Equilibrium configurationsEquilibrium configurationsof perfect fluidof perfect fluid

in Reissner-Nordstrin Reissner-Nordströöm-anti-de Sitter m-anti-de Sitter spacetimesspacetimes

Hana Kučáková, Zdeněk Stuchlík, Petr SlanýHana Kučáková, Zdeněk Stuchlík, Petr SlanýInstitute of Physics, Silesian University at OpavaInstitute of Physics, Silesian University at Opava

RAGtime 10RAGtime 1015.-17. September 2008, Opava15.-17. September 2008, Opava

IntroductionIntroduction

• investigating equilibrium configurations of perfect fluidinvestigating equilibrium configurations of perfect fluidin charged black-hole and naked-singularity spacetimes within charged black-hole and naked-singularity spacetimes withan attractive cosmological constant (an attractive cosmological constant ( << 0 0))

• the line element of the spacetimes (the geometric units the line element of the spacetimes (the geometric units c c == G G == 1)1)

• dimensionless cosmological parameter and dimensionless charge dimensionless cosmological parameter and dimensionless charge parameterparameter

• dimensionless coordinatesdimensionless coordinates

d sindd2

1d 2

1d 222

1

22

222

2

22 rrr

r

Q

r

Mtr

r

Q

r

Ms

M

Qe 2

3

1My

Mtt Mrr

TTypes of the ypes of the Reissner-NordstrReissner-Nordströöm-anti-de Sitter m-anti-de Sitter spacetimesspacetimes

• fourfour types with qualitatively different behavior of the effective types with qualitatively different behavior of the effective potential of the geodetical motion and the circular orbitspotential of the geodetical motion and the circular orbits

Black-hole spacetimesBlack-hole spacetimes

• AAdS-BH-1dS-BH-1 – one region of circular geodesics at – one region of circular geodesics at r r > > rrph+ ph+ with unstable with unstable

and and then stable geodesics (for radius growing)then stable geodesics (for radius growing)

2/12

ph 9

811

2

3)(

eer

TTypes of the ypes of the Reissner-NordstrReissner-Nordströöm-anti-de Sitter m-anti-de Sitter spacetimesspacetimes

Naked-singularity spacetimesNaked-singularity spacetimes

• AAdS-NS-1dS-NS-1 – – ttwo regions of circular geodesics, the inner wo regions of circular geodesics, the inner oneone ((r r << rrphph--) )

consists of stable geodesics onlyconsists of stable geodesics only, , the outer onethe outer one ( (r r > > rrph+ph+)) contains contains bothboth

unstableunstable and and then stable circular then stable circular geodesicsgeodesics

• AAdS-NS-2dS-NS-2 – – oone region of circular orbits, subsequently with stable, ne region of circular orbits, subsequently with stable, then then unstableunstable and finallyand finally stable orbitsstable orbits

• AAdS-NS-3dS-NS-3 – – oone region of circular orbitsne region of circular orbits with stable orbits exclusively with stable orbits exclusively

Test perfect fluidTest perfect fluid

• does not alter the geometrydoes not alter the geometry

• rotating in the rotating in the direction – its four velocity vector field direction – its four velocity vector field U U has, has, therefore, only two nonzero components therefore, only two nonzero components U U = ( = (U U tt, 0, 0 , , 0, 0 , U U ))

• the stress-energy tensor of the perfect fluid isthe stress-energy tensor of the perfect fluid is

(( and and pp denote the total energy density and the pressure of the fluid denote the total energy density and the pressure of the fluid))

• the rotating fluid can be characterized by the vector fields of the the rotating fluid can be characterized by the vector fields of the angular velocity angular velocity , and the angular momentum density , and the angular momentum density ll

pUUpT

tU

U

tU

U

Equipotential surfacesEquipotential surfaces

• the solution of the relativistic Euler equation can be given by Boyer’s the solution of the relativistic Euler equation can be given by Boyer’s condition determining the surfaces of constant pressure through the condition determining the surfaces of constant pressure through the “equipotential surfaces” of the potential “equipotential surfaces” of the potential WW ( (rr, , ))

• the equipotential surfaces are determined by the condition the equipotential surfaces are determined by the condition

• equilibrium configuration of test perfect fluid rotating around an axis equilibrium configuration of test perfect fluid rotating around an axis of rotation in a given spacetime are determined by the equipotential of rotation in a given spacetime are determined by the equipotential surfaces, where the gravitational and inertial forces are just surfaces, where the gravitational and inertial forces are just compensated by the pressure gradientcompensated by the pressure gradient

• the the equipotentialequipotential surfaces can be closed or open, moreover, there is surfaces can be closed or open, moreover, there isa special class of critical, self-crossing surfaces (with a cusp), which a special class of critical, self-crossing surfaces (with a cusp), which can be either closed or opencan be either closed or open

const , rW

Equilibrium configurationsEquilibrium configurations

• the closed equipotential surfaces determine stationary equilibrium the closed equipotential surfaces determine stationary equilibrium configurationsconfigurations

• the fluid can fill any closed surface – at the surface of the equilibrium the fluid can fill any closed surface – at the surface of the equilibrium configuration pressure vanish, but its gradient is non-zeroconfiguration pressure vanish, but its gradient is non-zero

• configurations with uniform distribution of angular momentum densityconfigurations with uniform distribution of angular momentum density

• relation for the equipotential surfacerelation for the equipotential surfacess

• in Reissner–Nordstrin Reissner–Nordströöm–(anti-)de Sitter spacetimesm–(anti-)de Sitter spacetimes

const , r

,ln , rUrW t

2/1222222

2/1222

//21sin

sin//21ln,;

yrrerr

ryrrereyrW

Behaviour of the equipotential surfaces, and the Behaviour of the equipotential surfaces, and the related potentialrelated potential

• according to the values ofaccording to the values of

• region containing stable circular geodesics -> accretion processesregion containing stable circular geodesics -> accretion processesin the disk regimein the disk regime are possible are possible

• behaviour of potential in the equatorial plane (behaviour of potential in the equatorial plane ( = = /2)/2)

• equipotential surfaces - meridional sectionsequipotential surfaces - meridional sections

const , r

1)1) openopen equipotential equipotential surfaces surfaces only only, no dis, no disks are possibleks are possible

2)2) an infinitesimally thin unstable ring existsan infinitesimally thin unstable ring exists

3)3) equilibrium configurations are possibleequilibrium configurations are possible,, closed closed equipotential equipotential surfaces exist, one surfaces exist, one with the cuspwith the cusp that enables accretion from the toroidal disk into the black hole that enables accretion from the toroidal disk into the black hole

AAdS-dS-BHBH--11: : MM = 1; = 1; ee = 0.99; = 0.99; yy = = - 0.0001- 0.0001

ll = 2.00 = 2.00 ll = 3.10048313 = 3.10048313 ll = 3. = 3.7700

4)4) the potential diverges, the cusp disappearsthe potential diverges, the cusp disappears, , accretion into the black-hole is accretion into the black-hole is impossibleimpossible

5)5) like in the previous caselike in the previous case,, equilibrium configurations are equilibrium configurations are still still possiblepossible,, closed closed equipotential equipotential surfaces existsurfaces exist

AAdS-dS-BHBH--11: : MM = 1; = 1; ee = 0.99; = 0.99; yy = = - 0.0001- 0.0001

ll = 4.03557287 = 4.03557287 ll = 5 = 5.00.00

1)1) closed closed equipotential equipotential surfaces exist, equilibrium configurations are possiblesurfaces exist, equilibrium configurations are possible, one , one disk (1) onlydisk (1) only

2)2) the center of the second disk the center of the second disk (2) (2) appearsappears, , one one equipotential equipotential surface with the cuspsurface with the cusp existexistss

3)3) the flow between the inner disk the flow between the inner disk (1) (1) and the outer oneand the outer one (2) (2) is possibleis possible

AAdS-NS-dS-NS-11: : MM = 1; = 1; ee = 0 = 0..99;99; yy = = - - 0.0.44

ll = 1 = 1..3300 ll = 1.448272709327 = 1.448272709327 ll = 1.465 = 1.465

4)4) the potential divergesthe potential diverges,, no equipotential surface with no equipotential surface with the cusp the cusp exists,exists, the disks are the disks are separatedseparated, , the flow between the disk the flow between the disk 1 1 and the and the disk 2 disk 2 is is imimpossiblepossible

5)5) like in the previous caselike in the previous case,, two separated disks existtwo separated disks exist

6)6) the the disk 2disk 2 is infinitesimally thin is infinitesimally thin

AAdS-NS-dS-NS-11: : MM = 1; = 1; ee = 0 = 0..99;99; yy = = - - 0.0.44

ll = 1.47233576 = 1.47233576 ll = 1 = 1.5.500 ll = 1.58113883 = 1.58113883

7)7) the the disk 1 exists onlydisk 1 exists only, , equilibrium configurations are equilibrium configurations are still still possiblepossible,, closed closed equipotential equipotential surfaces existsurfaces exist

AAdS-NS-dS-NS-11: : MM = 1; = 1; ee = 0 = 0..99;99; yy = = - - 0.0.44

ll = 1.60 = 1.60

1)1) closed closed equipotential equipotential surfaces exist, equilibrium configurations are possiblesurfaces exist, equilibrium configurations are possible, one , one disk (1) onlydisk (1) only

2)2) the center of the second disk the center of the second disk (2) (2) appearsappears, , one one equipotential equipotential surface with the cuspsurface with the cusp existexistss

3)3) the flow between the inner disk the flow between the inner disk (1) (1) and the outer oneand the outer one (2) (2) is possibleis possible

AAdS-NS-2dS-NS-2: : MM = 1; = 1; ee = = 1.01.07;7; yy = = - - 0.0 0.0000011

ll = 2 = 2.00.00 ll = 2.94183736 = 2.94183736 ll = 3 = 3..1100

4)4) the same values of the potential in the centers of both disksthe same values of the potential in the centers of both disks

5)5) the flow between the inner disk the flow between the inner disk (1) (1) and the outer oneand the outer one (2) (2) is possibleis possible

6)6) the diskthe disk 1 1 is infinitesimally thin is infinitesimally thin

AAdS-NS-2dS-NS-2: : MM = 1; = 1; ee = = 1.01.07;7; yy = = - - 0.0 0.0000011

ll = 3.2181567 = 3.2181567 ll = 3.41935796 = 3.41935796ll = 3.30 = 3.30

7)7) the the disk disk 22 exists only exists only, , equilibrium configurations are equilibrium configurations are still still possiblepossible,, closed closed equipotential equipotential surfaces existsurfaces exist

AAdS-NS-2dS-NS-2: : MM = 1; = 1; ee = = 1.01.07;7; yy = = - - 0.0 0.0000011

ll = 4 = 4.00.00

1)1) there is only one center and one disk in this case, closed there is only one center and one disk in this case, closed equipotential equipotential surfaces surfaces exist, equilibrium configurations are possibleexist, equilibrium configurations are possible

AAdS-NS-3dS-NS-3: : MM = 1; = 1; ee = = 1.1.1;1; yy = = - - 0.00.033

ll = 3.00 = 3.00

ConclusionsConclusions

• The Reissner–NordstrThe Reissner–Nordströöm–anti-de Sitter spacetimes can be separated m–anti-de Sitter spacetimes can be separated into four types of spacetimes with qualitatively different character of into four types of spacetimes with qualitatively different character of the geodetical motion. In all of them toroidal disks can exist, becausethe geodetical motion. In all of them toroidal disks can exist, becausein these spacetimes stable circular orbits exist.in these spacetimes stable circular orbits exist.

• The motion above the outer horizon of black-hole backgrounds has the The motion above the outer horizon of black-hole backgrounds has the same character as in the Schwarzschild–anti-de Sitter spacetimes.same character as in the Schwarzschild–anti-de Sitter spacetimes.

• The motion in the naked-singularity backgrounds has similar character The motion in the naked-singularity backgrounds has similar character as the motion in the field of Reissner–Nordstras the motion in the field of Reissner–Nordströöm naked singularities. m naked singularities. Stable circular orbits exist in all of the naked-singularity spacetimes.Stable circular orbits exist in all of the naked-singularity spacetimes.

ReferencesReferences

• Z. Stuchlík, S. Hledík. Properties of the Reissner-NordstrZ. Stuchlík, S. Hledík. Properties of the Reissner-Nordströöm m spacetimes with a nonspacetimes with a nonzero cosmological constant. zero cosmological constant. Acta Phys. SlovacaActa Phys. Slovaca, , 52(5):363-407, 200252(5):363-407, 2002

• Z. Stuchlík,Z. Stuchlík, P. Slan P. Slaný,ý, S. Hledík S. Hledík. Equilibrium configurations of perfect . Equilibrium configurations of perfect fluid orbiting Schwarzschild-de Sitter black holes. fluid orbiting Schwarzschild-de Sitter black holes. Astronomy and Astronomy and AstrophysicsAstrophysics, 363(2):425-439, 2000, 363(2):425-439, 2000

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