Another Kerr interior solution Fatemeh Zahra Majidi∗
arXiv:1705.00584v2 [gr-qc] 14 May 2017
Department of Physics, University of Tehran, Tehran 14395-547, Iran
Abstract A stationary axially symmetric solution for Einstein Field Equations(EFE) is proposed which describes a rotating anisotropic fluid and matches to exterior Kerr metric. As rotation parameter goes to zero a spherical source consisting of a normal and exotic matter is retrieved for the exterior Schwarzschild solution, the corresponding redshift function, however, is regular. The anisotropic source satisfies all energy conditions - weak, strong, and dominant - for a wide range of metric’s free parameters values. The resultant energy-momentum tensor components, and consequently energy density and pressure profiles, redshift function and angular velocity are singularity free and behave as expected. This is the first solution of its kind.
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Introduction
Ever since 1963 that R. P. Kerr presented his exact solution for EFE [1] numerous attempts for finding a physically reasonable interior solution as its source have been made [2-7]; however, all of them proved to be unsuccessful so far. One can find a valuable review of the proposed solutions until 1978 in [3]. One of the neglected solutions for Kerr metric in this review is Wahlquist fluid [2] which has a nonphysical prolate pressure profile and the other is Gurses and Gursey’s paper [13] which for the very first time suggested an anisotropic source for exterior Kerr solution. The best-known solution for exterior Kerr metric is [4] which describes a rigidly rotating disk of dust which approaches the extreme Kerr solution (J = M 2 ) at a certain limit of the metric parameters. The disk, however, exhibits a discontinuity which was interpreted by the authors as a phase transition from normal matter to the black hole state. One of the best analytic solutions proposed so far can be found in [5] which generates anisotropic disks that match the exterior Kerr metric smoothly and analytic expression were found for their energy density and pressure profiles. The only problem this solution is suffering from is the divergence of central redshift function. In this regard, both solutions experience ultra-relativistic limits at the center of the generated disks. Another impressive solution published recently [8] generates anisotropic oblate sources which match the exterior Kerr metric smoothly, in the spherically symmetric limit the Schwarzschild interior metric [10] is retrieved and for a wide ∗
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range of parameters the solution satisfies strong energy condition (SEC). In the specific ranges that this solution satisfies SEC violates weak energy condition (WEC) and for a specific range satisfying WEC it violates SEC. Another issue which is still in progress is the study of redshift function. As mentioned above and proposed earlier by Collas [9], checking the redshift function - aside from the energy-momentum tensor components - can be the very first test for verifying the validity of the solution. In section 2 of this paper, a stationary axisymmetric anisotropic source is proposed for the Kerr metric (as an ansatz) which matches the Kerr metric at the boundary surface. All the resultant physical variables are regular and behave as one expects from a rotating source; i.e. the angular velocity increases from zero as one moves outward from the center of the body, redshift function remains finite and positive everywhere, energy density and pressure profiles monotonically decrease as one moves from the center of the body toward its surface. In section 3, the spherically symmetric limit of the suggested solution is studied. The spherical source consists of two different fluids and matches the exterior Schwarzschild metric. The resultant redshift function is regular and positive everywhere, although one of the source fluids is exotic.
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The interior metric
The proposed metric is a stationary axisymmetric interior solution for EFE in Boyer-Lindquist [11] coordinates :
ds2 = −[
Σ∆ rs a2 r2 (1 − (2 + cosθ)2 ) 2 2ars r(1 − (2 + cosθ)2 ) − ]dt − dφdt B BΣ Σ +
Σ 2 B(1 − (2 + cosθ)2 ) dr + Σ dθ2 + dφ2 , ∆ Σ
(1)
where: Σ = rg2 − a2 (2 + cosθ)2 , ∆=
rg3 − r(−a2 − rs rg ) , rg
B = (rg2 − a2 )2 + a2 ∆(1 − (2 + cosθ)2 ) , and rs = 2GM/c2 , rg is the rotating body’s radius and a = J/M is the rotation parameter. According to above metric it’s easy to find the angular velocity and redshift function(for photons emitted in a locally non rotating frame) [9]: ω=
ars r , B
r
(2)
B −1. (3) Σ∆ In the same coordinates - for convenience in comparison - one can write the exterior Kerr metric as follows: Z=
2
ds2 = −[
Σ∆ rs a2 r2 sin2 θ 2 2ars rsin2 θ Σ 2 + ]dt − dφdt + dr B BΣ Σ ∆ Bsin2 θ + Σ dθ2 + dφ2 , Σ
(4)
where: Σ = r2 + a2 cos2 θ , ∆ = r2 − rs r + a2 , B = (r2 + a2 )2 − a2 ∆sin2 θ , in which rs = 2GM/c2 and a = J/M is the rotation parameter. Metric (1) can be matched to exterior Kerr metric via a thin shell [12] with curious features. This thin shell has jumps in the rotation parameter and cosθ function: it is easy to see that at boundary surface by substituting the exterior (cosθ)2 with (2 + cosθ)2 and a2 with −a2 the interior solution will be retrieved. The existence of this thin shell in turn implies that Tµν isn’t zero at the surface layer and the two metrics can’t be matched smoothly because of the change of sign in a2 . The change of variables must have a meaningful physical interpretation, but at the moment the suggested solution can be considered as a workable guess for EFE which matches the exterior Kerr metric and in spherically symmetric limit reduces to a source for the exterior Schwarzschild solution. For obtaining the energy density and pressure profiles of the source the following eigenvalue equation should be solved: |Tµν − λi gµν | = 0 ,
(5)
in which −λ0 indicates energy density and λi are pressure profiles to the first order of a. Solving (4) for metric (1) leads to [7] : Ttt − λgtt 0 0 Ttφ − λgtφ 0 Trr − λgrr Trθ 0 =0 0 Trθ Tθθ − λgθθ 0 Ttφ − λgtφ 0 0 Tφφ − λgφφ and the eigenvalues are given by: q 2 )(T T 2 P ± P 2 − 4(gtt gφφ − gtφ tt φφ − Ttφ ) , λ0,1 = 2 ] 2[gtt gφφ − gtφ P = (Ttt gφφ + gtt Tφφ − 2Ttφ gtφ ) , λ2,3 =
Q±
p
2 ) Q2 − 4grr gθθ (Trr Tθθ − Trθ , 2grr gθθ
Q = (Trr gθθ + grr Tθθ ) . Regarding energy conditions, for the weak energy condition eigenvalues should satisfy: 3
−λ0 ≥ 0 , −λ0 + λi ≥ 0 ; for the strong energy condition : −λ0 +
X
λi ≥ 0 , −λ0 + λi ≥ 0 ;
i
and for the dominant energy condition : −λ0 ≥ 0 , λ0 ≤ λi ≤ −λ0 . Determining an equation of state is a rather complicated task, so instead a practical example is studied. The free parameters are fixed according to the available information on planet earth(in SI units): M = 5.972 × 1024 kg , a = J/M = 1016 m2 /s , rg = 106 m , and therefore rs = 0.001 N s2 /kg . Regarding the change of variables, the usual 0 ≤ θ ≤ Π range should change to 0 ≤ θ ≤ 2Π. The energy density, pressure profiles, redshift function, and angular velocity are plotted according to fixed free parameters:
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Figure 1 : The energy density, pressure profiles, angular velocity and redshift function diagrams for a toy model with the same mass, radius, and angular momentum as planet earth. All diagrams are singularity free.
Now it’s easy to check the energy conditions via below graphs:
P Figure 2 : Graphs for checking WEC (−λ0 + λi ≥ 0) and SEC (−λ0 + i λi ≥ 0 and −λ0 + λi ≥ 0). Regarding that the energy density is positive everywhere and λ1 , λ2 , and λ3 are all smaller in amount compared to energy density, strong,weak and dominant energy conditions are satisfied.
Table(1) indicates a sample with the same mass as planet earth but different radius and rotation parameter. The aim of this table is to exhibit in what range of parameter values energy conditions are satisfied. 5
3
The spherically symmetric limit
As rotation parameter goes to zero an interior spherically symmetric metric is achieved: ds2 = −[
rg2 − rs r 2 rg2 ]dt + dr2 + rg2 dθ2 + r2 (1 − (2 + cosθ)2 ) dφ2 , (6) rg2 rg2 − rs r
that again matches the exterior Schwarzschild metric: ds2 = −(1 −
rs 2 rs )dt + (1 − )−1 dr2 + r2 (dθ2 + sin2 θdφ2 ) , r r
(7)
via a thin shell on which the exterior (cosθ)2 turns to (2 + cosθ)2 in spherical coordinates. Metric (6) describes a fluid which experiences two distinct regimes: rg ≤ rs and rg > rs which is evident from the metric structure and energymomentum tensor. The corresponding redshift function is:
Z=
q rg2 − rs r rg
−1.
(8)
This function generates negative values for rg ≤ rs regime, so it can be considered non-physical1 . As the metric is diagonal the energy density and pressure profiles can be read directly from energy momentum tensor components:
Ttt = ρ/c2 =
(rg2 − rs r)(3cosθ + 5cos2 θ + 3cos3 θ + cos4 θ − 2sin2 θ) , rg4 (1 − (2 + cosθ)2 )2
Trr = P = −
3cosθ + 5cos2 θ + 3cos3 θ + cos4 θ − 2sin2 θ , (rg2 − rs r)(1 − (2 + cosθ)2 )2
(9)
(10)
and the Tθθ and Tφφ components are both zero. The following graphs indicate the energy density, radial pressure, and redshift function for a spherical planet with rs = 10−6 N s2 /kg and rg = 106 m :
1 The
corresponding energy density and pressure profile are only different in a minus sign from the rg > rs regime. It is interesting that even in rg ≤ rs regime the redshift function is singularity free.
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Figure 3 : The energy density, radial pressure and redshift function for a spherical planet with rg = 106 m , and rs = 10−6 N s2 /kg . It is evident from the energy density and radial pressure graphs that the source consists of three sections, two which violate SEC, WEC, and DEC because the energy density is negative and one that satisfies all energy conditions. These sections can change from two to one and one to two according to rs and rg values. It is interesting that although it seems the source consists of one or two sections of exotic matter the redshift function behaves regularly everywhere. Therefore, the spherical body doesn’t experience an ultra-relativistic limit or a phase transition from normal matter to black hole state as explained in the introduction.
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conclusion
The proposed solution matches the Kerr metric on a thin shell - not smoothly - hence, and energy-momentum tensor is non- zero on the boundary surface. The corresponding redshift function is positive definite and regular everywhere. The resultant energy density and pressure profiles satisfy SEC, WEC, and DEC simultaneously; so the suggested solution can successfully describe a rotating stationary axisymmetric body. The energy-momentum tensor, redshift function, and angular velocity are all singularity free. In the spherically symmetric limit the interior metric matches exterior Schwarzschild metric on a thin shell and again the energy-momentum tensor isn’t zero on the surface layer. The source consists of three sections which at least one of them violates all energy conditions- strong, weak, and dominant- and one satisfies them all. The corresponding redshift, however, is regular and positive everywhere so the exotic matter can’t be interpreted as a phase transition from normal matter to the black hole state; i.e. the source doesn’t experience the ultra-relativistic limit nor in the spherical body neither on the surface.
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a (m2 /s)/rg (m)
1
10
102
103
104
105
106
107
108
103 (m) 104 (m) 105 (m) 106 (m) 107 (m) 108 (m)
V V V V V V
V V V V V V
V V V V V V
S S V V V V
S S V V V V
S S S V V V
S S S S V V
S S S S S V
S S S S S V
Table 1 : a versus rg for a rotating body with rs = 0.001 N s2 /kg . ”S” indicate that SEC, WEC, and DEC are satisfied while ”V” is the sign of energy conditions’ violation.
References [1] R.P. Kerr, Phys. Rev.Lett. 11 (1963) 273-238 . [2] H . Wahlquist , Phys. Rev. 172 ( 1968) 1291 . [3] A. Krasinski, Annals of physics, Vol.112, No.1.(1978) . [4] G. Neugebauer and R. Meinel , Phys.Rev.Lett. 73 (1994) 2166-2168 . [5] C. Pichon and D. Lynden-Bell , Mon. Not. R. Astron. Soc 280 (1996) 1007. [6] S. P. Drake and R. Turolla, Classical Quantum Gravity 14 ( 1997) 1883. [7] S. Viaggiu, Int.J.Mod.Phys.D15 (2006)1441-1453 . [8] J.L. Hernandez-Pastora and L. Herrera, Phys. Rev.D 95 (2017) 024003. [9] P. Collas, Lett.Nuovo Cim. 22 (1978) 472. [10] K. Schwarzschild, Sitzungsberichte der K¨oniglich-Preussischen Akademie der Wissenschaften (1916) 424–434. [11] R. H. Boyer and R. Lindquist, Journ. Math. Phys 8 (1967) 265. [12] E. Poisson, A Relativist’s Toolkit: The Mathematics Of Black-Hole Mechanics (2004) Chapter3. [13] M. Gurses and F. Gursey, J. Math. Phys. 16 (1975) 2385 .
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