Uniqueness of the static spacetimes with a photon sphere in Einstein-scalar field theory Stoytcho Yazadjiev ∗ 5 1 1Department of Theoretical Physics, Faculty of Physics, Sofia University 0 2 5 J. Bourchier Blvd., Sofia 1164, Bulgaria n 2Theoretical Astrophysics, Eberhard-Karls University of Tübingen, a J Tübingen 72076, Germany 7 2 ] c q Abstract - r g Inthepresentpaperweproveauniquenesstheoremforthestaticandasymptotically [ flat solutions to the Einstein-scalar field equations which possess a photon sphere. We 1 showthatsuchsolutionsareuniquelyspecifiedbytheirmassM andscalarchargeqand v 7 thattheyareisometric totheJanis-Newman-Winicour solution withthesamemassand 3 scalarcharge subjecttotheinequality q2 <3. 8 M2 6 0 . 1 Introduction 1 0 5 General relativity and the generalized gravitational theories predict the existence of space- 1 : time regions where light can be confined in closed orbits [1]-[10]. These regions called v i photon spheres play a very important role in gravitational lensing which is a standard and X powerful tool in modern observational astronomy and astrophysics [4],[6]. From the point r a ofview of gravitationallensing thephoton sphere can beconsidered as a timelikehypersur- face on which the bending angle of a light ray is unboundedly large [4],[6]. A more precise definition will be given below. In the presence of a photon sphere the gravitational lensing effect gives rise to the so-called relativistic images which are characteristic for relativistic gravity. It is commonlyexpected that ultracompactobjects such as black holes, neutron and boson stars, wormholes and naked singularities are surrounded by a photon sphere [1]-[10]. Apart from its significance for gravitational lensing the photon sphere is intimately related to thequasinormalmodesofultracompact objectsand iscrucially relevant fortheirstability [9], [11]-[14]. Thephotonspherepossessessomeveryspecificcharacteristicswhichmakeitaveryspe- cial and unique object. For example, in static spacetimes, as the explicit exact or numerical solutions show, the lapse function is constant on the photon sphere. Moreover, the photon [email protected] ∗ 1 sphere is a totally umbilic hypersurface with constant mean curvature and constant surface gravity[5],[7],[15]. Withthesepropertiesthephotonsphereresemblesverymuchtheblack holehorizon. It is wellknownthatthepresenceofan eventhorizonallowsus toclassifythe asymptotically flat spacetimes only in terms of their conserved asymptotic charges, such as the mass and the electric charge in the static case [16]. In analogy with the black hole case, thenaturalquestionthatarisesis: Doesthepresenceofaphotonsphereuniquelyspecifythe spacetimes with given asymptotic charges? In other words, we ask whether the spacetimes possessing a photon sphere can be classified only in terms of asymptotic charges (or other global physical quantities). The problem set in this way seems to be much more difficult that the classification of black hole spacetimes since the spectrum of spacetimes possessing a photon sphere is much richer than the spectrum of black hole spacetimes. Nevertheless, in some cases the desired classification can be achieved. Recently Cederbaum [15] proved that the static, asymptotically flat solutions to the vacuum Einstein equations with mass M possessingaphoton sphere are isometricto the Schwarzschild solutionwith the samemass. In the present paper we consider the more general case of static Einstein-scalar field equa- tions. We prove that the static and the asymptotically flat solutions to the Einstein-scalar fieldequationspossessingaphotonsphereareuniquelyspecifiedbytheirmassM andscalar charge q. Moreprecisely weprovethatthe staticand the asymptoticallyflat solutionsto the Einstein-scalar field equations with mass M and scalar charge q possessing a photon sphere are isometric to the Janis-Newman-Winicour solution [17] with the same mass and scalar chargeas well as withacertain restrictionontheratioq/M. 2 General definitions and equations In thepresent paperweconsiderEinstein-scalarfield theorydescribed bytheaction 1 S = 16p Z d4x g(4) R(4)−2g(4)µn (cid:209) (µ4)(cid:209)j (n 4)j , (1) q (cid:16) (cid:17) where j is the scalar field, (cid:209) (4) and R(4) are the Levi-Civitaconnection and theRicci scalar µ curvature with respect to the spacetime metric gµn . This action gives rise to the following field equationson thespacetimemanifoldM(4) Ric(µ4n) =2(cid:209) (µ4)(cid:209)j (n 4)j , (2) (cid:209) (4)(cid:209) (4)µj =0, µ (4) with Ricµn being the Ricci tensor. It is important to note the following. Although the Einstein-scalarfield equationscan be considered in theirownrightthey are in fact the"vac- uum" field equations of scalar-tensor theories presented in the so-called Einstein frame. In this way all results for (2) obtained in the present paper can be easily transformed as results foran arbitrary classofscalar-tensortheories. In the present work we focus on static and asymptotically flat spacetimes. A spacetime iscalledstaticifthereexistsasmoothRiemannianmanifold(M(3),g(3))and asmoothlapse functionN :M(3) R+ such that → 2 M(4) =R M(3), g= N2dt2+g(3). (3) × − In addition to the metric staticity we have to defined the scalar field staticity. The scalar field is called static if Lx j = 0 where Lx is the Lie derivative along the Killing field x = ¶ . We note that both notions of staticity are consistent since the Ricci 1-form Ric(4)[x ] = ¶ t x µRic(µ4n)dxn is zero duetothefield equationsandthefact that x µ(cid:209) (µ4)j =Lx j =0. We will adopt the following notion of asymptotic flatness. The spacetime is called asymptoticallyflatifthereexistsacompactsetK M(3) suchthatM(3) K isdiffeomorphic toR3 B¯ whereB¯ is theclosed unitballcentered a⊂t theorigininR3,and−such that \ M q g(3) =d +O(r−1), N =1 +O(r−2), j =j ¥ +O(r−2), (4) − r − r with respect to the standard radial coordinate r of R3. Here M, j ¥ and q are constants with M and q being themass and thescalarcharge. We willconsiderspacetimes withM >0 and withoutlossofgeneralitywewillset j ¥ =0. Thedimensionallyreduced staticEinstein-scalarfield equationsarethefollowing Ric(3) =N 1(cid:209) (3)(cid:209) (3)N+2(cid:209) (3)(cid:209)j (3)j , ij − i j i j (cid:209) (3)(cid:209) (3)iN =D (3)N =0, (5) i (cid:209) (3) N(cid:209) (3)ij =0, i (cid:16) (cid:17) where(cid:209) (3) andRic(3) aretheLevi-CivitaconnectionandtheRiccitensorwithrespecttothe i ij (3) metricg . ij Now we can define the notion of photon sphere. First we give the definition of photon surface[5],[15]. Definition An embedded timelike hypersurface P ֒ M(4) is called a photon surface if → andonlyifanynullgeodesicinitiallytangenttoP remainstangenttoP aslongasitexists. Thedefinitionofphotonsphereisanaturalextensionofthedefinitioninthepurevacuum case[15]. Definition Let P ֒ M(4) be a photon surface. Then P is called a photon sphere if the lapsefunctionN andt→hescalarfieldj areconstantalongP. It is worth noting that the definition of the photon sphere is very well justified by the known exact and numerical solutions in general relativity and the scalar-tensor theories as wellas thestudiesofthegravitationallensingin thesetheories. As an additional technical assumption we shall assume that the lapse function regu- (3) larly foliates the region of spacetime M exterior to the photon sphere. This means that ext g(3)((cid:209) (3)N,(cid:209) (3)N) = 0 everywhere on M(3). In what follows we will use the function ext 6 r :M(3) R+ defined by ext → 3 1/2 r = g(3)((cid:209) (3)N,(cid:209) (3)N) − . (6) h i Nowletusconsiderthe2-dimensionalintersectionS ofthephotonsphereP andthetime slice M(3). By the definition of the photon sphere, S which is the inner boundary of M(3), ext is given by N =N for some N R+. The metric induced on S will be denoted by s . It is 0 0 ∈ notdifficulttoseethatourassumptionsrestrictourconsiderationstothecaseofaconnected photon sphere. Moreover, all the level sets N =const, including S , are topological spheres whichisa directconsequencefrom ourassumptions. By the maximum principle for harmonic functions and by the asymptoticbehavior of N forr ¥ weobtainthatthevaluesofN onM(3) satisfy ext → N N <1. (7) 0 ≤ 3 Functional dependence between the lapse function and the scalar field Here we show that there is a functional dependence between the lapse function N and the scalarfield j . Morepreciselywehavethefollowing Lemma. The lapsefunctionand thescalarfield aresubjecttotherelation q j = ln(N), (8) M withM andq beingthemassand thescalarchargerespectively. Proof. LetusfirstusethefactthatN isharmoniconM(3). IntegratingD (3)N =0onM(3) ext ext andapplyingtheGausstheoremwe have 0= D (3)N g(3)d3x= (cid:209) (3)Nd2S i (cid:209) (3)Nd2S i, (9) ZMe(x3t) q IS¥2 i −IS i where S2 is the 2-dimensional sphere at infinity. This result and the asymptoticbehavior of ¥ N give 1 1 M = (cid:209) (3)Nd2S i = (cid:209) (3)Nd2S i. (10) 4p IS¥2 i 4p IS i Inthesamewaymakinguseofthefieldequationforj ,namely(cid:209) (3) N(cid:209) (3)ij =0,and i theasymptoticbehaviorofN and j , wefind (cid:16) (cid:17) 4 1 1 q= (cid:209) (3)j d2S i = N (cid:209) (3)j d2S i. (11) 4p IS¥2 i 4p 0IS i The next step is to consider J =(cid:209)j (3)N Nln(N)(cid:209) (3)j . As a consequence of the field i i i − equations for N and j , it is not difficult one to show that (cid:209) (3)Ji =0. Integrating (cid:209) (3)Ji =0 i i (3) on M , applying the Gauss theorem and taking into account the asymptotic behavior of N ext andj , weobtain (cid:209)j (3)Nd2S i = Nln(N)(cid:209) (3)j d2S i, (12) IS i IS i whichinview of(11)gives q j = ln(N ). (13) 0 0 M Thefinal stepis toconsiderthedivergenceidentity q N 1 w w i =(cid:209) (3) j ln(N) w i , (14) − i i −M h(cid:16) (cid:17) i (cid:0) (cid:1) where q w =N(cid:209) (3)j (cid:209) (3)N. (15) i i −M i As one can verify this identity is a consequence of the field equations for the lapse function andthescalarfield. By applyingtheGausstheoremto theaboveidentityweobtain q q N 1 w w i g(3)d3x= j ln(N) w d2S i j ln(N) w d2S i =0,(16) − i i i ZMe(x3t) q IS¥2 (cid:16) −M (cid:17) −IS (cid:16) −M (cid:17) (cid:0) (cid:1) where we have taken into account the asymptoticbehavior of N and j as well as (13) in the evaluation of the surface integrals. Since N > 0 on M(3) we conclude that w = N(cid:209) (3)j ext i i − q(cid:209) (3)N =0 on M(3). Therefore we obtain j = q ln(N)+C withC being a constant. From M i ext M theasymptoticbehaviorofN and j orfromeq.(13)wefind thatC=0 whichproves(8). 4 Some relations for the photon sphere and inequality for the scalar charge to mass ratio In thissectionweshallderivesomekey relationsforthephotonsphereandan importantin- equalityforthescalarchargetomassratio. Alltheseresultswillbedirectlyorindirectlyused 5 in the proof of the main theorem. We first recall the important result of Claudel-Virbhadra- Ellis[5]: Theorem Let P ֒ M(4) be an embedded timelike hypersurface. Then P is a photon → surfaceifand onlyifit istotallyumbilic(iff itssecondfundamentalformis puretrace). Denotingthemetricinducedon P by p theaboveresult can bewrittenintheform P H KP = p, (17) 3 where KP is the second fundamental form and HP is the mean curvature of P. It is easy to show that HP is constant on P. Using the contracted Codazzi equation for (P,p) ֒ → (M(4),g(4))wehave 2 Ric(4)(X,n)= X(HP), (18) 3 wheren is theunitnormal toP and X is a vectorfield tangentto P, i.e. X G (TP). Taking intoaccountthefieldequationswefindRic(4)(X,n)=2X(j )n(j )=0sinc∈ej isconstanton P. HenceweconcludethatX(HP)=0which meansthat HP isconstanton P. The same can be proven for the Ricci scalar curvature RP of P. Applying the Gauss equationfor(P,p)֒ (M(4),g(4)) → 2 2 R(4) 2Ric(4)(n,n)=RP TrKP +Tr KP , (19) − − (cid:16) (cid:17) (cid:16) (cid:17) andmakinguseofthefield equationsand (17), wefind 2 RP = (HP)2 2(n(j ))2. (20) 3 − Since HP is constant on P we have to show that n(j ) is constant on P. In view of the relation (8), it is enough to show that n(N) is constanton P. Since thespacetimeis static, it issufficientto provethat n(N)is constantonS . Thiswillbedonebelow. Forthesecond fundamental form KS of(S ,s )֒ (M(3),g(3))(with a unitnormal n)we → have P P H H KS (X,Y)=g(3)((cid:209) (3)n,Y)=g(4)((cid:209) n,Y)=KP(X,Y)= p(X,Y)= p(X,Y), (21) X X 3 3 where X,Y G (TS ). Therefore we find KS = HPs which also gives a simple relation ∈ 3 between themean curvatures HP and HS , namelyHS = 2HP. Usingthisinformationinthe 3 contracted Codazzi equationweeasilyfind that Ric(3)(X,n)=0. In order to show that n(N) is constant on S we follow [15]. For an arbitrary X G (TS ) ∈ wehave 6 X(n(N))=X(n(N)) ((cid:209) (3)n)(N)=((cid:209) (3)(cid:209) (3)N)(X,n)= X − N Ric(3)(X,n) 2X(j )n(j ) =0, (22) − h i wherewehavetaken intoaccountthedimensionallyreduced field equations(5)andthefact thatN and j are constantonS . Therefore n(N)isindeed constantonS . FromtheGaussequationfor(S ,s )֒ (P,p),andtakingintoaccountthatthespacetime isstatic,itis easy toshowthattheRicci→scalarcurvatureRS ofS isgivenby 2 RS =RP = (HP)2 2(n(j ))2. (23) 3 − TheGaussequation R(3) 2Ric(3)(n,n)=RS TrKS 2+Tr KS 2 (24) − − for(S ,s )֒ (M(3),g(3))gives (cid:0) (cid:1) (cid:0) (cid:1) → 1 R(3) 2Ric(3)(n,n)=RS (HS )2. (25) − −2 InordertofindRic(3)(n,n)wecanusethedimensionallyreduced fieldequations(5)and D (3)N =D (2)N+(cid:209) (3)(cid:209) (3)N(n,n)+HS n(N), (26) whichleads to N Ric(3)(n,n)= HS n(N)+2N (n(j ))2. (27) 0 0 − Using again the dimensionally reduced field equations one can easily show that R(3) = 2(n(j ))2 which combinedwith(25) and(27)gives 1 N RS =2HS n(N)+ N (HS )2 2N (n(j ))2. (28) 0 0 0 2 − WecaneliminateRS from(23)and(28)byintegratingoverS andusingtheGauss-Bonnet theorem RS √s d2x = 8p for the topological sphere S . The integration reduces (23) and S (28) tothRefollowingrelations 3 1 1= (HS )2AS (n(j ))2AS , (29) 16p −4p 1 1 1 N0 = 4p HS n(N)AS +16p N0(HS )2AS −4p N0(n(j ))2AS . 7 From theseequationsitis easy toshowthatthefollowingimportantrelationis satisfied S 2n(N)=N H , (30) 0 orequivalently N HS r =2, (31) 0 0 bytakingintoaccount thatn(N)=r 1. −0 Thepresenceofaphotonsphereimposesaveryimportantrestrictiononthescalarcharge to mass ratio. It is convenient to express this restriction in terms of the parameter n defined by q2 −1/2 n = 1+ . (32) M2 (cid:18) (cid:19) Forthispurposewefirst rewrite(10)and (11)intheform 1 AS M = n(N)AS = , (33) 4p 4pr 0 1 q= 4p N0n(j )AS . (34) Substitutingrelations(33)and (34)in (29)and after somealgebraweobtain 4n 2 1 r − =N2 0 >0. (35) n 2 0 M Thisinequalitycombinedwiththedefinitionofn givesthedesired inequalityforn , namely 1 <n 1, (36) 2 ≤ q2 whichisequivalentto <3. M2 5 Uniqueness theorem Themainresultofthepresent paperis thefollowing Theorem There can be only one static and asymptoticallyflat spacetime (M(4),g(4),j ), satisfying the static Einstein-scalar field equations, possessing a photon sphere P ֒ M(4) → asaninnerboundaryofM(4),withlapsefunctionN regularlyfoliatingM(4) andgivenmass 8 M and scalar charge q. Moreover, the solution is isometric to the Janis-Newman-Winicour solutionwith 1 <n (M,q) 1. 2 ≤ Proof: Thestrategyoftheproofisthefollowing. Thefirstandthemostdifficultstepisto provethatthespacetimeissphericallysymmetric. Thenweconstructthesolutionexplicitly. (3) Let usconsiderthe3-metrichon M defined by ext h =N2g(3). (37) ij ij In termsofthenewmetricthedimensionallyreduced equationsbecome R(h) =2D ln(N)D ln(N)+2D j D j , ij i j i j D Diln(N)=0, (38) i D Dij =0, i whereD andR(h) aretheLevi-ChivitaconnectionandtheRiccitensorwithrespecttoh , i ij ij respectively. Takingintoaccount thefunctionaldependancej = q ln(N) theequations(38) M can becast intheform R(h) =2D ln(N˜)D ln(N˜), (39) ij i j D Diln(N˜)=0, i wherewehaveintroducedanewfunction N˜ =N1/n . (40) Proceeding furtherweconsidertheBach tensor 1 R(h) =2D R(h) + h D R(h). (41) ijk [i j]k k[i j] 2 Usingeqs. (39)aftera longcalculationwefind 1 Di W 1D c = c 7W 3R(h) R(h)ijk, (42) − i − ijk 16 (cid:0) (cid:1) where 1 1 N˜ 4N˜ c = hijD HD H 4, H = − , W = . (43) i j 1+N˜ 1+N˜ 2 (cid:0) (cid:1) WecancombinetheequationforN˜ from(39)witheq. (4(cid:0)2)toob(cid:1)tainanotherdivergence identity 9 1 Di W 1(HDc c D H) = Hc 7W 3R(h) R(h)ijk. (44) − i i − ijk − 16 (cid:2) (cid:3) SinceH >0 weobtaintheinequality Di W 1(HD c c DH) √hd3x 0. (45) − i i ZM(3) − ≥ ext (cid:2) (cid:3) Anotherinequalitycan beobtainedbytakingintoaccount thatH <1, namely Di W 1D c √hd3x Di W 1(HD c c DH) √hd3x. (46) − i − i i ZM(3) ≥ZM(3) − ext ext (cid:0) (cid:1) (cid:2) (cid:3) In bothcases (45)and (46)theequalityholdsifandonly ifR(h) =0. ijk Calculating theintegralin (45) by usingthe Gausstheorem and takinginto account (31) wegettheinequality 2 2n 1 N˜2 =Nn − . (47) 0 ≤ 2n +1 In thesameway (46)gives 2 2n 1 N˜2 =Nn − . (48) 0 ≥ 2n +1 Hence we conclude that N˜2 = Nn2 = 2n 1 and therefore R(h) = 0. This means that the 0 2n +−1 ijk metric h is conformally flat. As an immediate consequence we get that g is conformally ij ij (3) flat too(i.e. R(g) =R =0). ijk ijk (3) SinceN regularlyfoliatesM wecanintroduceadaptedcoordinatesinwhichthemetric ext (3) g takes theform ij g(3) =r 2dN2+s , (49) where s is the 2-dimensional metric on the 2-dimensional intersections S of the level AB N (3) setsN =const withthetimesliceM . Usingtheformula ext R(3)R(3)ijk = 4 KS N 1HS Ns KS NAB 1HS Ns AB + 1 s AB¶ r¶ r , (50) ijk N4r 4 AB −2 AB −2 r 2 A B (cid:20)(cid:18) (cid:19)(cid:18) (cid:19) (cid:21) where KS N is the extrinsic curvature of (S ,s ) ֒ (M(3),g(3)) and HS N is its trace, we AB N → concludethat 10