Schwarzschild solution of the modified Einstein field equations
aa r X i v : . [ g r- q c ] M a r SCHWARZSCHILD SOLUTION OF THEMODIFIED EINSTEIN FIELD EQUATIONS
Carringtone Kinyanjui, Dismas S. WamalwaMarch 20, 2017
Department of PhysicsUniversity of NairobiChiromo, Nairobi, 30197-0100, KENYA
Abstract
A reformulation of the Schwarzschild solution of the linearized Einsteinfield equations in post-Riemannian Finsler spacetime is derived. The solu-tion is constructed in three stages: the exterior solution, the event-horizonsolution and the interior solution. It is shown that the exterior solution isasymptotically similar to Newtonian gravity at large distances implyingthat Newtonian gravity is a low energy approximation of the solution.Application of Eddington-Finklestein coordinates is shown to reproducethe results obtained from standard general relativity at the event hori-zon. Further application of Kruskal-Szekeres coordinates reveals that theinterior solution contains maximally extensible geodesics.
AMS Subject Classification:
Key Words and Phrases:
Einstein field equations, Schwarzschildmetric, Black holes, Finsler spaces
The general theory of relativity as proposed by Einstein [1] relates the curvatureof spacetime to gravity. The theory describes the relation between the curvatureof spacetime to the energy of an object. This can succinctly be described byEinstein field equations (stated without the cosmological constant) [2]: R µν − g µν R = 8 πGc T µν where R µν is the Ricci curvature tensor, R is the Ricci scalar, g µν is the metric,G is the gravitational constant, T µν is the stress energy tensor and c is the speedof light. The theory was soon tested observationally by Eddington in 1919 andfound to be correct [3]. The simplest analytical solution to the field equationsis the solution for a static uncharged and spherical mass. The solution wasproposed by Karl Schwarzschild in 1916 [4]. The Schwarzschild metric for pathsalong radial lines is given by [2]: d s = (cid:18) − m r (cid:19) − d t + (cid:18) − m r (cid:19) d r (1)1t can be seen that the metric is singular at r = 2 m and r = 0 [5]. However,a change of coordinates particularly proposed by Eddington [6] and later byFinklestein [7], showed that the singularity at r = 2 m can be removed. However,the curvature singularity due to spacetime structure at r = 0 persists and cannot be removed [5].The interior solution ( r < m ) describes objects called “blackholes” that“swallow” objects that come too close to them ( r = 2 m ) and have escape ve-locities greater than the speed of light. This means that at r < m , there is nopossibility of escape [2]. While scientific speculation on the existence of blackholes predates general relativity [8], they seem implicit within the description ofthe theory. Initially, blackholes were not accepted as physically feasible objects.According to Einstein, the solution was in fact only a mathematical curiosityand of no astrophysical importance [9]. Chandrasekhar [10], showed that be-yond some mass limit, stars at the endpoints of stellar evolution and thereforeundergoing gravitational collapse can not be held back by electron degeneracy.Later work by Oppenheimer, Volkoff and Tolmann generalized Chandrasekhar’swork [11] and proved that collapse to a blackhole is astrophysically feasible.Since then, the reality of astrophysical black holes has been confirmed [12], [13].Important theoretical work in the description of blackholes, both from gen-eral relativity and from a quantum field theoretic perspective has been done byHawking et. al. [14], [15]. It is hoped that quantum gravity will resolve thesingularity problem in the centre of stationary blackholes. Gambini and Pullin[16], [17] have proposed that Loop Quantum Gravity can lead to the descriptionof a non-singular quantised Schwarzschild metric . In this work, we shall take adifferent approach from the (relatively) standard gravity quantisation procedureas a correction of gravity at the scale of black hole energies. We shall rely on theresults of the model formulated within the frame work of finsler spaces. Finslerspaces [18], [19] can be thought of as generalizations of Riemann spaces. InParticular, we adopt the extended Einstein field equations in Finsler geometry[20]: (cid:18) R µν − g µν R (cid:19) + (cid:18) S µν − g µν S (cid:19) = 8 πGc ( T µν + τ µν ) (2)where S µν and S are additional Ricci tensor and Ricci scalar terms respectively.Furthermore, S µν and R µν are functions of position, x and velocity, y respec-tively. We hold the velocity terms constant so that R µν ( x , y ) = R µν ( x ) and S µν ( x , y ) = S µν ( x ). It should be noted that the field equations in equation (2)above are difficult to solve. We shall therefore proceed to solve the above fieldequations with an extra constraint of symmetry and hence develop a mathe-matical model of Schwarzschild blackholes appropriately. In this respect, weshall assume that components of the tensors R µν and S µν are related througha symmetrical linear transformation to be described later. We can rewrite Einstein field equations in equation (2) above as: Z αβ − Zg αβ = kQ αβ (3)2here Z αβ = R αβ + S αβ (4) Z = R + S (5) Q αβ = T αβ + τ αβ (6)Transforming equation (3) into covariant form, we obtain Z µν − Zg µν = kQ µν (7)We shall now proceed to find the vacuum field equations corresponding to equa-tion (7). Setting Q µν = 0 and performing contraction with metric tensor, it canbe shown that R µν + S µν = 0 (8)These are our desired vacuum field equations. We next consider a solution ofthe vacuum field equations for the Schwarzschild metric. The difficulty of solving Finsler extended Einstein field equations is evident. Inthis paper, we simply introduce a further constraint of symmetry by demandingthat the tensor terms R µν and S µν are linearly related. While this of courselimits the richness of the theory of Finsler spaces, it helps us develop a physi-cally relevant and realistic model. In order to solve equation (8), we follow thestandard procedure of finding the actual form of the metric by constraining itsfunctional parameters. Rewriting equation (8) for µ = ν = θ , we have: R ˆ θ ˆ θ + S ˆ θ ˆ θ = 0 (9)It has been shown that the value of R ˆ θ ˆ θ [2] is: R ˆ θ ˆ θ = 2 r dλd r e − λ + 1 − e − λ r (10)But R ˆ θ ˆ θ and S ˆ θ ˆ θ are linearly related, and therefore similar in structure. Indeedgiven two differential equations D ( x ) and D ( y ) such that: D ( x ) + D ( y ) = 0; y = α x or D ( x ) + D ( α x ) = 0therefore, D ( x ) = − αD ( x )implying that: D ( x ) D ( x ) = − α = γ (11)It thus, follows that D ( x ), D ( x ) are linearly related and similar in structure.Differential equations can be modelled as matrix eigenvalue problems wherebythe differential operator becomes the matrix and the solution the eigenfunction.Taking into account that for any eigenvalue problemˆ A X = β X , Y = g X , we have ˆ A Y = β Y where g is a constant. Implication of equation (11) is that the structure of S ˆ θ ˆ θ arises as: S ˆ θ ˆ θ = 2 r d L dr e − L + 1 − e − L r (12)where L is a function in space that is linearly related to λ . Equation (9) is,therefore, rewritten as:2 r dλd r e − λ + 1 − e − λ r + 2 r d L d r e − L + 1 − e − L r = 0 (13)Or 2 r dλd r e − λ + 1 − e − λ + 2 r d L d r e − L + 1 − e − L = 0 (14)Application of chain rule d L dλ = d L dλ dλd r (15)and trying to find dLdλ by assuming the most general case of “pseudopolynomial”functions, we have: L ( r ) = A r n + B r n − + C r n − + · · · + D r + E + F r − + G r − + H r − + · · · + I r − n λ ( r ) = J r n + K r n − + L r n − + · · · + M r + N + O r − + P r − + Q r − + · · · + R r − n where A to R are distinct constants. Therefore, d L dλ = A ′ r n − + B ′ r n − + C ′ r n − + · · · + D + F ′ r − + G ′ r − + H ′ r − + · · · I ′ r − n − J ′ r n − + K ′ r n − + L ′ r n − + · · · + M + O ′ r − + P ′ r − + Q ′ r − + · · · + R ′ r − n − where primed constants are new constants obtained after differentiation. Ne-glecting higher order terms for large r , we get: d L dλ = A ′ r n − + B ′ r n − + C ′ r n − + · · · + D + F ′ r − J ′ r n − + K ′ r n − + L ′ r n − + · · · + M We can express d L dλ as: d L dλ = b ( r ) r (16)where b ( r ) = A ′ r n + B ′ r n − + C ′ r n − + · · · + D r + E ′ r − J ′ r n − + K ′ r n − + L ′ r n − + · · · + M (17)Equations (16) when used in equation (13) yields:2 − e − λ + 2 r dλd r e − λ + − e − L + 2 r b r dλd r e − L = 0 (18)Since L = kλ , we have equation (18) as:2 − e − λ + 2 r dλd r e − λ + − e − kλ + 2 r b r dλd r e − k λ = 0 (19)4r expressing k as the logarithm of a certain constant a , we may write e − k λ = (cid:0) e − λ (cid:1) k = ae − λ (20)Using equation (20) in equation (19) and regarding the small constant a asunity, we have: − dλd r e − λ ( r + b ) + 2 e − λ = 2 (21)Equation (21) can easily be integrated to give: e − λ = (cid:18) − m r + b (cid:19) (22)Where m = GMc The standard Schwarzschild metric is written as: d s = e − λ d t − e λ d r − r ( dθ + sin φ ) (23)If we consider paths along radial lines for light cones around the singularity r = 2 m , the above metric reduces to: d s = e − λ d t − e λ d r (24)Using equation (22) in equation (24) yields: d s = (cid:18) − m r + b (cid:19) d t − (cid:18) − m r + b (cid:19) − d r (25)Comparison with equation (1) shows that as a result of our computation, thereis a correction to the term 2 m/ r present in equation (25).Let us now consider the asymptotic behaviour of equation (25) at the Schwarzschildradius, at the centre of the black hole and at large distances. At r = 2 m , a radius is defined such that nothing that goes into the blackholeever gets out [4]. In relativistic terms, the light cones of test particles arecompletely tipped over such that the geodesics are pointing towards the centerof the black hole [6] i.e., d s = 0 (26)Singularity at r = 2 m is due to poor choice of coordinate system hence, iscalled coordinate singularity. Transformation into Eddington-Finklestein coor-dinates removes the singularity. We now proceed to transform the metric intoEddington-Finklestein coordinates. These coordinates describe spacetime at the5vent horizon. They are derived from null geodesics where the metric is set tozero (equation 26). Combining equations (26) and (25), we obtain: (cid:18) − m r + b (cid:19) d t = (cid:18) − m r + b (cid:19) − d r This equation can be solved to give: t = r + 2 m ln ( r + b m −
1) + c where c = b − m + 2 m ln m is a constant. Relabelling t as r ∗ , we have: r ∗ = r + 2 m ln ( r + b m −
1) + c (27)so that d r ∗ d r = r + b r + b − m (28)Introducing “tortoise” coordinates coordinates [2] of the form: v = t + r ∗ (29) u = t − r ∗ , (30)it can easily be shown that: d r = 14 (cid:18) r + b − m r + b (cid:19) ( d v − d u d v + d v ) (31)and d t = 14 ( d u + 2 d u d v + d v ) (32)Using equations (22), (31) and (32) in equation (25), we obtain: d s = (cid:18) − m r + b (cid:19) ·
14 ( d u + 2 d u d v + d v ) − (cid:18) − m r + b (cid:19) − (cid:18) − m r + b (cid:19) ( d v − d u d v + d v ) (33)Equation (33) can be reduced to: d s = (cid:18) − m r + b (cid:19) dudv (34)Equation (34) is the metric restated in Eddington-Finklestein coordinates. Tak-ing asymptotic behaviour at r = 2 m and noting that 2 m >> b yields: d s | r =2 m = 0 (35)which is consistent with equation (25). The result is also consistent with thebehaviour of the standard Schwarzschild metric [2] and hence, with the math-ematical predictions of standard general relativity. Therefore, the theory isasymptotically similar to the standard theory outside the black hole and at theevent horizon. 6 .1.2 Asymptotic behaviour inside the Scwhwarzschild black hole By inspection, it can easily be seen that the metric in equation (25) is non-singular at r = 0 i.e., d s | r =0 = (1 − mb ) d t + (1 − mb ) − d r (36)However, to be completely sure that we have eliminated the singularity, we needto transform the metric into Kruskal-Szekeres coordinates [5]. Using equations(29) and (30) in equation (27), we get: r ∗ = r + 2 m ln ( r + b m −
1) + c = 12 ( v − u ) (37)Dividing equation (37) by 2 m , we obtain: r + b − m = 2 m e m ( v − u ) − m ( r + c ) (38)Rearranging equation (34) and applying equation (38), we obtain: d s = 2 m r + b e ( v − u )4 m e − ( r + c )2 m d u d v Using transformations of the form: U = − e − u m ; V = e − v m , it is easy to show that: d s = 32 m r + b e − ( r + c )2 m d U d V so that application of cooordinate transformation suggested by Kruskal [4] gives: d s = 32 m r + b e − ( r + c )2 m ( d T − d X ) (39)Taking asymptotic behaviour at r = 0, we obtain d s | r =0 = 32 m b e − r + c m ( d T − d X ) (40)showing that the metric has non-singular behaviour at r = 0. At large distances, equation (25) reduces to: d s = d t − d r (41)Equation (41) is just the Minkowski spacetime [5]. At large distances from themass, curvature is minimised, the general relativistic curvature corrections areabsent and thus the metric is asymptotically flat and similar to the standardmetric. 7 .2 Conclusion The metric corresponding to a Schwarzschild solution for the extended Einsteinfield equations has been derived. The metric has been shown to have the externalSchwarzschild solution as an aymptotic extension at long distances. At the eventhorizon, the metric is shown to be equal to the standard Schwarzschild metric.Further, and more interestingly, the metric is shown to be non-singular at r = 0.We invite further exploration of the work presented, including the calculationof the Kretschmann invariant and possible modification of the Kerr metric. Wehope in later work to explore a unification scheme based on the extended fieldequations that will assist in the determination of the constant b. References [1] A. Einstein,The Field Equations of Gravitation,
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