Equilibrium configurationsEquilibrium configurationsof perfect fluidof perfect fluid

in Reissner-Nordstrin Reissner-Nordströöm de Sitter m 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 9RAGtime 919.-21. September 2007, Hradec nad Moravicí19.-21. September 2007, Hradec nad Moravicí

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 witha repulsive cosmological constant (a repulsive 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 de Sitter m de Sitter spacetimesspacetimes

• seven types with qualitatively different behavior of the effective seven 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• dS-BH-1dS-BH-1 – one region of circular geodesics at – one region of circular geodesics at r r > > rrph+ ph+ with unstable with unstable

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

• dS-BH-2dS-BH-2 – one region of circular geodesics at – one region of circular geodesics at r r > > rrph+ ph+ with unstable with unstable

geodesics onlygeodesics only

2/12

ph 9

811

2

3)(

eer

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

Naked-singularity spacetimesNaked-singularity spacetimes• dS-NS-1dS-NS-1 – – ttwo regions of circular geodesics, the inner region consists wo regions of circular geodesics, the inner region consists

of stable geodesics onlyof stable geodesics only, , the outer one contains subsequently unstable, the outer one contains subsequently unstable, then stable and finally unstable circularthen stable and finally unstable circular geodesicsgeodesics

• dS-NS-2dS-NS-2 – – ttwo regions of circular orbits, the inner one consist of wo regions of circular orbits, the inner one consist of stable orbits, the outer onestable orbits, the outer one ofof unstable orbitsunstable orbits

• dS-NS-3dS-NS-3 – – oone region of circular orbits, subsequently with stable, ne region of circular orbits, subsequently with stable, unstable, then stable and finallyunstable, then stable and finally unstable orbitsunstable orbits

• dS-NS-4dS-NS-4 – – oone region of circular orbits with stable and then unstable ne region of circular orbits with stable and then unstable orbitsorbits

• dS-NS-5dS-NS-5 – – nno circular orbits allowedo circular orbits allowed

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–de Sitter spacetimesm–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) open surfacesopen surfaces only only, no dis, no disks are possibleks are possible, surface with the outer cusp exists, surface with the outer cusp exists

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

3)3) closed surfaces exist, many equilibrium configurations without cusps are possible, closed surfaces exist, many equilibrium configurations without cusps are possible, one with the inner cuspone with the inner cusp

dS-dS-BHBH--11: : MM = 1; = 1; ee = 0.5; = 0.5; yy = 10 = 10-6-6

ll = 3.00 = 3.00 ll = 3. = 3.5537805355378053 ll = 3. = 3.7575

4)4) there is an equipotential surface with both the inner and outer cusps, the there is an equipotential surface with both the inner and outer cusps, the mechanical nonequilibrium causes an inflow into the black hole, and an outflow mechanical nonequilibrium causes an inflow into the black hole, and an outflow from the disk, with the same efficiencyfrom the disk, with the same efficiency

5)5) accretion into the black-hole is impossible, the outflow from the disk is possibleaccretion into the black-hole is impossible, the outflow from the disk is possible

6)6) the potential diverges, the inner cusp disappearsthe potential diverges, the inner cusp disappears

dS-dS-BHBH--11: : MM = 1; = 1; ee = 0.5; = 0.5; yy = 10 = 10-6-6

ll = = 3.81364253.8136425 ll = = 4.004.00 ll = = 4.967975644.96797564

7)7) the closed equipotential surfaces still exist, one with the outer cuspthe closed equipotential surfaces still exist, one with the outer cusp

8)8) an infinitesimally thin, unstable ring exists (the center, and the outer cusp coalesce)an infinitesimally thin, unstable ring exists (the center, and the outer cusp coalesce)

9)9) open equipotential surfaces exist only, there is no cusp in this caseopen equipotential surfaces exist only, there is no cusp in this case

dS-dS-BHBH--11: : MM = 1; = 1; ee = 0.5; = 0.5; yy = 10 = 10-6-6

ll = = 6.006.00 ll = = 7.110013497.11001349 ll = = 10.0010.00

1)1) closed surfaces exist, one with the outer cusp, equilibrium configurations are closed surfaces exist, one with the outer cusp, equilibrium configurations are possiblepossible

2)2) the second closed surface with the cusp, and the center of the second disk appears, the second closed surface with the cusp, and the center of the second disk appears, the inner disk (1) is inside the outer one (2)the inner disk (1) is inside the outer one (2)

3)3) two closed surfaces with a cusp exist, the inner disk is still inside the outer onetwo closed surfaces with a cusp exist, the inner disk is still inside the outer one

dS-NS-dS-NS-11: : MM = 1; = 1; ee = = 1.021.02;; yy = 0.00001 = 0.00001

ll = = 2.002.00 ll = = 3.043274723.04327472 ll = = 3.153.15

4)4) closed surface with two cusps exists, two disks meet in one cusp, the flow between closed surface with two cusps exists, two disks meet in one cusp, the flow between disk 1 and disk 2, and the outflow from disk 2 is possibledisk 1 and disk 2, and the outflow from disk 2 is possible

5)5) the disks are separated, the outflow from disk 1 into disk 2 only, and the outflow the disks are separated, the outflow from disk 1 into disk 2 only, and the outflow from disk 2 is possiblefrom disk 2 is possible

6)6) the cusp 1 disappears, the potential diverges, two separated disks still existthe cusp 1 disappears, the potential diverges, two separated disks still exist

dS-NS-dS-NS-11: : MM = 1; = 1; ee = = 1.021.02;; yy = 0.00001 = 0.00001

ll = = 3.22268243.2226824 ll = = 3.553.55 ll = = 3.914848033.91484803

7)7) like in the previous case, the flow between disk 1 and disk 2 is impossible, the like in the previous case, the flow between disk 1 and disk 2 is impossible, the outflow from disk 2 is possibleoutflow from disk 2 is possible

8)8) disk 1 exists, also an infinitesimally thin, unstable ring exists (region 2)disk 1 exists, also an infinitesimally thin, unstable ring exists (region 2)

9)9) disk 1 exists only, there are no surfaces with a cuspdisk 1 exists only, there are no surfaces with a cusp

dS-NS-dS-NS-11: : MM = 1; = 1; ee = = 1.021.02;; yy = 0.00001 = 0.00001

ll = = 4.404.40 ll = = 4.94867084.9486708 ll = = 5.155.15

10)10) disk 1 is infinitesimally thindisk 1 is infinitesimally thin

11)11) no disks, open equipotential surfaces onlyno disks, open equipotential surfaces only

dS-NS-dS-NS-11: : MM = 1; = 1; ee = = 1.021.02;; yy = 0.00001 = 0.00001

ll = = 5.395744845.39574484 ll = = 6.006.00

1)1) there is only one center and one disk in this case, closed equipotential surfaces there is only one center and one disk in this case, closed equipotential surfaces exist, one with the cusp, the outflow from the disk is possibleexist, one with the cusp, the outflow from the disk is possible

2)2) the potential diverges, the cusp disappears, equilibrium configurations are possible the potential diverges, the cusp disappears, equilibrium configurations are possible (closed surfaces exist), but the outflow from the disk is impossible(closed surfaces exist), but the outflow from the disk is impossible

3)3) the situation is similar to the previous casethe situation is similar to the previous case

dS-NS-2dS-NS-2: : MM = 1; = 1; ee = = 1.021.02;; yy = 0.01 = 0.01

ll = = 4.004.00 ll = = 4.254031094.25403109 ll = = 5.005.00

4)4) the disk is infinitesimally thinthe disk is infinitesimally thin

5)5) no disk is possible, open equipotential surfaces onlyno disk is possible, open equipotential surfaces only

dS-NS-2dS-NS-2: : MM = 1; = 1; ee = = 1.021.02;; yy = 0.01 = 0.01

ll = = 6.407405256.40740525 ll = = 7.007.00

1)1) closed surfaces exist, one with the outer cusp, equilibrium configurations are closed surfaces exist, one with the outer cusp, equilibrium configurations are possiblepossible

2)2) the second closed surface with the cusp, and the center of the second disk appears, the second closed surface with the cusp, and the center of the second disk appears, the inner disk (1) is inside the outer one (2)the inner disk (1) is inside the outer one (2)

3)3) two closed surfaces with a cusp exist, the inner disk is still inside the outer onetwo closed surfaces with a cusp exist, the inner disk is still inside the outer one

dS-NS-3dS-NS-3: : MM = 1; = 1; ee = = 1.071.07;; yy = 0.0001 = 0.0001

ll = = 2.502.50 ll = = 2.937233422.93723342 ll = = 3.003.00

4)4) closed surface with two cusps exists, two disks meet in one cusp, the flow between closed surface with two cusps exists, two disks meet in one cusp, the flow between disk 1 and disk 2, and the outflow from disk 2 is possibledisk 1 and disk 2, and the outflow from disk 2 is possible

5)5) the disks are separated, the outflow from disk 1 into disk 2 only, and the outflow the disks are separated, the outflow from disk 1 into disk 2 only, and the outflow from disk 2 is possiblefrom disk 2 is possible

6)6) an infinitesimally thin, unstable ring exists (region 1), also disk 2an infinitesimally thin, unstable ring exists (region 1), also disk 2

dS-NS-3dS-NS-3: : MM = 1; = 1; ee = = 1.071.07;; yy = 0.0001 = 0.0001

ll = = 3.04116773.0411677 ll = = 3.203.20 ll = = 3.423317373.42331737

7)7) one cusp, and disk 2 exists only, the outflow from disk 2 is possibleone cusp, and disk 2 exists only, the outflow from disk 2 is possible

8)8) an infinitesimally thin, unstable ring exists (region 2)an infinitesimally thin, unstable ring exists (region 2)

9)9) no disk, no cusp, open equipotential surfaces onlyno disk, no cusp, open equipotential surfaces only

dS-NS-3dS-NS-3: : MM = 1; = 1; ee = = 1.071.07;; yy = 0.0001 = 0.0001

ll = = 3.590081263.59008126 ll = = 3.803.80ll = = 3.503.50

1)1) there is only one center and one disk in this case, closed equipotential surfaces there is only one center and one disk in this case, closed equipotential surfaces exist, one with the cusp, the outflow from the disk is possibleexist, one with the cusp, the outflow from the disk is possible

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

3)3) no disk is possible, no cusp, open equipotential surfaces exist onlyno disk is possible, no cusp, open equipotential surfaces exist only

dS-NS-4dS-NS-4: : MM = 1; = 1; ee = = 1.071.07;; yy = 0.01 = 0.01

ll = 3.00 = 3.00 ll = 3.63788074 = 3.63788074 ll = 3.80 = 3.80

ConclusionsConclusions• The Reissner–NordstrThe Reissner–Nordströöm–de Sitter spacetimes can be separated into m–de Sitter spacetimes can be separated into

seven types of spacetimes with qualitatively different character of the seven types of spacetimes with qualitatively different character of the geodetical motion. In five of them toroidal disks can exist, becausegeodetical motion. In five of them toroidal disks can exist, becausein these spacetimes stable circular orbits exist.in these spacetimes stable circular orbits exist.

• The presence of an outer cusp of toroidal disks nearby the static radius The presence of an outer cusp of toroidal disks nearby the static radius which enables outflow of mass and angular momentum from the which enables outflow of mass and angular momentum from the accretion disks by the Paczyaccretion disks by the Paczyńński mechanism, i.e., due to a violationski mechanism, i.e., due to a violationof the hydrostatic equilibrium. of the hydrostatic equilibrium.

• 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–de Sitter spacetimes for same character as in the Schwarzschild–de Sitter spacetimes for asymptotically de Sitter spacetimes. There is only one static radiusasymptotically de Sitter spacetimes. There is only one static radiusin these spacetimes. No static radius is possible under the inner black-in these spacetimes. No static radius is possible under the inner black-hole horizon, no circular geodesics are possible there.hole horizon, no circular geodesics are possible there.

• 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. However, in the case of Reissner–NordstrHowever, in the case of Reissner–Nordströöm–de Sitter, two static radii m–de Sitter, two static radii can exist, while the Reissner–Nordstrcan exist, while the Reissner–Nordströöm naked singularities contain m naked singularities contain one static radius only. The outer static radius appears due to the effect one static radius only. The outer static radius appears due to the effect of the repulsive cosmological constant. Stable circular orbits exist in all of the repulsive cosmological constant. Stable circular orbits exist in all of the naked-singularity spacetimes. There are even two separated of the naked-singularity spacetimes. There are even two separated regions of stable circular geodesics in some cases.regions of stable circular geodesics in some cases.

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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