Gen Relativ Gravit (2015) 47:70DOI 10.1007/s10714-015-1912-6RESEARCHARTICLEA note on spherically symmetric, static spacetimesin KannoSoda on-brane gravitySayan Kar1Sayantani Lahiri2,3Soumitra SenGupta4Received: 11 April 2015 / Accepted: 16 May 2015 / Published online: 29 May 2015 Springer Science+Business Media New York 2015Abstract Spherically symmetric, static on-brane geometries in the KannoSoda (KS)effective scalartensor theory of on-brane gravity are discussed. In order to avoid branecollisions and/or an innite inter-brane distance, at nite values of the brane coordi-nates, it is necessary that the radion scalar be everywhere nite and non-zero. Thisrequirement constrains the viability of the standard, well-known solutions in generalrelativity (GR), in the context of the KS effective theory. The radion for the Schwarz-schild solution does not satisfy the above requirement. For the ReissnerNordstrom(RN) naked singularity and the extremal RN solution, one can obtain everywherenite, non-zero radion proles, though the required on-brane matter violates the WeakEnergy Condition. In contrast, for the RN black hole, the radion prole yields a diver-gent inter-brane distance at the horizon, which makes the solution unphysical. Thus,both the Schwarzschild and the RN solutions can be meaningful in the KS effectivetheory, only in the trivial GR limit, i.e. with a constant, non-zero radion.BSayan Karsayan@iitkgp.ac.inSayantani Lahirisayantani.lahiri@gmail.comSoumitra SenGuptatpssg@iacs.res.in1Department of Physics and Center for Theoretical Studies, Indian Institute of Technology,Kharagpur 721 302, India2Institute for Physics, University Oldenburg, 26111 Oldenburg, Germany3ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany4Department of Theoretical Physics, Indian Association for the Cultivation of Science, 2A and 2BRaja S.C. Mallick Road, Jadavpur, Kolkata 700 032, India1 370 Page 2 of 13 S. Kar et al.Keywords Braneworld gravity General relativity Radion Energy conditions1 IntroductionEffective, on-brane theories of gravity have been in vogue ever since the RandallSundrum warped braneworld scenario was proposed [1, 2]. Among such four dimen-sional gravity theories, the most well-known one is due to Shiromizu, Maeda andSasaki (SMS) [3, 4] which considers a bulk with an innite extra-dimension and asingle brane. There have been proposals on effective theories in a two-brane set-up. Inthis article, we consider one such effective theory due to Kanno and Soda (KS) [5, 6].A major difference between the SMS and the KS effective theories is the presence ofa non-local (bulk dependent) term in the former and the absence of any non-localityin the latter. Our objective is to look for solutions in the KS effective theory, keepingin mind that the radion eld, which is linked to the distance between the branes, (1)is never zero in value (thus, avoiding brane collisions) and (2) does not diverge at anynite value of the brane coordinates.Recently [7], we have discussed some cosmological and spherically symmetric,static spacetimes in this four dimensional, effective, on-brane, scalartensor theory ofgravity. The spherically symmetric, static solutions obtained in [7] turned out to be theMajumdarPapapetrou solution [8, 9] with the source being the effective scalar eld(radion) energy-momentum and additional on-brane matter. Here, we ask a broaderquestion: are the standard general relativity (GR) solutions like the Schwarzschild orthe ReissnerNordstrom (RN) permissible in the KS theory? Obviously, we do notexpect the GR solutions to arise in the KS theory with the same matter content asin GR. Rather, we would like to nd out if the radion energy momentum and extraon-brane matter can conspire in unison to allow the Schwarzschild or the RN solutionin the KS effective theory.It should be noted that there are several aspects related to this question, a couple ofwhich we have already mentioned. In addition to having a radion which is nite andnon-zero everywhere, the on-brane matter must also be physically reasonable in theclassical sense, i.e. it must satisfy one of the well-known energy conditions, such asthe Weak Energy Condition or the Null Energy Condition [11].We may recall that the ReissnerNordstromsolution does arise as a solution [10] ofthe ShiromizuMaedaSasaki (SMS) single brane effective Einstein equations [3, 4],where the non-local contribution from the bulk Weyl tensor (the traceless E) actsas its source without any explicit on-brane matter. The functional form of the Edepends on the bulk Weyl tensor and other features of the bulk geometry. It cannot bedetermined uniquely from the knowledge of four dimensional, on-brane physics. Incontrast, in the KS effective theory, the inuence of the bulk is exclusively through theradion eld which depends only on the brane coordinates. It is therefore meaningful toask whether the equations which arise in the effective, on-brane KannoSoda theory(which are local and different from those obtained in the single brane SMS effectivetheory) also admit a RN or a Schwarzschild solution, in some way.We will mainly work with the ReissnerNordstrom solution written in isotropiccoordinates. After obtaining the radion proles in the various cases, we will see if the1 3A note on spherically symmetric, static spacetimes in Page 3 of 13 70radion satises the necessary requirements. Subsequently, we will analyse the natureof required on-brane matter with reference to the Weak Energy Condition [11].2 The KannoSoda effective theory: main equationsThe effective on-brane scalartensor theories developed by Kanno and Soda [5, 6] inthe context of the RandallSundrum two-brane model leads to the following Einstein-like equations on the visible b brane [5, 6],G =2lTb +2(1 +)lTa +1

_ g

_32(1 +)_ 12g

_(1)Here g is the on-brane metric, the covariant differentiation is dened with respectto g and we have taken the ve dimensional line element as,ds25 = e2(x)dy2+ g(y, x)dxdx(2)2is the 5D gravitational coupling constant.Ta, Tbare the stress-energy on thePlanck brane and the visible brane respectively. The appearance of Ta (matter energymomentum on the a brane) in the eld equations on the b brane, inspired the usageof the term quasi-scalartensor theory. However, if we assumeTa =0 then wehave the usual scalartensor theory.We denote d(x) as the proper distance between branes located at y = 0 and y = l.d(x) is dened as,d(x) =_l0e(x)dy (3)We further dene= e2dl 1. It may be observed that the viability of such a modelwith a everywhere nite and non-zero brane separation, implies that (1) the minimumof d(x) is not equal to zero and (2) d(x) is never innity at any nite value of thebrane coordinates. This, in turn indicates that the value of (x) is always greater thanzero and (x) never becomes innity.Note that the radion contribution on the R.H.S. of the eld equation is traceless,which is reminiscent of the traceless E in the SMS effective theory.The scalar eld equation of motion on the visible brane is given as, =2lTa+ Tb2 +312 +3dd_)(

_(4)where Ta, Tbare the traces of energy momentum tensors on Planck (a) and visible(b) branes, respectively. The coupling function () expressed in terms ofis,() = 32(1 +)(5)1 370 Page 4 of 13 S. Kar et al.We know that gravity on both the branes are not independent. Dynamics on the Planckbrane at y = 0 is linked to that on the visible brane through the relation [5, 6]:(x) =

1 (6)where is the radion eld as dened on Planck brane. The induced metric on thevisible brane can be expressed in terms ofas,gbbrane=(1 )_h + g(1)_h, , Ta, Tb, y = l__(7)where g(1) is the rst order correction term(see [5, 6] for details). It is possible to workwith the gravity theory and theeld equation on the a brane.In our work here, we assume that the on-brane stress energy is nonzero only onthe b brane (visible brane). We also assume that the on-brane matter is traceless andtherefore, since the effective radion stress energy is also traceless, the Ricci scalar ofthe spacetime geometry is identically zero. Conversely, if we assumeR = 0, the on-brane matter is traceless. This choice of R = 0 enables us to propose the standard GRsolutions (like Schwarzschild and ReissnerNordstrom) as solutions in the KannoSoda effective theory with on-brane matter. The two main hurdles we need to addressare therefore:Are the standard GR solutions also solutions in the effective theory, with a non-zero, everywhere nite radion?What is the nature of the on-brane matter required to support such standard GRsolutions?3 Spherically symmetric, static solutions3.1 Line element, eld equations and the radionLet us assume a four dimensional line element on the visible brane, in isotropic coor-dinates, given asds2= f 2(r)U2(r)dt2+U2(r)_dr2+r2d2+r2sin2d2_(8)where U(r) andf (r) are the unknown functions to be determined from the Einstein-like equations. Using the above line element ansatz and the assumption thatisa function of ralone, we get the following eld equations from the Einstein eldequations mentioned above.2U

U+_U

U_24 U

Ur =

24(1 +)+_U

Uf

f_

+2l (9)1 3A note on spherically symmetric, static spacetimes in Page 5 of 13 70_U

U_2+2 f

f_U

U+ 1r_ = 324(1 +) U

U

2

rf

f

+2l (10)_U

U_2+f

f2f

fU

U+f

f1r =

24(1 +)+ U

U

+

r +2lp (11)where , and p correspond to on-brane matter and using the tracelessness conditionwe have + +2p = 0. We have absorbed a factor of U2in the denitions of , andp.On the other hand, the scalar () eld equation gives

+f

f

+2

r=

22(1 +)(12)which can be integrated once to get

1 += 2C1r2f(13)whereC1is a positive, non-zero constant. It is useful to note here that the radiondepends only on the metric function f (r) and not on U(r). We also know that theexistenceofahorizoninastatic, sphericallysymmetricgeometryislinkedwiththe existence of zeros in f (r). Thus,

will always diverge at the horizon of anyspherically symmetric, static spacetime.The eld equations with the requirement of traceless on-brane matter, leads to thefollowing equation for the metric functionf and U:U

U+f

ff

fU

U+2f

f r +2 U

Ur = 0 (14)A solution for f and U which satises the above tracelessness condition can be foundby recalling the ReissnerNordstrom solution written in isotropic coordinates. Forsuch a solution we have,f (r) = 1 M24r2 +e24r2(15)U(r) = 1 +Mr+M24r2 e24r2(16)where M2and e2are constants. We have retained the notation of e and M used inthe standard GR ReissnerNordstrom solution where they represent charge and mass,respectively. However, here, e and M may not carry the same physical meaning asin ReissnerNordstrom. It is easy to check that the above-written functional forms off and U satisfy the tracelessness criterion.1 370 Page 6 of 13 S. Kar et al.0 5 10 15 20 25 3021012345Fig. 1 (r) versus r for a2> 0; C1 = 1, a = 2, C4 = 3 (blue), C4 = 3 (red) (colour gure online)We nowlook at the equation for the scalar . Assume 1+ = 2. The rst integralof the scalar wave equation then becomes

=C1r2f=C1r2+a2(17)where a2=e2M24and C1, an integration constant. It is clear that there will be twodifferent solutions for a2> 0 (e2>M2) and a2< 0(e2M2(i.e. the nakedly singular solution) we obtain(r) =_C1atan1ra + C42_21 (18)This solution remains valid for all r 0 with a choice of C4> 2 necessary to ensurethe positivity condition (r) > 0. In addition, note that the radion is nite everywhereincluding the asymptotic region r . In the limit a 0, this solution forwillreduce to that for the extremal case, given as(r) =_C1r+ C42_21 (19)HereC1>0 andC4>2 is a requirement for positivity of the brane separation.Figure 1 shows the cases with C1 = 1, a = 2, C4 = 3 (blue curve) and C4 = 3 (redcurve). It may be noted from the gure that for C4 = 3 the radion ends up havingzeros and is therefore such a choice of C4 is not permitted.When e2M2): In Figs. 4, 5 and 6 we have chosen e = 5, M = 3 (nakedsingularity ar r = 1) so that e2 M2= 16. We also choose = 1 which means thattheM in the metric functions is the same as the C1 in the radion eld solution. Theplot of versus r (Fig. 4) demonstrates that is indeed positive over the entire domainof r. Also, from the expression for , it is clear that the dominant term which varies as1r2 as r has a positive coefcient. However, the + 0 inequality is violatedin a nite region around r = 1 (Fig. 5). In the same way, we note that the + p 0inequality is violated for large r, a fact which we demonstrate in Fig. 6. In Fig. 6,they-axis is scaled by a factor of 107and we plot from r = 100 to r = 1000. The1 3A note on spherically symmetric, static spacetimes in Page 9 of 13 701 2 3 4 5 6701020304050Fig. 4 M = 3, e = 5, C4 = 3, = 1, versus r1 2 3 4 5 6 72.01.51.00.50.00.51.01.52.0Fig. 5 M = 3, e = 5, C4 = 3, = 1, ( +) versus rnegativity of + p at large values of r, is evident from this gure. Thus the on-branematter must necessarily violate the WEC if the naked singularity is a viable solution.To see this more explicitly let us go back to the compact expression for + quotedin (25), with = 1. Here, note that the second term is positive but the positivity ofthe rst term depends crucially on the sign of

as well as the value of r. Now, wecan have a solution (for e2 M2> 0), with an always positive

(see Fig. 1, bluecurve) which, in turn will ensure the positivity of the brane separation. Further, notethat in this case, f is never zero. Hence the + 0 inequality is never violated aslong as r> acri t. For r< acri t there is a violation. The value of acri t is determined bythe r for which the term in square brackets in (25) turns negative. acri t< a, as seen inFig. 5. In general, the value of acri t can be found from a solution of the transcendentalequation:t an1acri ta+ C4a2M =aacri ta2a2cri t(27)1 370 Page 10 of 13 S. Kar et al.200 400 600 800 100020151050Fig. 6 M = 3, e = 5, C4 = 3, = 1, 107( + p) versus rFurther, it is easy to see that near the naked singularity one cannot avoid a violation ofthe WEC by any choice of the parameters. Let us evaluate the term in square bracketsin (25) at r =eM2(the location of the naked singularity). One nds that[. . .] =16M( 1)3_1 C42221 tan1_ 1 +1_(28)where =eM>1. Note, that earlier we foundC4>2 from the requirement ofpositive brane separation. Thus the R.H.S. of (28) is always negative for any> 1and C4> 2. One can control the amount of violation by increasing M (since it appearsin the denominator) such thatM2 e2is negative. Similarly, by adjusting C4, M, eone can control the extent of the region (the value of acri twhere the WEC will beviolated). But there is no way to avoid the violation of the + inequality though itmay be less in value or conned to a small region.In a similar manner, the +p inequality must necessarily be violated for larger. Thelarge r limit of the + p expression clearly shows this feature. We have demonstratedthis violation of the + p WEC inequality in Fig. 6.Case 2(e2 M2> 0, = 1): In contrast, if we choose = 1(i.e.M =C1), theEq. (29) becomes[. . .] =16M( 1)3_ C42221 tan1_ 1 +1_(29)Here, with appropriate choices of , C4 and one can satisfy the , + inequalitiesover the required domain, i.e. from r =eM2to innity. For example, withM =6, e =10, =6 and C4 =3 we nd that in the domain 2 r there is noviolation of the , + inequalities (the naked singularity is at r = 2). This is shownin Figs. 7 and 8. However, the + p inequality still remains violated at large r, a factwe show in Fig. 9.1 3A note on spherically symmetric, static spacetimes in Page 11 of 13 702 3 4 5 67050100150200250300Fig. 7 M = 6, e = 10, C4 = 3, = 6, versus r2 4 6 8 1010123456Fig. 8 M = 6, e = 10, C4 = 3, = 6, ( +) versus r400 500 600 700 800 900 1000151050Fig. 9 M = 6, e = 10, C4 = 3, = 6, 106( + p) versus r1 370 Page 12 of 13 S. Kar et al.By tuning , C4 and one can move around and reduce the extent of WECviolationthough it cannot be avoided completely for the +p inequality. Thus, for a nakedsingularity, WEC violation of on-brane matter is necessary and this conforms with theCosmic Censorship Hypothesis [11].Recall that when e2 M2 0, the diverges at the black hole horizon r =awhich makes the RNblack hole solution unphysical. The divergence of also impliesa divergence of , andp, as is evident from (22), (23). Thus, we do not discuss thiscase any further here.In our earlier paper, we had noted that the extremal limit solution also does requireon-brane matter which violates the energy condition inequality + 0. Here wehave seen that this violation persists for all e2>M2.4 ConclusionIn this article, we have discussed the viability of the various well known GR solutionslike Schwarzschild and ReissnerNordstromin the context of the KS theory of gravity.The crucial element in this work is related to nding a stable radion which is niteand non-zero everywhere. The RN black hole solution requires an innite inter-branedistance at the horizona fact which makes it unphysical. On the other hand, theSchwarzschild solution requires a radion which diverges at the horizon, vanishes attwo values of routside the horizon and is negative between these values. The RNnaked singularity and the extremal RN solution do have a non-zero and nite radion,but our analysis shows that the required on-brane matter violates the Weak and the NullEnergy Condition. For the naked singularity, this feature conforms with the CosmicCensorship Conjecture [11].A possible way out of the problems mentioned here is to look for solutions whichare non-singular in nature and see if the radion is nite and non-zero everywhereand the on-brane matter satises the energy conditions. The niteness of the radionseems to be in conict with the existence of a black hole horizon. Does this mean thatthere are no eternal black hole solutions in this theory? We have not proved any suchstatement but the analysis on the RN and Schwarzschild solutions seem to suggestsuch an outcome.Finally, it may be useful to assume a specic form of the on-brane matter (on eitheror both branes), and then nd the radion and the metric functions. This will genuinelybe like nding an exact solution, given the matter content on the branes. However,knowing the complicated nature of the eld equations, this will not be easy to do.Further, if we remove the traceless requirement, the equations will become even moredifcult to solve.One might view the KS theory as a scalartensor theory in its own right. Then,of course, the radion is just another scalar eld without any reference to braneworldsand it need not satisfy the requirements we have mentioned in this paper. However,such an approach is not the main motivation of this article where we have chosen toview the radion as related to the proper distance between branes located in a higherdimensional bulk spacetime.1 3A note on spherically symmetric, static spacetimes in Page 13 of 13 70Acknowledgments One of the authors, Sayantani Lahiri, would like to acknowledge the support by DFGResearch Training Group 1620 Models of Gravity.References1. Randall, L., Sundrum, R.: Phys. Rev. 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