A note on spherically symmetric, static spacetimes in Kanno-Soda on-brane gravity
aa r X i v : . [ h e p - t h ] A p r A note on spherically symmetric, static spacetimes inKanno-Soda on-brane gravity
Sayan Kar , Sayantani Lahiri † , ‡ and Soumitra SenGupta ∗∗ Department of Physics and Center for Theoretical StudiesIndian Institute of Technology, Kharagpur, 721 302, India, † Institute for Physics, University Oldenburg, D-26111 Oldenburg, Germany, ‡ ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany and ∗∗ Department of Theoretical Physics,Indian Association for the Cultivation of Science2A and 2B Raja S.C. Mallick Road, Jadavpur, Kolkata 700 032, India.
Abstract
Spherically symmetric, static on-brane geometries in the Kanno-Soda (KS) effective scalar-tensortheory of on-brane gravity are discussed. In order to avoid brane collisions and/or an infiniteinter-brane distance, at finite values of the brane coordinates, it is necessary that the radion scalarbe everywhere finite and non-zero. This requirement constrains the viability of the standard, well-known solutions in General Relativity (GR), in the context of the KS effective theory. The radionfor the Schwarzschild solution does not satisfy the above requirement. For the Reissner–Nordstrom(RN) naked singularity and the extremal RN solution, one can obtain everywhere finite, non-zeroradion profiles, though the required on-brane matter violates the Weak Energy Condition. Incontrast, for the RN black hole, the radion profile yields a divergent inter-brane distance at thehorizon, which makes the solution unphysical. Thus, both the Schwarzschild and the RN solutionscan be meaningful in the KS effective theory, only in the trivial GR limit, i.e. with a constant,non-zero radion. ∗ Electronic address: [email protected],[email protected],[email protected] . INTRODUCTION Effective, on-brane theories of gravity have been in vogue ever since the Randall-Sundrumwarped braneworld scenario was proposed[1]. Among such four dimensional gravity theories,the most well-known one is due to Shiromizu, Maeda and Sasaki (SMS) [2] which considersa bulk with an infinite extra-dimension and a single brane. There have been proposals oneffective theories in a two-brane set-up. In this article, we consider one such effective theorydue to Kanno and Soda (KS) [3]. A major difference between the SMS and the KS effectivetheories is the presence of a non-local (bulk dependent) term in the former and the absenceof any non-locality in the latter. Our objective is to look for solutions in the KS effectivetheory, keeping in mind that the radion field, which is linked to the distance between thebranes, (i) is never zero in value (thus, avoiding brane collisions) and (ii) does not divergeat any finite value of the brane coordinates.Recently [4], we have discussed some cosmological and spherically symmetric, static space-times in this four dimensional, effective, on-brane, scalar-tensor theory of gravity. The spher-ically symmetric, static solutions obtained in [4] turned out to be the Majumdar-Papapetrousolution [5] with the source being the effective scalar field (radion) energy-momentum andadditional on-brane matter. Here, we ask a broader question: are the standard General Rel-ativity (GR) solutions like the Schwarzschild or the Reissner–Nordstrom (RN) permissible inthe KS theory? Obviously, we do not expect the GR solutions to arise in the KS theory withthe same matter content as in GR. Rather, we would like to find out if the radion energymomentum and extra on-brane matter can conspire in unison to allow the Schwarzschild orthe RN solution in the KS effective theory.It should be noted that there are several aspects related to this question, a couple of whichwe have already mentioned. In addition to having a radion which is finite and non-zeroeverywhere, the on-brane matter must also be physically reasonable in the classical sense,i.e. it must satisfy one of the well-known energy conditions, such as the Weak EnergyCondition or the Null Energy Condition [7].We may recall that the Reissner-Nordstrom solution does arise as a solution [6] of theShiromizu-Maeda-Sasaki (SMS) single brane effective Einstein equations [2], where the non-local contribution from the bulk Weyl tensor (the traceless E µν ) acts as its source withoutany explicit on-brane matter. The functional form of the E µν depends on the bulk Weyl2ensor and other features of the bulk geometry. It cannot be determined uniquely from theknowledge of four dimensional, on-brane physics. In contrast, in the KS effective theory,the influence of the bulk is exclusively through the radion field which depends only on thebrane coordinates. It is therefore meaningful to ask whether the equations which arise in theeffective, on-brane Kanno-Soda theory (which are local and different from those obtained inthe single brane SMS effective theory) also admit a RN or a Schwarzschild solution, in someway.We will mainly work with the Reissner–Nordstrom solution written in isotropic coordinates.After obtaining the radion profiles in the various cases, we will see if the radion satisfiesthe necessary requirements. Subsequently, we will analyse the nature of required on-branematter with reference to the Weak Energy Condition [7]. II. THE KANNO-SODA EFFECTIVE THEORY: MAIN EQUATIONS
The effective on-brane scalar-tensor theories developed by Kanno and Soda [3] in the contextof the Randall–Sundrum two-brane model leads to the following Einstein-like equations onthe visible ‘b’ brane [3], G µν = κ l Φ T bµν + κ (1 + Φ) l Φ T aµν + 1Φ ( ∇ µ ∇ ν Φ − g µν ∇ α ∇ α Φ) − (cid:18) ∇ µ Φ ∇ ν Φ − g µν ∇ α Φ ∇ α Φ (cid:19) (1)Here g µν is the on-brane metric, the covariant differentiation is defined with respect to g µν and we have taken the five dimensional line element as, ds = e φ ( x ) dy + ˜ g µν ( y, x µ ) dx µ dx ν (2) κ is the 5 D gravitational coupling constant. T aµν , T bµν are the stress-energy on the Planckbrane and the visible brane respectively. The appearance of T aµν (matter energy momentumon the ‘a’ brane) in the field equations on the ‘b’ brane, inspired the usage of the term ‘quasi-scalar-tensor theory’. However, if we assume T aµν = 0 then we have the usual scalar-tensortheory.We denote d ( x ) as the proper distance between branes located at y = 0 and y = l . d ( x ) isdefined as, d ( x ) = Z l e φ ( x ) dy (3)3e further define Φ = e dl −
1. It may be observed that the viability of such a model witha everywhere finite and non-zero brane separation, implies that (i) the minimum of d ( x ) isnot equal to zero and (ii) d ( x ) is never infinity at any finite value of the brane coordinates.This, in turn indicates that the value of Φ( x ) is always greater than zero and Φ( x ) neverbecomes infinity.Note that the radion contribution on the R. H. S. of the field equation is traceless, which isreminiscent of the traceless E µν in the SMS effective theory.The scalar field equation of motion on the visible brane is given as, ∇ α ∇ α Φ = κ l T a + T b ω + 3 − ω + 3 dωd Φ ( ∇ α Φ)( ∇ α Φ) (4)where T a , T b are the traces of energy momentum tensors on Planck (‘a’) and visible (‘b’)branes, respectively. The coupling function ω (Φ) expressed in terms of Φ is, ω (Φ) = − y = 0 is linked to that on the visible brane through the relation [3] :Φ( x ) = Ψ1 − Ψ (6)where Ψ is the radion field as defined on Planck brane. The induced metric on the visiblebrane can be expressed in terms of Ψ as, g b − braneµν = (1 − Ψ) [ h µν + g (1) µν ( h µν , Ψ , T aµν , T bµν , y = l )] (7)where g (1) µν is the first order correction term (see [3] for details). It is possible to work withthe gravity theory and the Ψ field equation on the ‘a’ brane.In our work here, we assume that the on-brane stress energy is nonzero only on the ‘b’brane (visible brane). We also assume that the on-brane matter is traceless and therefore,since the effective radion stress energy is also traceless, the Ricci scalar of the spacetimegeometry is identically zero. Conversely, if we assume R = 0, the on-brane matter is traceless.This choice of R = 0 enables us to propose the standard General Relativity solutions (likeSchwarzschild and Reissner-Nordstrom) as solutions in the Kanno-Soda effective theory withon-brane matter. The two main hurdles we need to address are therefore:4 Are the standard GR solutions also solutions in the effective theory, with a non-zero,everywhere finite radion? • What is the nature of the on-brane matter required to support such standard GRsolutions?
III. SPHERICALLY SYMMETRIC, STATIC SOLUTIONSA. Line element, field equations and the radion
Let us assume a four dimensional line element on the visible brane, in isotropic coordinates,given as ds = − f ( r ) U ( r ) dt + U ( r ) (cid:2) dr + r dθ + r sin θdφ (cid:3) (8)where U ( r ) and f ( r ) are the unknown functions to be determined from the Einstein-likeequations. Using the above line element ansatz and the assumption that Φ is a functionof r alone, we get the following field equations from the Einstein field equations mentionedabove. − U ′′ U + (cid:18) U ′ U (cid:19) − U ′ U r = − Φ ′ (cid:18) U ′ U − f ′ f (cid:19) Φ ′ Φ + κ l Φ ρ (9) − (cid:18) U ′ U (cid:19) + 2 f ′ f (cid:18) U ′ U + 1 r (cid:19) = − ′ − U ′ U Φ ′ Φ − ′ Φ r − f ′ f Φ ′ Φ + κ l Φ τ (10) (cid:18) U ′ U (cid:19) + f ′′ f − f ′ f U ′ U + f ′ f r = Φ ′ U ′ U Φ ′ Φ + Φ ′ Φ r + κ l Φ p (11)where ρ , τ and p correspond to on-brane matter and using the tracelessness condition wehave − ρ + τ + 2 p = 0. We have absorbed a factor of U in the definitions of ρ , τ and p .On the other hand, the scalar (Φ) field equation givesΦ ′′ + f ′ f Φ ′ + 2 Φ ′ r = Φ ′ ′ √ C r f (13)where C is a positive, non-zero constant. It is useful to note here that the radion dependsonly on the metric function f ( r ) and not on U ( r ). We also know that the existence of ahorizon in a static, spherically symmetric geometry is linked with the existence of zeros5n f ( r ). Thus, Φ ′ will always diverge at the horizon of any spherically symmetric, staticspacetime.The field equations with the requirement of traceless on-brane matter, leads to the followingequation for the metric function f and U : U ′′ U + f ′′ f − f ′ f U ′ U + 2 f ′ f r + 2 U ′ U r = 0 (14)A solution for f and U which satisfies the above tracelessness condition can be found by re-calling the Reissner–Nordstrom solution written in isotropic coordinates. For such a solutionwe have, f ( r ) = 1 − M r + e r (15) U ( r ) = 1 + Mr + M r − e r (16)where M and e are constants. We have retained the notation of ‘ e ’ and ‘ M ’ used inthe standard GR Reissner-Nordstrom solution where they represent charge and mass, re-spectively. However, here, ‘ e ’ and ‘ M ’ may not carry the same physical meaning as inReissner-Nordstrom. It is easy to check that the above-written functional forms of f and U satisfy the tracelessness criterion.We now look at the equation for the scalar Φ. Assume 1 + Φ = ξ . The first integral of thescalar wave equation then becomes ξ ′ = C r f = C r + a (17)where a = e − M and C , an integration constant. It is clear that there will be two differentsolutions for a > e > M ) and a < e < M ). Both these solutions must convergeto the solution for e = M which gives the extremal limit.When e > M (i.e. the nakedly singular solution) we obtainΦ( r ) = (cid:18) C a tan − ra + C (cid:19) − r ≥ C > r ) >
0. In addition, note that the radion is finite everywhere includingthe asymptotic region r → ∞ . In the limit a →
0, this solution for Φ will reduce to that forthe extremal case, given as Φ( r ) = (cid:18) C r + C (cid:19) − C > C > C = 1, a = 2, C = 3 (blue curve) and C = − C = − C is not permitted. - - FIG. 1: Φ( r ) vs. r for a > C = 1 , a =2 , C = 3( blue ) , C = − red ) - FIG. 2: Φ( r ) vs. r for a < C = 1 , a =2 , C = − red ) , C = 3( blue ) When e < M (i.e. the black hole solution with horizon at r = a , a = √− a ), we obtainΦ( r ) = (cid:18) C a ln | r − ar + a | + C (cid:19) − r ≥ a . It may be observed that at the location of the horizon,the inter-brane distance becomes infinitely large (see Figure 2). As shown later, this diver-gence implies a divergent matter stress energy at the horizon, thereby making the solutionphysically disallowed. Note also that for C = 3 the radion has zeros and therefore such achoice of the parameters is not permissible. In the limit a → a < a = 0.Finally, let us obtain the radion for the Schwarzschild case, i.e. when e = 0. This turns outto be: Φ = (cid:18) C M ln 2 r − M r + M + C (cid:19) − r = M , Φ divergesmaking the solution again physically unacceptable. We illustrate this behaviour of radionprofile in Figure 3. We note that apart from the divergence of the radion at the horizon, theradion profile hits Φ = 0 at locations outside the horizon. This is a generic feature. Coupledwith the radion divergence at the horizon, one is forced to conclude that a physically allowedradion is not possible in a Schwarzschild spacetime.7 .5 1.0 1.5 2.0 2.5 3.0 - - FIG. 3: Φ( r ) vs. r for Schwarzschild; C = 2 , C = 2 , M = 1 B. The Weak Energy Condition inequalities
Using the expressions of U , Φ and f mentioned in the previous subsection, in Eqns. (9)-(11)we obtain non-zero ρ , τ and p , i.e. we have non-trivial on-brane matter.It is now necessary to see if the matter is reasonable in terms of satisfying the energyconditions [7] and whether the a → κ ℓ ρ = Φ e U r + α M r f − Φ ′ U f (cid:18) − Mr + 4 a r (cid:18) M r (cid:19)(cid:19) (22) κ ℓ τ = − Φ e U r + 3 α M r f + Φ ′ (cid:18) fU r + 1 r − a r f (cid:19) (23)where we have used C = αM where α is a proportionality constant.The pressure p can be obtained from the tracelessness condition. It is given as, κ ℓ p = 12 ( ρ − τ ) (24)Recall that ρ ≥ ρ + τ ≥ ρ + p ≥ ρ + τ ≥ ρ + p ≥ ρ + τ and ρ + p in a compact form. They are8iven as: κ ℓ ( ρ + τ ) = 2 f r (cid:20) Φ ′ (cid:18) − a r (cid:19) + 2 α M r (cid:21) (25) κ ℓ ( ρ + p ) = 1 r (cid:20) e U + Φ ′ U f (cid:8) r ( U − f ) − ra (cid:9)(cid:21) (26)We now analyse the various cases separately. Case 1 ( α = 1 , e > M ): In Figs. 4,5,6 we have chosen e = 5 , M = 3 (naked singularityar r = 1) so that e − M = 16. We also choose α = 1 which means that the M in themetric functions is the same as the C in the radion field solution. The plot of ρ vs. r (Figure 4) demonstrates that ρ is indeed positive over the entire domain of r . Also, fromthe expression for ρ , it is clear that the dominant term which varies as r as r → ∞ has apositive coefficient. However, the ρ + τ ≥ r = 1 (Figure 5). In the same way, we note that the ρ + p ≥ r , a fact which we demonstrate in Figure 6. In Fig. 6, the y -axis is scaled by a factorof 10 and we plot from r = 100 to r = 1000. The negativity of ρ + p at large values of r ,is evident from this figure. Thus the on-brane matter must necessarily violate the WEC ifthe naked singularity is a viable solution. FIG. 4: M = 3 , e = 5 , C = 3, α = 1, ρ vs. r . - - - - FIG. 5: M = 3 , e = 5 , C = 3, α = 1, ( ρ + τ ) vs. r .
200 400 600 800 1000 - - - - FIG. 6: M = 3 , e = 5 , C = 3, α = 1, 10 ( ρ + p ) vs. r . To see this more explicitly let us go back to the compact expression for ρ + τ quoted in (25),with α = 1. Here, note that the second term is positive but the positivity of the first termdepends crucially on the sign of Φ ′ as well as the value of r . Now, we can have a solutionΦ (for e − M > ′ (see Figure 1, blue curve) which, in turnwill ensure the positivity of the brane separation. Further, note that in this case, f is neverzero. Hence the ρ + τ ≥ r > a crit . For r < a crit there is a violation. The value of a crit is determined by the r for which the term in square9rackets in (25) turns negative. a crit < a , as seen in Figure 5. In general, the value of a crit can be found from a solution of the transcendental equation: tan − a crit a + C a M = aa crit a − a crit (27)Further, it is easy to see that near the naked singularity one cannot avoid a violation of theWEC by any choice of the parameters. Let us evaluate the term in square brackets in (25)at r = e − M (the location of the naked singularity). One finds that[ .... ] = 16 M ( ν − " − C − √ ν − − r ν − ν + 1 (28)where ν = eM >
1. Note, that earlier we found C > ν > C >
2. One cancontrol the amount of violation by increasing M (since it appears in the denominator) suchthat M − e is negative. Similarly, by adjusting C , M , e one can control the extent of theregion (the value of a crit where the WEC will be violated). But there is no way to avoid theviolation of the ρ + τ inequality though it may be less in value or confined to a small region.In a similar manner, the ρ + p inequality must necessarily be violated for large r . The large r limit of the ρ + p expression clearly shows this feature. We have demonstrated this violationof the ρ + p WEC inequality in Figure 6.
Case 2 ( e − M > , α = 1 ): In contrast, if we choose α = 1 (i.e. M = C ), the Eqn. (29)becomes [ .... ] = 16 αM ( ν − " α − C − α √ ν − − r ν − ν + 1 (29)Here, with appropriate choices of α , C and ν one can satisfy the ρ , ρ + τ inequalities overthe required domain, i.e. from r = e − M to infinity. For example, with M = 6, e = 10, α = 6and C = 3 we find that in the domain 2 ≤ r ≤ ∞ there is no violation of the ρ , ρ + τ inequalities (the naked singularity is at r = 2). This is shown in Figures 7,8. However, the ρ + p inequality still remains violated at large r , a fact we show in Figure 9.By tuning α , C and ν one can move around and reduce the extent of WEC violation thoughit cannot be avoided completely for the ρ + p inequality. Thus, for a naked singularity, WECviolation of on-brane matter is necessary and this conforms with the Cosmic CensorshipHypothesis [7]. 10 FIG. 7: M = 6 , e = 10 , C =3, α = 6, ρ vs. r . - FIG. 8: M = 6 , e = 10 , C =3, α = 6, ( ρ + τ ) vs. r .
400 500 600 700 800 900 1000 - - - FIG. 9: M = 6 , e = 10 , C =3, α = 6, 10 ( ρ + p ) vs. r . Recall that when e − M ≤
0, the Φ diverges at the black hole horizon r = a which makesthe RN black hole solution unphysical. The divergence of Φ also implies a divergence of ρ , τ and p , as is evident from (22),(23). Thus, we do not discuss this case any further here.In our earlier paper, we had noted that the extremal limit solution also does require on-branematter which violates the energy condition inequality ρ + τ ≥
0. Here we have seen thatthis violation persists for all e > M . IV. CONCLUSION
In this article, we have discussed the viability of the various well known GR solutions likeSchwarzschild and Reissner–Nordstrom in the context of the KS theory of gravity. Thecrucial element in this work is related to finding a stable radion which is finite and non-zeroeverywhere. The RN black hole solution requires an infinite inter-brane distance at thehorizon – a fact which makes it unphysical. On the other hand, the Schwarzschild solutionrequires a radion which diverges at the horizon, vanishes at two values of r outside thehorizon and is negative between these values. The RN naked singularity and the extremalRN solution do have a non-zero and finite radion, but our analysis shows that the requiredon-brane matter violates the Weak and the Null Energy Condition. For the naked singularity,this feature conforms with the Cosmic Censorship Conjecture [7].A possible way out of the problems mentioned here is to look for solutions which are non-singular in nature and see if the radion is finite and non-zero everywhere and the on-branematter satisfies the energy conditions. The finiteness of the radion seems to be in conflictwith the existence of a black hole horizon. Does this mean that there are no eternal blackhole solutions in this theory? We have not proved any such statement but the analysis on11he RN and Schwarzschild solutions seem to suggest such an outcome.Finally, it may be useful to assume a specific form of the on-brane matter (on either orboth branes), and then find the radion and the metric functions. This will genuinely be like finding an exact solution , given the matter content on the branes. However, knowing thecomplicated nature of the field equations, this will not be easy to do. Further, if we removethe traceless requirement, the equations will become even more difficult to solve.One might view the KS theory as a scalar-tensor theory in its own right. Then, of course,the radion is just another scalar field without any reference to braneworlds and it need notsatisfy the requirements we have mentioned in this paper. However, such an approach isnot the main motivation of this article where we have chosen to view the radion as relatedto the proper distance between branes located in a higher dimensional bulk spacetime. [1] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370 (1999); ibid , 4690 (1999).[2] T. Shiromizu, K. Maeda and M. Sasaki, Phys. Rev. D62 , 024012 (2000), R. Maartens, Braneworld gravity, Living Rev. Relativity , 5 (2010).[3] S. Kanno, J. Soda, Phys.Rev. D 66 , 083506 (2002); T. Shiromizu and K. Koyama, Phys. Rev.
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