Schwarzschild 1/r -singularity is not permissible in ghost free quadratic curvature infinite derivative gravity
aa r X i v : . [ g r- q c ] S e p Schwarzschild /r -singularity is not permissible in ghost free quadratic curvatureinfinite derivative gravity Alexey S. Koshelev , , , , Jo˜ao Marto , , Anupam Mazumdar , Departamento de F´ısica, Universidade da Beira Interior,Rua Marquˆes D’ ´Avila e Bolama, 6201-001 Covilh˜a, Portugal. Centro de Matem´atica e Aplica¸c˜oes da Universidade da Beira Interior (CMA-UBI),Rua Marquˆes D’ ´Avila e Bolama, 6201-001 Covilh˜a, Portugal. Theoretische Natuurkunde, Vrije Universiteit Brussel. The International Solvay Institutes, Pleinlaan 2, B-1050, Brussels, Belgium. Van Swinderen Institute, University of Groningen, 9747 AG, Groningen, The Netherlands and Kapteyn Astronomical Institute, University of Groningen, 9700 AV Groningen, The Netherlands. (Dated: September 13, 2018)In this paper we will study the complete equations of motion for a ghost free quadratic curvatureinfinite derivative gravity. We will argue that within the scale of non-locality, Schwarzschild-typesingular metric solution is not permissible . Therefore, Schwarzschild-type vacuum solution whichis a prediction in Einstein-Hilbert gravity may not persist within the region of non-locality. Wewill also show that just quadratic curvature gravity, without infinite derivatives, always allowsSchwarzschild-type singular metric solution.
I. INTRODUCTION
Arguably Einstein’s theory of general relativity is one of the most successful description of spacetime which has seennumerous confirmations of observational tests at different length scales, predominantly in the infrared (IR) (far awayfrom the source and at late time scales) [1], including the fascinating detection of gravitational waves [2]. In spite ofthis success, at short distances and at small time scales, i.e. in the ultraviolet (UV), the Einstein-Hilbert action leadsto well-known singular solutions, in terms of blackhole solutions in the vacuum, and the cosmological singularity in atime dependent background [3]. The nature of latter singularity is indeed very different from the former, which bringsuncertainty in the cosmological models at the level of initial conditions for inflation and the big bang cosmology. Inreality, one would expect that nature would avoid any kind of classical singularities, whether they are covered by anevent horizon or they are naked - a stronger version of the cosmic censorship hypothesis [4, 5]. In this respect, itcan be argued that the singularities present in the Einstein-Hilbert action are mere artefacts of the action, and theremust be a way to ameliorate the singularities in nature. Indeed, removing the singularities is one of the foremostfundamental questions of gravitational physics.Recently, Biswas, Gerwick, Koivisto and Mazumdar (BGKM) have shown that the quadratic curvature infinitederivative theory of gravity in 4 spacetime dimensions can be made ghost free and avoid both cosmological and black-hole singularities at the linearised level around the Minkowski background [6] , while the cosmological singularity canbe resolved even at the full non-linear level [9–13]. At the linear level (around asymptotically Minkowski background),resolution of blackhole singularities has been studied both in the context of static [6, 14, 16–20], and in a rotatingcase [21] by various groups. Furthermore, lack of formation of singularity at the linear level has also been studied ina dynamical context by Frolov and his collaborators [22, 23].In Ref. [6], the authors have shown that for a ghost free quadratic curvature gravitational form factors , at shortdistances the gravitational metric-potential tends to be a constant, while at large distances from the source the metricpotential takes the usual form of 1 /r behavior in the IR. Furthermore, the gravitational force quadratically vanishestowards the center in the UV. Such a system behaves very much like a compact object, but by construction there is nocurvature singularity, nor there is an event horizon. The gravitational entropy calculated by the Wald’s formalism [24]leads to the Area -law [25]. Since, all the interactions are purely derivative in nature, the gravitational form factors give rise to non-local interactions for such a spacetime [26–30]. The non-locality is indeed confined within the scale M s , which has very a interesting behavior.The aim of this short note is to show that the full non-linear metric solution of the BGKM gravity will not permit1 /r -type metric potential, i.e. Schwarzschild type solution, for the static background. Note that what is relevant forus is indeed 1 /r part of the metric potential, be it in isotropic coordinates or Schwarzschild’s coordinates. Near thevicinity of singularity, at r = 0, what dominates is indeed the 1 /r part of the metric potential in Einstein’s theory of See previous to this work other relevant references [7–9], where the authors have argued absence of singularity in infinite derivativegravity motivated from the string theory, however, the full quadratic curvature action including the Weyl term with two gravitationalmetric potentials were first presented in [6]. gravity. Also, such a singular solution exists in quadratic curvature gravity as well [31], see for instance [32], thereforeit is a pertinent question to ask whether 1 /r -kind of metric potential would survive infinite derivative theory of gravityor not? II. THE INFINITE COVARIANT DERIVATIVE ACTION
The most general quadratic curvature action (parity invariant and free from torsion) has been derived aroundconstant curvature backgrounds in Refs. [6, 14, 15], given by S = 116 πG Z d x √− g (cid:0) R + α c (cid:2) R F ( (cid:3) s ) R + R µν F ( (cid:3) s ) R µν + W µνλσ F ( (cid:3) s ) W µνλσ (cid:3)(cid:1) , (2)where G = 1 /M p is the Newton’s gravitational constant, α c ∼ /M s is a dimensionful coupling, (cid:3) s ≡ (cid:3) /M s , where M s signifies the scale of non-locality at which new gravitational interaction becomes important. In the limit M s → ∞ ,the action reduces to the Einstein-Hilbert term. The d’Alembertian term is (cid:3) = g µν ∇ µ ∇ ν , where µ , ν = 0 , , ,
3, andwe work with a metric convention which is mostly positive ( − , + , + , +). The F i ’s are three gravitational form-factors , F i ( (cid:3) s ) = X n f i,n (cid:3) ns , (3)reminiscence to any massless theory possessing only derivative interactions. In this theory, the graviton remainsmassless with transverse and traceless degrees of freedom. However, the gravitational interactions are non-local dueto the presence of the form factors F i ’s, see [6]. These form factors contain infinite covariant derivatives, which showsthat the interaction vertex in this class of theory becomes non-local. In fact, the gravitational interaction in thisclass of theory leads to smearing out the point source by modifying the gravitational potential, as shown in [6]. Thenon-local gravitational interactions are also helpful to ameliorate the quantum aspects of the theory, which is believedto be UV finite [26–29]. The scale of non-locality is governed by M − s .Around the Minkowski background the three form factors obey a constraint equation, in order to maintain only the transverse-and traceless graviton degrees of freedom, i.e. the perturbative tree-level unitarity [6, 14] F ( (cid:3) s ) + 3 F ( (cid:3) s ) + 2 F ( (cid:3) s ) = 0 . (4)Let us first discuss very briefly the linear properties of this theory around an asymptotically Minkowski backgroundbefore addressing the non-linear equations of motion. The linear solutions are indeed insightful and provides a lot ofunderstanding of the solutions within BGKM gravity. Even though, we will not discuss explicitly non-linear solution,but any non-linear solution should have a limit in the linear regime. Note that the mass of the source is the relevantparameter, which plays a crucial role in determining linear and non-linear solutions. In Refs. [6, 16, 17], it was shownthat for a ( (cid:3) s ) = e γ ( (cid:3) s ) , where γ is an entire function , the central singularity is avoided, while recovering the correct1 /r dependence in the metric potential in the IR. For a specific choice of a ( (cid:3) s ) = e (cid:3) s , and assuming the Dirac-deltamass distribution, mδ ( r ) at the center, the gravitational metric potential, i.e. the Newtonian potential remainslinear, as long as: mM s ≤ M p , (5)with the gravitational metric potential in static and isotropic coordinates is given by [6]: φ ( r ) = − Gmr
Erf (cid:18) rM s (cid:19) , (6) The original action was first written in terms of the Riemann, but it is useful to write the action in terms of the Weyl term which isrelated to the Riemann as: W µανβ = R µανβ −
12 ( δ µν R αβ − δ µβ R αν + R µν g αβ − R µβ g αν ) + R δ µν g αβ − δ µβ g αν ) (1) In order to make sure that the full action Eq. (2) contains the same original dynamical degrees of freedom as that of the masslessgraviton in 4 dimensions. This is to make sure that the action is ghost free , there are no other dynamical degrees of freedom in spite ofthe fact that there are infinite derivatives. The graviton propagator for the above action gives rise toΠ( k ) = 1 a ( k ) Π( k ) GR = 1 a ( k ) " P (2) k − P (0) k , where P (2) and P (0) are spin-2 and 0 projection operators, and a ( k ) = e γ ( k /M s ) , is exponential of an entire function - γ , which doesnot contain any poles in the complex plane, therefore no new degrees of freedom other than the transverse and traceless graviton, seefor details [6, 33]. The gravitational form factors F i ( (cid:3) s ) cannot be determined simultaneously in terms of a ( (cid:3) s ), if we switch one ofthe F i = 0, then we can express the other form factors in terms of a ( (cid:3) s ), for instance for F = 0 yields, F = − [( a ( (cid:3) s ) − / (cid:3) s ] and F = [( a ( (cid:3) s ) − / (cid:3) s ], see [14]. approaches to be constant with a magnitude less than 1 for r < /M s . Since the error function goes linearly in r for r < /M s , the metric potential becomes finite in this ultraviolet region. For r > /M s , the metric potential followsas ∼ Gm/r , in the infrared region. However, in our case the typical scale of non-locality is actually larger than theSchwarzschild’s radius as shown in [19, 20] r NL ∼ M s ≥ r sch = 2 mM p , (7)thus avoiding the event horizon as well . Now, for the rest of the discussion, let us focus on the full non-linearequations for the above action Eq. (2). III. TOWARDS IMPOSSIBILITY OF THE SCHWARZSCHILD METRIC SOLUTION
The complete equations of motion have been derived from action Eq. (2), and they are given by [14], P αβ = − G αβ πG + α c πG (cid:18) G αβ F ( (cid:3) s ) R + g αβ R F ( (cid:3) s ) R − (cid:0) ▽ α ∇ β − g αβ (cid:3) s (cid:1) F ( (cid:3) s ) R − αβ + g αβ (Ω σ σ + ¯Ω ) + 4 R αµ F ( (cid:3) s ) R µβ − g αβ R µν F ( (cid:3) s ) R νµ − ▽ µ ▽ β ( F ( (cid:3) s ) R µα ) + 2 (cid:3) s ( F ( (cid:3) s ) R αβ )+ 2 g αβ ▽ µ ▽ ν ( F ( (cid:3) s ) R µν ) − αβ + g αβ (Ω σ σ + ¯Ω ) − αβ − g αβ W µνλσ F ( (cid:3) s ) W µνλσ + 4 W αµνσ F ( (cid:3) s ) W βµνσ − R µν + 2 ▽ µ ▽ ν )( F ( (cid:3) s ) W βµνα ) − αβ + g αβ (Ω γ γ + ¯Ω ) − αβ (cid:19) = − T αβ , (8)where T αβ is the stress energy tensor for the matter components, and we have defined the following symmetric tensors,for the detailed derivation, see [14]:Ω αβ = ∞ X n =1 f n n − X l =0 ∇ α R ( l ) ∇ β R ( n − l − , ¯Ω = ∞ X n =1 f n n − X l =0 R ( l ) R ( n − l ) , (9)Ω αβ = ∞ X n =1 f n n − X l =0 R µ ; α ( l ) ν R ν ; β ( n − l − µ , ¯Ω = ∞ X n =1 f n n − X l =0 R µ ( l ) ν R ν ( n − l ) µ , (10)∆ αβ = ∞ X n =1 f n n − X l =0 [ R ν ( l ) σ R ( βσ ; α )( n − l − − R ν ; α ( l ) σ R βσ ( n − l − ] ; ν , (11)Ω αβ = ∞ X n =1 f n n − X l =0 W µ ; α ( l ) νλσ W νλσ ; β ( n − l − µ , ¯Ω = ∞ X n =1 f n n − X l =0 W µ ( l ) νλσ W νλσ ( n − l ) µ , (12)∆ αβ = ∞ X n =1 f n n − X l =0 [ W λν ( l ) σµ W βσµ ; α ( n − l − λ − W λν ; α ( l ) σµ W βσµ ( n − l − λ ] ; ν . (13)The notation R ( l ) ≡ (cid:3) l R has been used for the curvature tensors and their covariant derivatives. The trace equationis much more simple, and just for the purpose of illustration, we write it below [14]: P = R πG + α c πG (cid:18) (cid:3) s F ( (cid:3) s ) R + 2 (cid:3) s ( F ( (cid:3) s ) R ) + 4 ▽ µ ▽ ν ( F ( (cid:3) s ) R µν )+ 2(Ω σ σ + 2 ¯Ω ) + 2(Ω σ σ + 2 ¯Ω ) + 2(Ω σ σ + 2 ¯Ω ) − σ σ − σ σ (cid:19) = − T ≡ − g αβ T αβ . (14) This could potentially resolve the information-loss paradox, since there is no event horizon and the graviton interactions for r NL ∼ /M s becomes nonlocal, therefore for interacting gravitons the spacetime ceases to hold any meaning in the Minkowski sense. The Bianchi identity has been verified explicitly in Ref. [14]. Here we briefly sketch the Weyl part, since this will bethe most important part of our discussion. To accomplish this, note that the computations are simplified if one usesthe following tricks by rewriting the equations of motion with one upper and one lower index, express Ricci tensorthrough the Einstein tensor (who’s divergence is zero due to the Bianchi identities), and recalling the fact that thedivergence of the Weyl tensor is the third rank Cotton tensor, which can be expressed through the Schouten tensor: ∇ γ W αµνγ = −∇ α S µν + ∇ µ S αν , where the Schouten tensor in four dimensions is given by: S µν = 12 (cid:18) R µν − g µν R (cid:19) . With this in mind the rest of the computations amount to careful accounting of the symmetry properties of the Weyltensor (which are identical to those of the Riemann tensor). We should also observe that the Bianchi identities shouldhold irrespectively of the precise form of functions F i , and independently for each and every coefficient f i,n , becausethese are mere numerical coefficients, which are required to make the theory ghost free [14] Technically, this meansthat we should not bother about the summation over n , but rather concentrating on the inner summation over l inEqs. (9-13). Finally, the symmetry with respect to α ↔ β permutation in the equations of motion can be accountedby re-arranging the summation over l in the inverse order from n − asymptotically Minkowski background for a staticcase, R = 0 , R µν = 0 . (15)In this case the energy momentum tensor vanishes in all the region except at r = 0, where the source mδ ( r ) islocalized. One of the properties of such a vacuum solution is the presence of 1 /r - static and spherically symmetricmetric solution, similar to the Schwarzschild metric, given by: ds = − b ( r ) dt + b − ( r ) dr + r (cid:0) dθ + sin ( θ ) dφ (cid:1) , (16)where b ( r ) = 1 − Gm/r with the presence of a central singularity at r = 0, and also the presence of an event horizon.As we have already discussed, for r < Gm , what dominates is the 1 /r part of b ( r ), which dictates the rise in thegravitational potential all the way to r = 0. Note that, although the vacuum solution permits R = 0 , R µν = 0, theWeyl-tensor is non-vanishing in the case of a Schwarzschild metric, where W µνλσ W µνλσ → ∞ , as r →
0. Now in our case, indeed the full equations of motion are quite complicated, nevertheless, we might be ableto test this hypothesis of setting R = 0 , R µν = 0, and study whether the Schwarzschild metric, or 1 /r type metricpotential is a viable metric solution of our theory of gravity or not?Let us then demand that the above action, Eq. (2), along with the equations of motion Eq. (8), permits a solutionwhich is Schwarzschild metric with P αβ = 0, and R = 0 and R µν = 0. In fact, in the region of non-locality wherehigher derivative terms in the action are dominant, it suffices to demand that R = constant and R µν = constant .Let us now concentrate on the full equations of motion (8) with the Weyl part of the full equations of motion: P αβ = 0 = P αβ = α c πG (cid:18) − g αβ W µνλσ F ( (cid:3) s ) W µνλσ + 4 W αµνσ F ( (cid:3) s ) W βµνσ − R µν + 2 ▽ µ ▽ ν )( F ( (cid:3) s ) W βµνα ) − αβ + g αβ (Ω γ γ + ¯Ω ) − αβ (cid:19) . (17)Indeed, we would expect that in order to fulfill the necessary condition (but not sufficient) for the Schwarzschildmetric to be a solution of Eq. (8), we would have both the left and the right hand side of the above equation vanishes We have checked that the Bianchi identity holds true at each and every order of (cid:3) s . identically. The failure of this test will imply that the Schwarzschild metric cannot be the permissible solution of theequation of motion for Eq. (8).There are couple of important observations to note, which we summarize below:1. F i ( (cid:3) s ) contain an infinite series of (cid:3) s .2. The Bianchi identity holds for each and ever order in (cid:3) s , as we have already discussed.3. The right hand side of Eq. (17) should vanish at each and every order in (cid:3) s . This is due to the fact that whenwe compare the terms, assigned to coefficients f i,n (where the box operator has been applied n times, i.e. (cid:3) ns )with terms where the box operator has been applied n + 1 times ( (cid:3) n +1 s , assigned to coefficient f i,n +1 ), then the1 /r n dependence would at least be changed to 1 /r n +2 in this process. Note that the box operator has roughlytwo covariant derivatives in r . Therefore, if we are not seeking for any miraculous cancellation, between differentorders in (cid:3) s , it is paramount that each and every order in (cid:3) s , the right hand side must vanish to yield theSchwarzschild-like metric solution.4. In fact, we could repeat the same argument for higher order singular metric ansatzs, such as 1 /r α , for α > P αβ with one (cid:3) s only ,such that F ( (cid:3) s ) = ( f + f (cid:3) s ) . Therefore, Eq. (17) becomes P αβ = α c πG (cid:18) − g αβ W µνλσ ( f + f (cid:3) s ) W µνλσ + 4 W αµνσ ( f + f (cid:3) s ) W βµνσ − R µν + 2 ▽ µ ▽ ν )(( f + f (cid:3) s ) W βµνα ) − f ∇ α W µνργ ∇ β W µνργ + g αβ f ( ∇ α W µνργ ∇ β W µνργ + W µνργ (cid:3) s W µνργ ) − f ( W γνρµ ∇ α W γβρµ − W γβρµ ∇ α W γνρµ ) ; ν (cid:19) . (18)In the static limit, after some computations, we can infer the following:1. All the terms combining f terms cancel each other from the above expression in Eq. (18). This is indeedreminiscence, and agrees to the earlier computations performed in this regard in Ref. [32], where the actioncorresponds to just the quadratic in curvature, but with local quadratic curvature action: S = 116 πG Z d x √− g (cid:0) R + α c [ R + R µν R µν + W µνλσ W µνλσ ] (cid:1) . (19)Such an action indeed provides singular solutions with metric coefficients b ( r ) ∼ /r for r << r sch , as theleading order contribution, in spite of the fact that the above action has been shown to be renormalizable, butwith an unstable vacuum, due to spin-2 ghost [31]. The BGKM action indeed attempts to address the ghostproblem of quadratic curvature gravity.2. The first non-trivial result comes from the fact that the only terms which do not cancel , and survive from theright hand side of Eq. (18) are those proportional to f , and one can show explicitly that they go as1 /r , in the UV ( r ≪ /M s ), for details see the Appendix. This means, that indeed 1 /r as a metric solution doesnot pass through our test, since the right hand side of the above equation of motion is non-vanishing, but theleft hand side ought to vanish in lieu of the vacuum condition, P αβ = 0.3. In fact, we may be able to generalize our results to any orders in (cid:3) s by noting that the higher orders beyond onebox would contribute, at least, two more covariant derivatives in r in going from (cid:3) ns to (cid:3) n +1 s terms (assumingthat (cid:3) s ∼ M s ∂ r ). This means that the full computation for the right hand side of Eq. (18) would yield: P αβ ∽ g αβ (cid:18) f O ( 1 r ) + f O ( 1 r ) + ... + f n O ( 1 r n ) + ... (cid:19) , (20)( g αβ is defined from the metric (16), see the exact definition of P αβ in the appendix) which would require toomuch fine tuning to cancel each and every term, while keeping in mind that f n are mere constant coefficients.Barring such unjustified cancellation, it is fair to say that indeed 1 /r for r ≪ /M s as a metric potential for theBGKM gravity is not a valid solution, if F has a nontrivial dependence on (cid:3) s .Similar conclusions have already been drawn in Ref. [20], with a complementary arguments. In Ref. [20], theargument was based on taking a smooth limit from non-linear solution of Eq. (2) to the linear solution. For anyphysical solution to be valid, the non-linear solution must pave to the linear solution smoothly.At the linear level (where the metric potential is bounded below 1), it was shown that the Weyl term vanishesquadratically in r [20], for a non-singular metric solution given by a metric potential Eq. (6). Therefore, at thefull non-linear level 1 /r -type metric potential cannot be promoted as a full solution for the non-linear equations ofmotion for the BGKM action, since there is no way it can be made to vanish quadratically at the linear level. Similarconclusions can be made for any metric potential which goes as 1 /r α for α > r = 0, but also gives rise to a metric potential which is bounded below one in the entire spacetime regime. Thenotion about the physical mechanism which avoids forming a trapped surface, also yields a static metric solution ofgravity, which has no horizon, see [34]. The only viable solution of Eq. (2) remains that of the linear solution, aroundthe Minkowski background, already described by Eq. (6). Indeed, this last step has to be shown more rigorously,which we leave for future investigation.Another important conclusion arises due to the non-local interactions in the gravitational sector, which yields anon-vacuum solution, such that R = 0 and R µν = 0, within length scale ∼ /M s [20]. This is due to the fact thatthe BGKM gravity smears out the Dirac-delta source, and therefore a vacuum solution does not exist any more likein the case of the Einstein’s gravity, or any f ( R ) gravity, or even in the context of local quadratic curvature gravity,see Eq. (19). IV. CONCLUSION
The conclusion of this paper is very powerful. We have argued that Schwarzschild metric or 1 /r -type metricpotentials cannot be the solution of the full BGKM action given by Eq. (2), and the full non-linear equations ofmotion Eq. 8). By 1 /r -type metric potential, we mean the non-linear part of the Schwarzschild metric, for r < Gm ,where m is the Dirac delta source. The presence or absence of singularity is judged by the Weyl contribution. Inthe pure Einstein-Hilbert action, indeed the Weyl term in the Schwarzschild metric is non-vanishing, and contributestowards the Kretschmann singularity at r = 0. In the case of infinite derivatives in 4 dimensions, we have shown herethat this is not the case, and the infinite derivative Weyl contribution contradicts with 1 /r being the metric solutionfor a vacuum configuration, for which the energy momentum tensor vanishes, for a static and spherically symmetricsolution for the BGKM action. By itself the result does not prove or disprove a non-singular metric potential, butit provides a strong hint that the full equations of motion cannot support Schwarzschild type of 1 /r type metricpotential. We have also argued that on a similar basis even 1 /r α for α > For the Schwarzschild metric this takes the form (cid:3) s = M s g νµ ∇ ν ∇ µ = M s (cid:20)(cid:18) − mr (cid:19) ∂ r − (cid:18) − mr + (cid:18) − mr (cid:19) r (cid:19) ∂ r (cid:21) . V. APPENDIXA. Explicit non-vanishing contributions from the Weyl term
Here we show the relevant terms, present in Eq. (18), assuming b ( r ) = 1 − Gm/r in the metric (16). The explicitenumeration of each term is important to understand how the coefficient f and f appear and how they mightcancel. Let us define P αβ = α c πG X i =1 F αβi
1. For the first term, F αβ = − g αβ W µνλσ ( f + f (cid:3) s ) W µνλσ , the calculation yields F αβ = g αβ G r m M s − ( f M s r − f Gm ) r − ( f M s r − f Gm ) r − ( f M s r − f Gm ) r
00 0 0 − ( f M s r − f Gm ) r . (21)2. The second term, F αβ = +4 W αµνσ ( f + f (cid:3) s ) W βµνσ , is given by F αβ = − g αβ G r m M s − ( f M s r − f Gm ) r − ( f M s r − f Gm ) r − ( f M s r − f Gm ) r
00 0 0 − ( f M s r − f Gm ) r . (22)We can verify, at this point, that the first two terms cancel each other.3. The third term, F αβ = − (cid:0) R µν + ∇ µ ∇ ν ) ( f + f (cid:3) s ) W βµνα , is given by F αβ = g αβ G r m M s f − (5 r − Gm ) r − ( r − Gm ) r (3 r − Gm ) r
00 0 0 (3 r − Gm ) r , (23)which only depends on the f coefficient.4. The fourth term, F αβ = − f ∇ α W λµνσ ∇ β W µνσλ , is given by F αβ = g αβ G r m M s f − r − Gm ) r − ( r − Gm ) r
00 0 0 − ( r − Gm ) r . (24)5. The fifth term, F αβ = + g αβ f ( ∇ α W µνργ ∇ β W µνργ + W µνργ (cid:3) s W µνργ ), is given by F αβ = g αβ G r m M s f (5 r − Gm ) r (5 r − Gm ) r (5 r − Gm ) r
00 0 0 (5 r − Gm ) r . (25)6. The sixth term, F αβ = − f ( W γνρµ ∇ α W γβρµ − W γβρµ ∇ α W γνρµ ) ; ν , is given by F αβ = g αβ G r m M s f − ( r − Gm ) r (3 r − Gm ) r
00 0 0 (3 r − Gm ) r . (26)Having computed each term of P αβ , we can conclude that the stress energy momentum tensor dependence onthe f coefficient is vanishing, and only the one box, (cid:3) s , contributions survive. Finally, we have the non vanishingcontribution, P αβ = g αβ G πr m M s f − r − Gm ) r − r − Gm ) r (21 r − Gm ) r
00 0 0 (21 r − Gm ) r . (27) Second order in (cid:3) s contributions : In order to strengthen our arguments, we present below the additionalcontribution for the second order in box, i.e., (cid:3) s : P αβ ( (cid:3) s ) = − g αβ G πr m M s f a a a
00 0 0 a . (28)with the dimensionless matrix elements, defined as a = (939 G m − Gmr + 140 r ) r ,a = (195 G m − Gmr + 20 r ) r ,a = − (789 G m − Gmr + 80 r ) r ,a = − (789 G m − Gmr + 80 r ) r . Let us now consider, for example, the P element at (cid:3) s , namely: P = α c πG (cid:20) f (cid:18) G m r M s − G m r M s (cid:19) + f (cid:18) G m r M s − G m r M s + 454464 G m r M s (cid:19) + · · · (cid:21) . (29)Demanding that P = 0, implies that f = f = 0. We can now ask what would happen for higher orders in (cid:3) s .Since (cid:3) s ∼ M s ∂ r , we have at the lowest third order contribution in box, in powers of r , is proportional to f G m r M s . (30)Therefore, since we already concluded that f = f = 0, we now have to demand that the contribution of f G m r M s vanishes identically. The lowest fourth order contribution, in powers of r , will go as f G m r M s , (31)we are left with the option that f = f = 0. Obviously, we do not claim that this is a rigorous mathematicaldemonstration, however we can hint, by dimensional analysis, that the lowest n nth order contribution will be alwaysproportional to f n G m r n M ns . (32)Indeed, the aforementioned analysis will make it hard to make all the expansion coefficient set to vanish, i.e. f n = 0. ACKNOWLEDGMENTS
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