Cent. Eur. J. Phys. • 6(1) • 2008 • 80-83DOI: 10.2478/s11534-008-0017-1

Central European Journal of Physics

Issue of the spherically symmetric static vacuummetric in the relativistic theory of gravity

Research Article

Vladimir L. Kalashnikov∗

Institut für Photonik, TU Wien, Gusshausstr. 27/387, A-1040 Vienna, Austria

Received 29 June 2007; accepted 26 July 2007

Abstract: The spherically symmetric vacuum metric in the relativistic theory of gravity is analyzed numerically. Itis found that there is no deviation of the numerical solution from that of general relativity except in thenear-horizon range. The solution obtained has the well-established analytical asymptotics for both thenear- and far-horizon limits. It satisfies the causality principle and does not impose a lower limit on thegraviton mass.

PACS (2008): 04.20.-q, 04.25.dg, 04.20.Ex, 04.70.s

Keywords: relativistic theory of gravity • Schwarzschild problem© Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

The relativistic theory of gravity (RTG) develops the fieldapproach to the gravitation and treats it as the symmet-ric second-rank tensor field on the Minkowski backgroundspace-time [1]. To be self-consistent, such an approach re-quires a massive graviton that modifies the field equations.This, as well as the bi-metric nature of RTG, complicatesan analysis substantially so that even highly-symmetricfield configurations can be considered only approximatelyor numerically.Recently A. Godizov [2] reported the numerical static vac-uum solution of the RTG equations under the condition ofspherical symmetry. Prior to this report, approximate an-alytical [1, 3] and numerical solutions [3, 4] of the problemhad been known. The solutions’ stability against axial∗E-mail: kalashnikov@tuwien.ac.at

perturbations has been proved [4]. Nevertheless, the so-lution presented in Ref. [2] differs from the known oneradically because the former predicts a lower limit on thegraviton mass: mg > 10−46 g.If such a limit exists, RTG has to be ruled out accordingto cosmological observations [5]. However, in this reportI shall show that the solution by A. Godizov is incorrectand the correct solution does not contradict cosmologi-cal observations, the causality principle, and analyticalapproximations.2. Static spherically symmetric va-cuum in RTG

Let us consider the static spherically symmetric interval ofthe effective Riemann space-time in spherical coordinates:80

Vladimir L. Kalashnikov

ds2 = U (r)dt2 − V (r)dr2 −W (r)2 [dθ2 + sin2 θdφ2] .(1)The RTG field equations are [1] (signature corresponds toRef. [4]):Gµν −

m22[δµν + gµλγλν −

12δµνgκλγκλ] = −8πχ

c4 T µν ,

Dµg̃µν = 0. (2)

Here gµν is the metric tensor of the effective Riemannspace-time, g̃µν = √−ggµν , γµν is the metric tensor ofthe background Minkowski space-time, Gµν is the Einstentensor, T µ

ν is the matter energy-momentum tensor , Dµ isthe covariant derivative over the Minkowski space-time,m2 = (mgc/h̄)2, and χ is the Newtonian gravitational con-stant. The second equation of (2), excluding unwantedspin-states, fixes the gauge [6].

Outside matter, Eqs. (1,2) lead to the system [2, 3]:

dUdr = U

Wy

{V[1 + m22 (

W 2 − r2)]− y2 − m2W 24(VU − 1)} ,

dVdr = V

Wy

{V[1 + m22 (

W 2 − r2)− 4yrW

]+ 3y2 − m2W 24(VU − 1)} ,

dWdr = y, (3)

dydr = V

[1 + m22 (W 2 − r2)− 2yrW

]+ y2W .

Numerical solutions of Eqs. (2) were presented in [2]. Theextraordinary property of the solution is that V < U ata macroscopic distance from horizon if mg is lower thansome critical value. This inequality means the breakingof causality. As a result, there exists a limitation on theminimum mg, which does not agree with cosmological ob-servations (see [5]).Since the gauge is fixed in RTG (Eqs. (2)), the reportedsolution has to be unique for a given asymptotic. How-ever, the solution does not agree with that reported previ-ously (including the well-grounded near-horizon analyti-cal approximation [1, 3]). Moreover, it is strange that theextremely small correction to the general relativity fieldequations (in the dimensionless form, the massive gravitonterm is about of 10−44 for mg ≈10−66 g and collapsar massM ≈10 M�) results in an extreme effect at a macroscopicdistance (neither microscopic nor cosmological) from thehorizon. The suspicion arises that the solution reportedis incorrect. Such a conclusion can be proved by the nu-merical analysis presented below.Eqs. (2) have been solved numerically using the Maple10 computer algebra system. Systematic analysis demon-strated that there is the need for a careful choice of thenumerical procedure due to the stiffness of the problemand accumulation of numerical errors. The latter results

in artefact solutions like those of [2]. The most appropri-ate choice is a procedure using a Taylor series method,which provides high accuracy within a wide region of r.When r approaches the horizon (rg), this method becomestime-consuming and the Livermore stiff ODE solver canbe more useful. The solution obtained is shown in Fig. 1(the boundary conditions at r/rg = 104 correspond to theSchwarzschild metric). One can see that the solution doesnot differ from the Schwarzschild one within an extremelywide region of r. Obviously, V > U and y = 1.The near-horizon approximation leads to [3]:W ≡ xrg, r ≡ ζrg, rg ≡

2χMc2 , ε ≡ 12

(2χMmg

h̄c

)2,

dζ/dx ≈ 1, x, ζ � 1√2ε . (4)Then, Eqs. (2) result in [3]:

2ηd2ηdx2 − 3(dηdx

)2− εxη

(1− x2η) dηdx = 0, (5)where

U ≡ 1/xηA, V ≡ x/A(dζdx

)2.

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Issue of the spherically symmetric static vacuum metric in the relativistic theory of gravity

Figure 1. Numerical solutions for the RTG spherically symmetricstatic vacuum interval coefficients: V (green, solid), y(red, dotted), U (blue, dashed). mg = 1.6 × 10−66 g,M = 10M�.

The extremely-near-horizon solution of Eq. (5) is shown inFig. 2. A difference from the Schwarzschild solution ap-pears at extreme proximity to horizon: V tends to infinityat some r ≈ rg(1+1.65ε) > rg, but U has a non-zero min-imum ε/2 [1, 3]. This means physically that gravitationalrepulsion appears in the vicinity of horizon. The solutioncoincides with the approximated analytical metric of Ref.[4] both in the far- and near-horizon limits. There is noa violation of the causality principle and no constraint onthe minimum graviton mass.3. ConclusionThe spherically symmetric static vacuum metric in RTGis analyzed numerically. It is shown that this metric dif-fers from the Schwarzschild one only at extreme proximityto horizon. Such a difference results in strong gravita-tional repulsion, which prohibits the black hole formation.The solution reported agrees with the analytical approx-imations both in the vicinity of horizon and at the limitr � rg. There is no violation of the causality principle

Figure 2. Approximated numerical solutions for the RTG spheri-cally symmetric static vacuum interval coefficients: V(green, upper picture), U (blue, lower picture). TheSchwarzschild coefficients are shown by black curves.Circle corresponds to the analytical limit. r is normal-ized to rg.

as well as no constraint on the minimum graviton massreported in [2]. One can conclude that the solution by A.Godizov [2] is incorrect and RTG cannot be ruled out onthe basis of the constraint on the minimum graviton mass.AcknowledgmentIt is my pleasure to thank Dr. S.L. Cherkas for valuablediscussions.

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Vladimir L. Kalashnikov

References

[1] A.A. Logunov, Relativistic theory of gravity, Nova Sc.Publ., 1998[2] A.A. Godizov, Cent. Eur. J. Phys. 5, 201 (2007)

[3] V.L. Kalashnikov, arXiv:astro-ph/0304466[4] V.L. Kalashnikov, arXiv:gr-qc/0405032[5] V.L. Kalashnikov, arXiv:gr-qc/0208070[6] V.Fock, Rev. Mod. Phys. 29, 325 (1957)

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