Cosmological isotropic matter-energy generalizations of Schwarzschild and Kerr metrics

0.11 MB·

by  Metin Arik

Cosmological isotropic matter-energy generalizations of
Schwarzschild and Kerr metrics
Metin Arik, Yorgo Senikoglu
Department of Physics, Bo˘gazi¸ci University,Bebek, Istanbul, Turkey
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A Abstract

0 We present a time dependent isotropic fluid solution around a Schwarzschild
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black hole. We offer the solutions and discuss the effects on the field equations

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c andthehorizon. Wederivetheenergydensity,pressureandtheequationofstate
q
- parameter. Inthe secondpart,we generalizethe rotatingblackhole solutionto
r
g
anexpandinguniverse. Wederivefromtheproposedmetricthespecialsolutions
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of the field equations for the dust approximation and the dark energy solution.
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We show that the presence of a rotating black hole does not modify the scale
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4 factor b(t)=t2/3 law for dust, nor b(t)=eλt and p= ρ for dark energy.
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4 Keywords: Physics of Black Holes, General Formalism, Exact
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. Solutions
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0 PACS: [2010]04.20.Cv,04.20.Jb, 04.70.-s
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1. Introduction
i
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r In order to describe relativistic spheres in general relativity, one must con-
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sider time dependent solutions. Earlier works for an expanding universe as
exact solutions are widely known to us [1],[2]. Arakida[3] presented the dom-
inant effects due to cosmological expansion, and the use of a time dependent
spacetime model. Bhattacharya et al., [4] introduced a cosmological model
that describes isotropic expansion of inhomogeneous universe, and showed the
Email addresses: [email protected] (MetinArik),
[email protected] (YorgoSenikoglu)
Preprint submitted to- April 21, 2016
energy-momentumtensorthatcreatesthespatialinhomogeneitymaynotaffect
theuniformexpansionscalingfactora(t)intheFLRW-likemetrics. Zhangetal,
[5] obtained anexact time-dependent spherically symmetric solutiondescribing
gravitationalcollapsetoastaticscalar-hairyblackhole. Joshiet al.,[6]pointed
thatthe blackholesandnakedsingularities,whichareconsequencesofthe gen-
eral theory of relativity as we consider the gravitational collapse of a massive
star, would appear to be the leading candidates to explain the very high en-
ergyastrophysicalphenomenabeing observed. McVittie[7],[8]incorporatedtwo
solutions to depict a spherically symmetric metric that describes a point mass
embedded in an expanding spatially-flat universe. The motivation that drove
us to this part of the paper was the possibility if a solution satisfying only the
isotropy condition existed in a non-static spacetime.
Onthe otherhand, it is wellknownthat the gravitationalfieldofa rotating
black hole is described by the Kerr metric[9],[10]. Vaidya et al., [11] have ob-
tained an exact solution of the Einstein’s equations which describes the field of
a radiating Kerr particle embedded in Einstein static universe. Vaidya [12] has
alsogivenaverygeneralformoftheKerr-SchildmetricsatisfyingtheEinstein’s
fieldequations. Tzounisetal.,[13]exploredthepropertiesoftheergoregionand
the location of the curvature singularities for the Kerr black hole distorted by
the gravitational field of external sources and they also studied the scalar cur-
vature invariants of the horizon and compared their behaviour with the case
of the isolated Kerr black hole. Lake et al., [14] examined the global structure
of the family of Kerr-de Sitter spacetimes. Abdelqader et al., [15] presented
an invariant characterization of the physical properties of the Kerr spacetime
and introduced two dimensionless invariants, constructed out of some known
curvatureinvariants,thatactasdetectorsfortheeventhorizonandergosurface
of the Kerr black hole. Gibbons et al., [16] discussed the global structure of
the metrics,andobtainformulaeforthe surfacegravitiesandareasofthe event
horizons. There is a considerable interest in generalizing the Kerr metric to
describe the non-static field of a rotating star, that is why the main aim of the
secondpartistoderiveanexactsolutionofEinstein’sequationswhichdescribes
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the field of a Kerr source embedded in a rotating expanding universe.
The organization of this article is as follows. First we will derive a time
dependent isotropic metric with an energy momentum tensor whose diagonal
partisisotropicandcontainsaradialmomentumandinthe secondpartatime
dependentparametersolutionthatgeneralizestheKerrsolutioninanexpanding
universe.
2. A cosmological generalization of the Schwarzschild black hole
We assume a form of the metric motivated by the Schwarzschild black hole in
isotropic coordinates by replacing the mass parameter by an arbitraryfunction
of r and t. Then we will impose the equality of the spatial diagonal elements
of the energy-momentum tensor. Let us describe this by considering the fol-
lowing metric in a non-static spacetime generalized by a function f(r,t) and a
cosmologicalscale factor a(t).
(1 f(r,t))2
ds2 = − dt2 a(t)2(1+f(r,t))4(dr2+r2dΩ2)
(1+f(r,t))2 −
where dΩ2 =dθ2+sin2θdφ2. (1)
The calculations lead us to the following non-vanishing components of the Ein-
stein tensor in the orthonormal basis G , G , G and G =G .
tt tr rr θθ φφ
We will not bother the reader with the long and exhausting details of these
components at this point. The only condition that we impose is the following,
in the orthonormalbasis:
G =G =G . (2)
rr θθ φφ
The non-vanishing components of the Einstein tensor are presented in the Ap-
pendix A. The unique solution that satisfies (2) is:
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f(r,t)= . (3)
m(t)2r2+2n(t)
p
We notice that at r=r we have f(r,t)=1, where r is the horizon.
H H
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We denote p and ρ respectively the pressure and density. As r r , we
H
→
set f˙(r,t) 0 so that p is non-singular at the horizon, this gives:
→ ρ
1
f(r,t)= . (4)
m(t)2(r2 r2 )+1
− H
p
With (1) and (4) the non-vanishing components of the Einstein tensor in the
orthonormalbasis are presented in the Appendix B.
We note that as r we have:
→ ∞
a˙2 a¨ a˙2
G =3 and G =G =G = 2 . (5)
tt a2 rr θθ φφ − a − a2
And as r r :
H
→
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G = (f˙a+a˙)2 and G =G =G = af˙(f˙a+a˙).
tt a2(1 f)2 rr θθ φφ −a2(1 f)3
− −
(6)
For further purposes we denote the equation of state parameter:
p 2 af˙
ν = =
ρ −3(1 f)f˙a+a˙
−
with G =G =G =p and G =ρ.
rr θθ φφ tt
What we observe for the equation of state parameter and the function m(t) as
r r is:
H
→
3ν
m2(t)= ln(a(t)). (7)
−2r2
H
We can see that for this solution to exist for an expanding universe we need
ν 0. e.g. the dark energy solution p= ρ.
≤ −
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3. A cosmological generalization of the Kerr black hole
The Kerr metric describes the geometry of spacetime in the vicinity of a
mass M rotating with angular momentum J. We add to this metric in Boyer-
Lindquist coordinates a scale factor that we denote b(t). The metric can be
written as follows:
2Mr 4Mrαsin2θ ρ2
ds2 = (1 )dt2 b(t) dφdt+b(t)2 dr2
− − ρ2 − ρ2 ∆
(cid:0) (cid:1) (cid:0) (cid:1)
2Mrα2sin2θ
+b(t)2(ρ2)dθ2+b(t)2(r2+α2+ )sin2θdφ2. (8)
ρ2
where α= J , ρ2 = r2+α2cos2θ and ∆= r2 2Mr+α2. The non-vanishing
M −
components of the Einstein Tensor in the orthonormal basis are: G , G =
tt rr
G =G , G ,G , G , G and G .
θθ φφ tr tθ tφ rφ θφ
Dust approximation. Matter dominated Universe is modeled by dust approxi-
mation. The matter is approximated as dust particles which produce no pres-
sure. Solving for:
G =G =G =0. (9)
rr θθ φφ
we obtain:
b(t)=b t2/3 (10)
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where b is a constant. Now for the non-vanishing components of the Einstein
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Tensor in the orthonormal basis we get:
4 ρ2 4 M(α2+r2)(α2cos2θ r2)
G = ,G = −
tt 3t2 r2 2Mr+α2cos2θ tr −3b1t5/3 ∆(r2 2Mr+α2cos2θ)ρ4
− −
4 Mrα2sin2θ p
G = ,G =0
tθ −3b1t5/3 r2 2Mr+α2cos2θ)ρ4 tφ
−
4 αMsin2pθ(3r4+α2r2+α2r2cos2θ α4cos2θ)
G = − and
rφ −3b1t5/3ρ4 ∆sin2θ((α2cos2θ)∆+r4+2Mrα2+α2r2)
4 p α3Mr(sin2θ)(sin2θ)
G = . (11)
θφ 3b1t5/3ρ4 sin2θ((α2cos2θ)∆+r4+2Mrα2+α2r2)
p
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Dark Energy. For a dark energy dominated universe let us solve the equation:
G = G = G = G . (12)
tt rr θθ φφ
− − −
we obtain the solution:
b(t)=b0e√Λ3 t. (13)
whereb is aconstant. Thecomponentsofthe EinsteinTensorinthe orthonor-
0
mal basis are:
Λ (∆)(α2cos2θ)+r4+2Mrα2+α2r2
G = G = G = G = ,
tt − rr − θθ − φφ (cid:0) ρ2 ∆ (cid:1)
2 M(α2+r2)(α2cos2θ r2)(√3Λe−√Λ3 t)
G = − ,
tr −3b0 ∆(r2 2Mr+α2cos2θ)ρ4
−
2 p(Mrα2sin2θ)(√3Λe−√Λ3 t)
G = ,
tθ −3b0 √r2 2Mr+α2cos2θρ4
−
(2αMrΛsin2θ)((α2cos2θ)∆+r4+2Mrα2+α2r2)
G = ,
tφ
ρ2∆ sin2θ((α2cos2θ)∆+r4+2Mrα2+α2r2)(r2 2Mr+α2cos2θ)
−
2pαMsin2θ(3r4+α2r2+α2r2cos2θ α4cos2θ)(√3Λe−√Λ3 t)
G = − and
rφ −3b0 ρ4 ∆sin2θ((α2cos2θ)∆+r4+2Mrα2+α2r2)
p2 α3Mr(sin2θ)(sin2θ)(√3Λe−√Λ3 t)
G = . (14)
θφ 3b0ρ4 sin2θ((α2cos2θ)∆+r4+2Mrα2+α2r2)
p
4. Discussion of Solutions
For the Schwarzschild black hole solution that we have presented, a diag-
onal energy-momentum tensor satisfying G = G = G corresponds to
rr θθ φφ
an isotropic and homogeneous fluid. An energy momentum tensor satisfying
G = G = G with the only non-diagonal non-vanishing term G means
rr θθ φφ tr
thatwecanstilltalkofapressurebutthereisanon-vanishingradialmomentum
density corresponding to the infalling matter.
For the Kerr black hole, it is desirable to introduce a mass M rotating with
angularmomentumJ where bothJ andM aretime dependentbut inthis case
the equations do not yield any solution. By calculating the energy-momentum
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tensor, we have obtained some off-diagonal terms G ,G ,G which are mo-
tr tθ tφ
mentumdensitycomponents,andsomeshearstresslikeG andG . We have
rφ θφ
presented our work as a first attempt to describe also these shortcomings.
Ontheotherhand,itiswell-knownthatintroducingacosmologicalconstant
is equivalent to adding a part to the energy-momentum tensor satisfying G =
tt
G = G = G . The solutions that we get by solving the Einstein
rr θθ φφ
− − −
equations with a cosmological constant are the same; only the interpretation
of the energy-momentum tensor is different. In our presentation we choose a
language where there is no cosmological constant but where a term satisfying
G = G = G = G in the energy-momentum tensor is allowed.
tt rr θθ φφ
− − −
In the recent observations that have demonstrated the existence of a binary
stellar-massblackholesystem[17]andthefirstobservationofabinaryblackhole
merger,it has been pointedout that the angularmomentum observedwasvery
high. Thismaysupportadetailofourworkconcerningtheangularmomentum
components that we have noted in calculating the Einstein tensor. Since we
believe that there is a black hole at the center of each galaxy and that the
rotation axes of galaxies are not isotropically distributed in space, an isotropic
and homogeneous cosmology does not seem to be realistic.
5. Conclusion
We have studied the effects of a time dependent isotropic metric with an
energymomentumtensorwhosediagonalpartis isotropicandcontainsaradial
momentum. Consequently, field equations were presented for various cases to
illustrate our point of view. We have derived the energy, pressure and the
equation of state parameter ν that satisfied the condition G = G = G
rr θθ φφ
and obtained a non-singular expression for p that binds the scale factor, r , ν
ρ H
and the function figuring in the metric proposed for the latter condition.
FromamoregeneralizedKerrmetricinBoyer-Lindquistcoordinatesthatwe
haveproposedinthispaper,wehaveshownthat,dependingongivenconditions,
wecanobtainthecharacteristicsofthescalefactorb(t)forthematterdominated
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universe(p=0,matterdominateddustapproximation)andthevacuum(dark)
energydominateduniverse(p= ρ). Wenoticetheknownresultsthatforα
− →
0theBoyer-LindquistlineelementreproducestheSchwarzschildlineelementin
an expanding universe and for M 0 the flat expanding Minkowski space.
→
For a matter dominated universe, we observe G = 0 indicating no angu-
tφ
lar momemtum in the azimuthal direction. We note that as r , G is
tt
→ ∞
independent of r and G 1 , G 1 , G 1 , G 1 and
tt ∼ t2 tr ∼ r2t53 tθ ∼ r4t53 rφ ∼ r3t53
G 1 . For a dark energy dominated universe, we see that G , G ,
θφ 5 tt rr
∼ r5t3
G , G and G are independent of time. G 1 , G 1 ,
θθ φφ tφ tr ∼ r4e√Λ3 t tθ ∼ r4e√Λ3 t
G 1 , G 1 and G 1 .
tφ ∼ r2 rφ ∼ r3e√Λ3 t θφ ∼ r5e√Λ3 t
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6. Appendices
Appendix A.
(1 f(r,t))2
ds2 = − dt2 a(t)2(1+f(r,t))4(dr2+r2dΩ2) where dΩ2 =dθ2+sin2θdφ2.
(1+f(r,t))2 −
Weobtainthefollowingnon-vanishingcomponentsofthe Einsteintensorinthe
orthonormalbasis:
1 a˙2 a˙ a˙2 a˙ a˙2
G = 3 (f7)+(12 f˙+21 )(f6)+(12f˙2+72 f˙+63 )(f5)
tt (1+f)5(1 f)2(cid:18) a2 a a2 a a2
−
a˙ a˙2 a˙ a˙2
+(60f˙2+180 f˙+105 )(f4)+(120f˙2+240 f˙+105 )(f3)
a a2 a a2
a˙ a˙2 f f a˙ a˙2 f f
+(120f˙2+180 f˙+63 4 r,r 8 r )(f2)+(60f˙2+72 f˙+21 +8 r,r+16 r )(f)
a a2− a2 − ra2 a a2 a2 ra2
a˙ a˙2 f f
+(12f˙2+12 f˙+3 4 r,r 8 r )
a a2 − a2 − ra2 (cid:19)
4af f +af (1 f)+a˙f
G = r t r,t − r
tr − a2(1+f)2(1 f)2
−
1 a¨ a˙2 a˙ a¨ a˙2 a˙ a¨ a˙2
G = (2 + )(f8)+(4f¨+12 f˙+12 +6 )(f7)+(8f˙2+20f¨+56 f˙+28 +14 )(f6)
rr −(1+f)5(1 f)3(cid:18) a a2 a a a2 a a a2
−
a˙ a¨ a˙2 a˙ a¨ a˙2 f
+(24f˙2+36f¨+84 f˙+28 +14 )(f5)+(20f¨)(f4)+( 80f˙2 20f¨ 140 f˙ 28 14 4 r )(f3)
a a a2 − − − a − a− a2− ra2
a˙ a¨ a˙2 f f2 a˙ a¨ a˙2 f f2
+( 120f˙2 36f¨ 168 f˙ 28 14 +8 r 4 r)(f2)+( 72f˙2 20f¨ 84 f˙ 12 6 4 r +8 r)(f)
− − − a − a− a2 ra2− a2 − − − a − a− a2− ra2 a2
a˙ a¨ a˙2 f2
+( 16f˙2 4f¨ 16 f˙ 2 4 r)
− − − a − a − a2 − a2 (cid:19)
1 a¨ a˙2 a˙ a¨ a˙2
G =G = (2 + )(f8)+(4f¨+12 f˙+12 +6 )(f7)
θθ φφ −(1+f)5(1 f)3(cid:18) a a2 a a a2
−
a˙ a¨ a˙2
+(8f˙2+20f¨+56 f˙+28 +14 )(f6)
a a a2
a˙ a¨ a˙2 a˙ a¨ a˙2 f f
+(24f˙2+36f¨+84 f˙+28 +14 )(f5)+(20f¨)(f4)+( 80f˙2 20f¨ 140 f˙ 28 14 2 r 2 rr)(f3)
a a a2 − − − a − a− a2− ra2− a2
a˙ a¨ a˙2 f f2 f
+( 120f˙2 36f¨ 168 f˙ 28 14 +4 r +2 r +4 rr)(f2)
− − − a − a − a2 ra2 a2 a2
a˙ a¨ a˙2 f f2 f
+( 72f˙2 20f¨ 84 f˙ 12 6 2 r 4 r 2 rr)(f)
− − − a − a − a2 − ra2 − a2 − a2
a˙ a¨ a˙2 f2
+( 16f˙2 4f¨ 16 f˙ 2 +2 r)
− − − a − a − a2 a2 (cid:19)
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Appendix B. f(r,t)= 1
√m(t)2(r2−rH2)+1
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G = f3(2mrm˙ a+a˙2m2r)+f4( 2mm˙ ra)+f5( 3m3r(r2 r2 )m˙ a)+
tr a2(1+f)2(1 f)2(cid:18) − − − H
−
f6(2m3r(r2 r2 )m˙ a)
− H (cid:19)
1 a˙2 a˙2 a˙2 a˙2 a˙ m2
G = 3 +(21 )f+(63 )f2+(105 12 mm˙ (r2 r2 )+12 )f3
tt (1+f)5(1 f)2(cid:18) a2 a2 a2 a2− a − H a2
−
a˙2 a˙ m2 a˙2 a˙ m2 r2m4
+(105 72 mm˙ (r2 r2 ) 24 )f4+(63 180 mm˙ (r2 r2 )+12 12 )f5
a2− a − H − a2 a2− a − H a2 − a2
a˙2 a˙ r2m4
+(21 240 mm˙ (r2 r2 )+12m2m˙ 2(r4+r4 ) 24r2m2m˙ 2r2 +24 )f6
a2 − a − H H − H a2
a˙2 a˙ r2m4
+(3 180 mm˙ (r2 r2 )+60m2m˙ 2(r4+r4 ) 120r2m2m˙ 2r2 12 )f7
a2 − a − H H − H− a2
a˙
+( 72 mm˙ (r2 r2 ) 240r2m2m˙ 2r2 +120m2m˙ 2(r4+r4 ))f8
− a − H − H H
a˙
+( 12 mm˙ (r2 r2 ) 240r2m2m˙ 2r2 +120m2m˙ 2(r4+r4 ))f9
− a − H − H H
+( 120r2m2m˙ 2r2 +60m2m˙ 2(r4+r4 ))f10
− H H
+( 24r2m2m˙ 2r2 +12m2m˙ 2(r4+r4 ))f11 and
− H H (cid:19)
1 a¨ a˙2 a¨ a˙2 a¨ a˙2
G =G =G = 2 + +6(2 + )f+14(2 + )f2
rr θθ φφ −(1+f)5(1 f)3(cid:18) a a2 a a2 a a2
−
a¨ a˙2 a˙
+(14(2 + ) 4(r2 r2 )(m¨m+m˙ 2+4mm˙ ))f3
a a2 − − H a
a˙ m2
4((r2 r2 )(5m¨m+5m˙ 2+21mm˙ )+ )f4+
− − H a a2
a¨ a˙2 a˙ m2
( 14(2 + ) 4(r2 r2 )(9m¨m+9m˙ 2+42mm˙ )+12m2m˙ 2(r4+r4 ) 24m2m˙ 2r2r2 +8 )f5
− a a2 − − H a H − H a2
a¨ a˙2 a˙ m2
+( 14(2 + ) 20(r2 r2 )(m¨m+m˙ 2+7mm˙ )+76m2m˙ 2(r4+r4 ) 152m2m˙ 2r2r2 +4 (m2r2+1))f6
− a a2 − − H a H − H a2
a¨ a˙2 r2m4
+( 6(2 + ) 20(r2 r2 )(m¨m+m˙ 2)+180m2m˙ 2(r4+r4 ) 360m2m˙ 2r2r2 8 )f7
− a a2 − − H H − H− a2
a¨ a˙2 a˙ r2m4
+( (2 + )+4(r2 r2 )(9m¨m+9m˙ 2+21mm˙ )+180m2m˙ 2(r4+r4 ) 360m2m˙ 2r2r2 +4 )f8
− a a2 − H a H − H a2
a˙
+(4(r2 r2 )(5m¨m+5m˙ 2+14mm˙ )+20m2m˙ 2(r4+r4 ) 40m2m˙ 2r2r2 )f9
− H a H − H
a˙
+(4(r2 r2 )(m¨m+m˙ 2+3mm˙ ) 108m2m˙ 2(r4+r4 ) 216m2m˙ 2r2r2 )f10
− H a − H − H
+(168m2m˙ 2r2r2 84m2m˙ 2(r4+r4 ))f11+(40m2m˙ 2r2r2 20m2m˙ 2(r4+r4 ))f12 .
H− H H− H (cid:19)
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