0.99979, 2.0123)
2.005
B = (0.99998, 2.00949)
2
C = (1, 2)
0.9997
C
0.9998
0.9999
1
a
FIG. 3. Kerr spacetimes with the change of sign of the gradient
of LNRF velocity. In the ‘–a plane, the functions ‘exmax a
(upper solid curve), ‘exmin a (lower solid curve), ‘ms a
(dashed curve) and ‘mb a (dashed-dotted curve) are given. For
pairs of a; ‘ from the shaded region, the gradient of the orbital
velocity of tori changes its sign twice inside the tori. Between the
points A, B, the ‘exmax a curve determines an inflex point of
V r; a; ‘. The inflex points determined by the curve
‘exmin a are irrelevant being outside of the definition region
for constant specific angular momentum tori, ‘ 2
‘ms a; ‘mb a. The point I corresponds to the inflex point of
‘ex r; a; cf. Figure 2(a).
Keplerian discs [1]. For a > actori the relevance of
‘ex r; a is limited from below by ‘ms a. There is another
critical value of the rotational parameter, a acmb 0:99998,
_
where ‘mb a ‘exmax a; for a >
acmb the relevance of ‘ex r; a is limited from above by
‘mb a.
The character of the region, where @V =@r changes
sign, can be properly illustrated by considering the functions ‘ex r; a and ‘K r; a simultaneously. First, we show
that there is no common point of those functions in blackhole spacetimes with a < 1. Indeed, the condition
‘ex r; a ‘K r; a implies an equation quartic in a, which
has four solutions
p
a a1 r r r;
(34)
p
a a2 r r2 r;
(35)
p
a ah r r2 r;
(36)
a aph r p
r
3 r:
2
(37)
The solution a1 r > 1 at r > 1, i.e., it corresponds to
naked singularities at r > 1, the solution a2 r is negative
everywhere, the solution a3 ah r determines radius of
the event horizon, while the solution a4 aph r determines radius of the corotating photon circular geodesic.
None of the solutions is relevant for the stationary tori. We
can conclude that above the photon circular orbit there is
always ‘K r; a > ‘ex r; a; therefore, the innermost local
maximum of V r; a for a > acbh , and the only local
maximum of V r; a for a < acbh , is always physically
irrelevant in constant specific angular momentum tori.
For black-hole spacetimes, behavior of the functions
‘ex r; a and ‘K r; a can then be classified into six classes
which are illustrated in Fig. 4:
(1) 0 < a < acbh : No extrema of ‘ex r; a [Fig. 4(a)].
(2) a acbh : An inflex point of ‘ex r; a [Fig. 4(b)].
(3) acbh < a < actori : Two local extrema of ‘ex r; a
present, but out of the region allowing the existence
of tori [Fig. 4(c)].
(4) actori < a < acmb : Two local extrema of ‘ex r; a
allowed in the region of ‘ 2 ‘ms ; ‘exmax [Fig. 4(d)].
(5) acmb < a < 1: Two local extrema of ‘ex r; a allowed in the region ‘ 2 ‘ms ; ‘mb [Fig. 4(e)].
(6) a 1: The minimum of ‘ex r; a coincides with the
marginally bound geodesic with ‘mb 2 at rmb 1. The curves ‘ex r; a 1 and ‘K r; a 1 intersect at r 1 [Fig. 4(f)].
Clearly, three changes of sign of @V =@r can occur for
Kerr black holes with the rotational parameter a >
acbh 0:99964.
_
However, the effect is relevant for con-
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ASCHENBACH EFFECT: UNEXPECTED TOPOLOGY. . .
PHYSICAL REVIEW D 71, 024037 (2005)
.
a = ac(bh) = 0 .99964
2.05
a = 0 .996
2.3
✙
✟
2.04
mb
2.2
✠
2.03
ms
2.02
❄
2.1
✁✕
2.01
1.2
mb
1.4
1.6
r
1.8
✻
ms
I
1.1
1.2
r
1.4
1.99
1.9
(a)
2.03
1.3
✙
✟
(b)
mb
2.025
✠
2.02
mb
a = 0 .9999
2.015
2.02
2.01
✁✕
2.01
✻
2.005
ms
ms
a = 0 .99975
1.1
1.2
1.1
1.2
r
1.4
r 1.995
1.3
(c)
(d)
2.015
2.015
a = 0 .99999
2.01
✠
2.005
1.3
2.0125
mb
2.0075
ms
2.005
❄
2.0025
1.1
a=1
2.01
1.2
1.3
r
1.4
0.9
(e)
I
❅
1.1
mb, ms
1.2
1.3
1.4
r
(f)
FIG. 4. Classification of the Kerr black-hole spacetimes according to the properties of the functions ‘ex r; a (solid curves) and
‘K r; a (dashed curves). The functions are plotted for six cases corresponding to the classification. The constant specific angular
momentum tori can exist in the shaded region only along ‘ const lines. Their inner edge (center) is determined by the decreasing
(increasing) part of ‘K r; a. The local extrema of the orbital velocity relative to LNRF relevant for tori are given by the intersections of
‘ const line with the curve of ‘ex r; a in the shaded region. Notice that the region corresponding to the allowed values of ‘ for the
discs is narrowing with a ! 1, it is degenerated into the ‘ 2 line for a 1 as ‘ms ‘mb 2 in this case. In the case (e), the gradient
@V =@r changes sign for all values of ‘ 2 ‘ms ; ‘mb allowed for the tori, while in the case (d), it is allowed for a region restricted
from above by the value ‘exmax a. In the cases (a) –(c), the change of sign of @V =@r cannot occur in the disc. It is directly seen
from cases (d)–(f) that the gradient @V =@r changes the sign closely above the center of the disc.
stant specific angular momentum tori only if a >
actori 0:99979.
_
The interval of corresponding values of
the specific angular momentum ‘ 2 ‘ms a; ‘exmax a
grows with a growing up to the critical value of
acmb 0:99998.
_
For a > acmb , the interval of relevant
values of ‘ 2 ‘ms a; ‘mb a is narrowing with growing
of the rotational parameter up to a 1, which corresponds
to a singular case where ‘ms a 1 ‘mb a 1 2.
Notice that the situation becomes to be singular only in
terms of the specific angular momentum; it is shown [5]
that for a 1 both the total energy E and the axial angular
momentum L differ at rms and rmb , respectively, but their
combination, ‘ L=E, giving the specific angular momentum, coincides at these radii.
024037-5
STUCHLÍK, SLANÝ, TÖRÖK, AND ABRAMOWICZ
PHYSICAL REVIEW D 71, 024037 (2005)
IV. VON ZEIPEL SURFACES
Till now, our study was focused on the properties of
LNRF velocity profile in the equatorial plane of the constant specific angular momentum tori. However, it is useful
to obtain global characteristics of the phenomenon that is
shown to be manifested in the equatorial plane as the
existence of a small region with positive gradient of the
LNRF velocity.
It is well known that rotational properties of perfect fluid
equilibrium configurations in strong gravity are well repf introduced in the case
resented by the radius of gyration R,
of spherically symmetric Schwarzschild spacetimes in [2],
f defines a local outward
as the direction of increase of R
direction of the dynamical effects of rotation of the fluid. In
the stationary and axisymmetric spacetimes, the radius of
gyration was defined by the relation [7]
~ 1=2
‘
f
R
;
(38)
e
e ! is the angular velocity relative to the
where LNRF. However, ‘~ L=Ee is not the specific angular
momentum ‘ L=E with L p ; E p t
being the 4-momentum projections on the Killing vector
e
~ , where
fields and t t , but E p ~ ! is not a Killing vector field, i.e., Ee is
t
related to the LNRF and it is not a constant of motion.
Important consequence of such a definition is given by the
f and V :
relation between R
LNRF
V
LNRF
f:
eR
(39)
Here, we shall use another physically reasonable way of
defining a global quantity characterizing rotating fluid
configurations by using directly the LNRF orbital velocity.
We define, so-called, von Zeipel radius by the relation
R ‘
V LNRF
;
(40)
which generalizes the Schwarzschildian definition of gyration radius. In static spacetimes, the von Zeipel radius
(40) coincides with the radius of gyration defined by the
relation (38), however, in stationary, axisymmetric spacetimes, relation between the both radii has the form
f
R 1 !‘R:
(41)
In the case of tori with ‘r; const, the von Zeipel
surfaces, i.e., the surfaces of Rr; ; a; ‘ const, coincide with the equivelocity surfaces V LNRF r; ; a; ‘ const. For the tori in the Kerr spacetimes, there is
p
A 2a‘r sin
: (42)
R r; ; a; ‘ A a2 sin2 4a2 r2 sin2 Topology of the von Zeipel surfaces can be directly determined by the behavior of the von Zeipel radius (42) in the
equatorial plane
R r; =2; a; ‘ rr2 a2 2a‘ a
p
:
r (43)
The local minima of the function (43) determine loci of the
cusps of the von Zeipel surfaces, while its local maximum
(if it exists) determines a circle around which closed toroidally shaped von Zeipel surfaces are concentrated
(Fig. 5). Notice that the minima (maximum) of Rr; =2; a; ‘ correspond(s) to the maxima (minimum) of
V LNRF r; =2; a; ‘, therefore, the inner cusp is always physically irrelevant being located outside of the
toroidal configuration of perfect fluid cf. Figure 4.
V. DISCUSSION AND CONCLUSIONS
It is useful to discuss both the qualitative and quantitative aspects of the phenomenon of the positive gradient of
LNRF orbital velocity. In the Kerr spacetimes with a >
actori , changes of sign of the gradient of V r; a; ‘ must
occur closely above the center of relevant toroidal discs, at
radii corresponding to stable circular geodesics of the
spacetime (cf. Fig. 4).
For a actori 0:99979,
_
except the irrelevant local
maximum located always outside the disc, an inflex point
_
for the disc with
of V r; a; ‘ occurs at rinf 1:24143
‘ ‘ms 2:0123.
_
With rotational parameter growing (a >
actori ), the local maximum of V r; a; ‘ is succesively
shifted up to values of r 1:4, while the local minimum of
V r; a; ‘ is shifted down to r 1 in the limit of a 1
[Fig. 7(a)]. Notice that the function V r; a; ‘ can possess two local maxima even for a > acbh 0:99964
_
but for
irrelevant values of the parameter ‘, as it does not enters
the interval corresponding to the constant specific angular
momentum tori, ‘2
6 ‘ms ; ‘mb . The loci of these extreme
points can be directly inferred from Fig. 4, where the
regions corresponding to the constant specific angular
momentum tori are shaded.
For some representative cases corresponding to the classification of Kerr spacetimes given in Sec. III, behavior of
V r; a; ‘ is illustrated in Fig. 6, which enables us to
make some conclusions on the quantitative properties of
the orbital velocity and its gradient. For comparison, profiles of the Keplerian velocity V K r; a are included.
With a growing in the region of a 2 actori ; 1, the differ
ence V V max V min grows as well as the difference of radii, r rmax rmin , where the local extrema of
V r; a; ‘ occur, see Figs. 7(a) and 7(b). Recall that the
innermost local maximum of V r; a; ‘ must be located, necessarily, under the disc structure. The value of
V r rin ; a; ‘ at the inner edge of the toroid (where
‘ ‘K rin ; a) is located closer and closer to the local
024037-6
ASCHENBACH EFFECT: UNEXPECTED TOPOLOGY. . .
a = 0 .99
= 2 .14
1
3.0
PHYSICAL REVIEW D 71, 024037 (2005)
3.6
3.6
3.95
4.5
4.5
0.5
0.5
Θ
3.46922
Θ
0
r cos
r cos
a = 0 .99995
=2
1
3.96
− 0.5
3.82929
0
3.85
− 0.5
3.6
−1
4.5
3.0
0.5
1
1.5
r sin
Θ
4.5
3.96
2
2.5
−1
3
3.6
0.5
(a)
a = 0 .99995
= 2 .0054
1
1
1.5
r sin
Θ
3.95
2
2.5
(b)
3.6
a = 0 .99995
.
= 2 .00568
1
3.96
3.6
3.96
4.5
4.5
0.5
0
✲
Θ
3.82215
✙
✟
3.80967
r cos
r cos
Θ
0.5
❅
I
✯
✟
❅3.83838 ✟
− 0.5
3.80857
3.80857
3.82
0
3.82
❅
I
✯
✟
❅3.83386 ✟
− 0.5
4.5
−1
3.6
0.5
1
1.5
r sin
Θ
4.5
−1
3.96
2
2.5
3
3.6
0.5
(c)
a = 0 .99995
= 2 .006
1
1
1.5
r sin
Θ
3.96
2
2.5
3
(d)
3.6
1
3.96
a = 0 .99995
= 2 .012
3.7
3.96
2.8
4.5
4.5
0.5
3.79119
3.79119
r cos
3.80727
3.82
3.82
0
3.20189
Θ
Θ
0.5
r cos
3
❅
I
✯
✟
❅ 3.82894 ✟
− 0.5
0
3.20189
4.5
3.79
− 0.5
4.5
4.5
−1
3.6
0.5
1
1.5
r sin
Θ
2
2.5
2.8
−1
3.96
3
3.7
0.5
(e)
1
1.5
r sin
Θ
3.96
2
2.5
3
(f)
FIG. 5. Von Zeipel surfaces (meridional sections). (a) For a < acbh and any ‘, only one surface with a cusp in the equatorial plane
and no closed (toroidal) surfaces exist. The cusp is, however, located outside the toroidal equilibrium configurations of perfect fluid.
(b)–(f) For a > acbh and ‘ appropriately chosen, two surfaces with a cusp ((c), (e)), or one surface with both the cusps (d), together
with closed (toroidal) surfaces, exist. Moreover, if a > actori , both the outer cusp and the central ring of closed surfaces are located
inside the toroidal equilibrium configurations corresponding to constant specific angular momentum discs (cases (c)–(e)). If ‘ is
sufficiently low/high ((b)/(f)), there is only one surface with a cusp outside the configuration. Shaded region corresponds to the black
hole.
024037-7
STUCHLÍK, SLANÝ, TÖRÖK, AND ABRAMOWICZ
PHYSICAL REVIEW D 71, 024037 (2005)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
FIG. 6. Profiles of the equatorial orbital velocity of constant specific angular momentum tori in LNRF in terms of the radial BoyerLindquist coordinate. The profiles are given for typical values of a corresponding to the classification of the Kerr black-hole
spacetimes. For comparison, the profiles are given for the orbital velocity of Keplerian discs in Kerr spacetimes with the same
rotational parameter a. For tori, values of ‘ const are appropriately chosen; commonly, ‘ ‘ms is used giving the maximal value of
the velocity difference in between the local extrema, and representing the limiting case of constant specific angular momentum tori.
024037-8
ASCHENBACH EFFECT: UNEXPECTED TOPOLOGY. . .
PHYSICAL REVIEW D 71, 024037 (2005)
(c)
(b)
(a)
FIG. 7. (a) Positions of local extrema of V LNRF (in B-L coordinates) for the constant specific angular momentum tori with ‘ ‘ms
in dependence on the rotational parameter a of the black-hole. (b) Velocity difference V V max
V min
as a function of the
rotational parameter a of the black-hole for both the Keplerian disc and the constant specific angular momentum (non-Keplerian) disc
with ‘ ‘ms . (c) orbital-velocity curves in the limiting case of the extreme black-hole. At r 1, the Keplerian orbital velocity V K
has a local minimum, whereas the orbital velocity V ‘ms of the constant specific angular momentum disc has an inflex point. In both
_
V _
cases, the velocity difference V reaches its maximal values: V K 0:06986;
‘ms 0:02241.
minimum of V r; a; ‘ when a ! 1. For a 1, there is
an inflex point of V r; a 1; ‘ at r 1 where the
local minimum and the ‘‘forbidden’’ local maximum of
V r; a; ‘ for ‘ 2 coincide, Fig. 7(c). Moreover, the
velocity difference V is smaller but comparable in the
tori in comparison with Keplerian discs. We can see that
for a ! 1, the velocity difference in the tori V tori
0:02, while for the Keplerian discs it goes even up to
0:07, see Fig. 7(c). These are really huge velocV K
ity differences, being expressed in units of c.
We can conclude that the Aschenbach effect, i.e., the
change of sign of gradient of the LNRF orbital velocity, in
the case of constant specific angular momentum tori occurs
for the discs orbiting Kerr black holes with the rotational
parameter a > actori . In terms of the redefined rotational
parameter, 1 a, its value of 1 actori 2:1
_
104 is
more than one order lower than the value 1 acK 4:7
_
103 found by Aschenbach for the changes of sign of the
gradient of orbital velocity in Keplerian discs. In constant
specific angular momentum tori, the Aschenbach effect is
elucidated by topology changes of the von Zeipel surfaces.
In addition to one self-crossing von Zeipel surface existing
for all values of the rotational parameter a, for a > actori
the second self-crossing surface together with toroidal
surfaces occur. Toroidal von Zeipel surfaces exist under
the newly developing cusp, being centered around the
circle corresponding to the minimum of the equatorial
LNRF velocity profile.
[1] B. Aschenbach, Astron. Astrophys. 425, 1075 (2004).
[2] M. A. Abramowicz, J. C. Miller, and Z. Stuchlı́k, Phys.
Rev. D 47, 1440 (1993).
[3] M. A. Abramowicz, M. Jaroszyński, and M. Sikora,
Astron. Astrophys. 63, 221 (1978).
[4] M. Kozłowski, M. Jaroszyński, and M. A. Abramowicz,
Astron. Astrophys. 63, 209 (1978).
[5] J. M. Bardeen, W. H. Press, and S. A. Teukolsky,
Astrophys. J. 178, 347 (1972).
[6] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation
(Freeman, San Francisco, 1973).
[7] M. A. Abramowicz, P. Nurowski, and N. Wex, Classical
Quantum Gravity 12, 1467 (1995).
ACKNOWLEDGMENTS
The authors were supported by the Czech GAČR grant
205/03/H144 and by the grant of the Czech government
MSM4781305903. The main parts of the work were done
at the Department of Astrophysics of Chalmers University
at Göteborg and at Nordita at Copenhagen. The authors
Z. S., P. S., and G. T. would like to express their gratitude to
the staff of the Chalmers University and Nordita for perfect
hospitality.
024037-9