A classification (uniqueness) theorem for rotating black holes in 4D Einstein-Maxwell-dilaton theory Stoytcho Yazadjiev ∗ 1 1 Department of Theoretical Physics, Faculty of Physics, Sofia University 0 2 5 J. Bourchier Blvd., Sofia 1164, Bulgaria n a J 1 3 Abstract ] In the present paper we prove a classification (uniqueness) theorem for stationary, h t asymptoticallyflatblackholespacetimeswithconnectedandnon-degeneratehorizonin - p 4DEinstein-Maxwell-dilaton theorywithanarbitrarydilatoncouplingparametera . We e show that such black holes are uniquely specified by the length of thehorizon interval, h [ angular momentum, electric and magnetic charge and the value of the dilaton field at infinity when the dilaton coupling parameter satisfies 0 a 2 3. The proof is based 2 ≤ ≤ v onthenonpositivity oftheRiemanncurvatureoperatoronthespaceofthepotentials. A 2 generalization of the classification theorem for spacetimes with disconnected horizons 4 isalsogiven. 4 2 . 9 0 1 Introduction 0 1 : Key results in the 4D General relativity are the black hole classification (uniqueness) the- v orems [1]-[10]. Besides their intrinsic theoretical and mathematical value, these theorems i X haveplayedimportantroleinstudyingawiderangeoftopicsfromastrophysicalblackholes r a to thermodynamics of black holes. The advent of the theoretical physics leads to gravity theories and models generalizing general relativityand oneof theimportantproblemsis the study of black holes and in particular the classification of black hole solutions in these new theories. An example of such a theory is the so-called Einstein-Maxwell-dilaton gravity which naturally arises in the context of the low energy string theory [11, 12] and Kaluza- Kleingravity[13]. TheactionfortheEinstein-Maxwell-dilatongravityisgivenby A = d4x√ g R 2gab¶ ¶j j e 2ja F Fab (1) a b − ab Z − − − (cid:16) (cid:17) where R is the Ricci scalar curvature with respect to the spacetime metric g , F is the ab ab Maxwell field and j is the dilaton (scalar) field. The parameter a governs the coupling between the dilaton and electromagnetic field and is called a dilaton coupling parameter. It [email protected] ∗ 1 is worth mentioning that the Einstein-Maxwell-dilaton gravity arises also in some theories withgradientspacetimetorsion[14]. The black holes in 4D Einstein-Maxwell-dilatontheory were extensivelystudied in var- ious aspects during the last two decades. The static asymptotically flat Einstein-Maxwell- dilatonblackholeswereclassifiedin[15]. However,noclassificationisknownforthemore generalcaseofrotatingblackholes. Inthepresentpaperweproveauniquenesstheoremclas- sifyingall stationary,asymptoticallyflat black holes in 4D Einstein-Maxwell-dilatontheory with a certain restriction on the dilaton coupling parameter. In proving the classification theorems in pure 4D general relativity one strongly relies on the high degree of symmetries of the dimensionally reduced stationary and axisymmetric field equations. In both cases of vacuum Einstein and Einstein-Maxwell equations the space of potentials is a symmetric space which insures the existence of nice properties and, in particular, insures the existence oftheso-calledMazuridentity[5],whichisakeypointintheuniquenessproof. Contraryto the vacuum Einstein and Einstein-Maxwell gravity, the space of potentials associated with thedimensionallyreducedstationaryandaxisymmetricEinstein-Maxwell-dilatonfieldequa- tions is not a symmetric space in the general case. This forces us to use in the proof of the classification theorem other intrinsic geometric properties of space of potentials instead of theabsentsymmetries(isometries). SuchanintrinsicpropertyisthenonpositivityoftheRie- manniancurvatureoperatorofthespaceofpotentialswhichalsoinsuresthenonpositivityof the sectional curvature. The mentioned property together with the Bunting identity [6],[16] andnatural boundaryconditionsleads totheproofofdesiredclassificationtheorem. Thepaperisorganizedasfollows. InSec. 2and3wegiveinconciseformthenecessary mathematical background. The main result is presented in Sec. 4. In the Discussion we commenton possibleextensionsof the classification theorem and givean explicit extension forspacetimeswithdisconnectedhorizons. 2 Stationary Einstein-Maxwell-dilaton black holes Let (M,g,F,j ) be a 4-dimensional, analytic, stationary black hole spacetime satisfying the Einstein-Maxwel-dilatonequations g R =2(cid:209) (cid:209)j j +2e 2ja F F c abF Fcd , (2) ab a b − ac b cd − 4 (cid:16) (cid:17) (cid:209) e 2ja Fab =0=(cid:209) F , (3) a − [a bc] (cid:16) a (cid:17) (cid:209) (cid:209) aj = e 2ja F Fab, (4) a − ab −2 derived from the action (1). Let x be the asymptotically timelike complete Killing field, £x g=0, which we assume is normalized so that limg(x ,x )= 1 near infinity. We assume also that the Maxwell tensor and the dilaton field are invarian−t under x , in the sense that £x F =0=£x j . Weconsider4-dimensionalspacetimewithasymptoticregionM¥ =R3,1 andasymptotic metric g= dt2+dx2+dx2+dx2+O(r 1) (5) 1 2 3 − − 2 where x are the standard Cartesian coordinates on R3. Here O(r 1) stands for all metric i − componentsthatdrop offatleast as r 1 in theradial coordinater= x2+x2+x2. − 1 2 3 q Denote by H =¶ B the horizon of the black hole B=M/I (J+) where J are the null − ± infinitiesofspacetime. As apart ofourtechnicalassumptionsweassumethat: (i) H is non-degenerate and the horizon cross section is compact connected manifold of dimension2. (ii)Thedomainofoutercommunication M isgloballyhyperbolic. hh ii Accordingtothetopologicalcensorshiptheorem[17], M isasimplyconnectedman- ifold with boundary ¶ M = H. Moreover, one can shohhw tihiat the horizon cross section hh ii has spherical topology. In the present paper we shall consider the case when x is not tan- gent to the null generators of the horizon1. In this case, according to the rigidity theorem [18, 19, 20] there exists an additional Killing field h which generates a periodic flow with period 2p , commutes with x , has nonempty axis and is such that £h F =£h j = 0. In other words,thespacetimeisaxisymmetricwithan isometrygroup G =R U(1). In theasymp- × toticregionM¥ theKillingfield h takes thestandard form h =x ¶ /¶ x x ¶ /¶ x . (6) 1 2 2 1 − Thereexistsalsoalinearcombination K =x +W h (7) H so that the Killing field K is tangent and normal to the null generators of the horizon and g(K ,h ) =0. Thesurfacegravity k oftheblackholemaybedefined byK , namely H | (cid:209) n(cid:209) an k =lim a (8) H n where n = (cid:209) aK b(cid:209) K . It is well known that k is constant on the horizon. The horizon a b non-degeneracyconditionimpliesthat k >0. Due to the symmetriesof thespacetimethe natural space to work on is theorbit (factor) spaceMˆ = M /G,whereG istheisometrygroup. Thestructureofthefactorspacein4D hh ii iswellknownand isdescribed bythefollowingtheorem[3], [9],[21]: Theorem: Let (M ,g) be a stationary, asymptotically flat Einstein-Maxwell-dilaton black i hole spacetime with isometry group G = R U(1) satisfying the technical assumptions × stated above. Then the orbit space Mˆ = M /G is a simply connected 2-dimensional hh ii manifold with boundaries and corners homeomorphic to a half-plane. More precisely, the boundary consists of one finite interval I corresponding to the quotient of the horizon, H I = H/G and two semi-infinite intervals I and I corresponding to the axis of h . The H + − corners correspondto thepointswheretheaxisintersectsthehorizon. 1Whenx istangenttothenullgeneratorsofthehorizon,thespacetimetimemustbestatic. 3 In the interior of Mˆ there is a naturally induced metric gˆ which has signature ++. We denote derivative operator associated with gˆ by Dˆ. Let us now consider the Gramm matrix of the Killing fields G =g(K ,K ), where K =x and K =h . Then the determinant r 2 = IJ I J 1 2 detG defines a scalar function r on Mˆ which, as well known, is harmonic, DˆaDˆ r =0 as a −a consequence of the Einstein-Maxwell-dilatonfield equations. It can be shown that r >0, Dˆ r = 0 in the interior of Mˆ and that r = 0 on ¶ Mˆ. We may define a conjugate harmonic a 6 functionzonMˆ bydz=⋆ˆdr ,where⋆ˆ istheHodgedualonMˆ. Thefunctionsr andzdefine globalcoordinatesonMˆ identifyingtheorbitspacewiththeuppercomplexhalf-plane Mˆ = z+ir C,r 0 (9) { ∈ ≥ } with the boundary corresponding to the real axis. The induced metric gˆ is given in these coordinatesby gˆ=S 2(r ,z)(dr 2+dz2), (10) S 2(r ,z)being aconformalfactor. In otherwordstheorbitspaceisthehalfplaneMˆ = z+ir ,r >0 and itsboundary¶ Mˆ { } isdividedintotheintervalsI =( ¥ ,z ],I =[z ,z ]and I =[z ,+¥ ): 1 H 1 2 + 2 − − ( ¥ ,z ],[z ,z ],[z ,+¥ ). (11) 1 1 2 2 − Letusnotethattherequirementfornon-degeneratehorizonisequivalenttoanon-zerolength of the horizon interval, l(I ) =z z >0. The parameter l(I ) is invariantly defined and H 2 1 H − willplayimportantrolein theclassificationtheorem. 3 Dimensionally reduced Einstein-Maxwell-dilaton equations In the context of the present paper it is important to perform the dimensional reduction in terms of the axial Killing field h . This insures positively definite metric in the space of the potentials and in this formulation we also avoid the ergosurface "singularities". First we considerthetwist1-form w associated withh anddefined by w =⋆(h dh )=ih ⋆dh . (12) ∧ By definition, w is invariant under the symmetries and naturally induces corresponding 1- formwˆ ontheorbitspaceMˆ. Takingtheexteriorderivativeofw weobtain: dw =dih ⋆dh =(£h ih d)⋆dh = ih d⋆dh = ih ⋆d†dh = 2ih ⋆R[h ] (13) − − − − whereR[h ]istheRicci 1-form. Makinguseofthefield equationswefind 4 dw =4ih F ih e−2ja ⋆F . (14) ∧ (cid:0) (cid:1) As a consequence of the symmetries and the field equations the 1-forms f = ih F and f⋆ =ih e−2ja ⋆F are closed, i.e. df =0 and df⋆ =0. Since f and f⋆ are invariant under the sym(cid:0)metries the(cid:1)y naturally induce corresponding 1-form fˆ and fˆ on the orbit space Mˆ ⋆ which are stillclosed. Thedomain ofoutercommunications M is simplyconnected and hh ii therefore there exist globally defined potentials F and Y such that f =dF and f =dY on ⋆ M . hh ii Proceeding furtherwehave dw =4dF dY =2d(F dY Y dF ). (15) ∧ − Usingagainthefact that M issimplyconnectedweconcludethatthereexistsapotential hh ii c such that w = dc +2F dY 2Y dF on M . Obviously, the potentials c , F and Y are − hh ii invariantunderthesymmetriesand theyarenaturallydefined ontheorbitspaceMˆ. ThepotentialF ,Y andc playimportantroleinwritingdownthedimensionallyreduced Einstein-Maxwell-dilaton equations on the orbit space. Let X and h be functions on Mˆ defined by X =g(h ,h ), e2hX 1 =g(dr ,dr ). (16) − ThenthestationaryandaxisymmetricEinstein-Maxwell-dilatonequationsareequivalent tothefollowingsetofequationsontheorbitspaceMˆ : r 1Dˆ r DˆaX =X 1Dˆ XDˆaX X 1 Dˆ c +2F Dˆ Y 2Y Dˆ F Dˆac +2F DˆaY 2Y DˆaF − a − a − a a a − − − 2e 2ja(cid:0) Dˆ F D(cid:1)ˆaF 2e2ja Dˆ Y DˆaY , (cid:0) (cid:1)(cid:0) (17(cid:1)) − a a − − Dˆ r X 2 Dˆac +2F DˆaY 2Y DˆaF =0, (18) a − − (cid:2) (cid:0) (cid:1)(cid:3) r 1Dˆ X 1r e 2ja DˆaF =X 2Dˆ Y Dˆac +2F DˆaY 2Y DˆaF , (19) − a − − − a − (cid:0) (cid:1) (cid:0) (cid:1) r 1Dˆ X 1r e2ja DˆaY = X 2Dˆ F Dˆac +2F DˆaY 2Y DˆaF , (20) − a − − a − − (cid:0) (cid:1) (cid:0) (cid:1) r 1Dˆ r Dˆaj = a X 1 e 2ja Dˆ F DˆaF e2ja Dˆ Y DˆaY , (21) − a − − a a − − (cid:0) (cid:1) (cid:0) (cid:1) togetherwith 5 X 2 X 2 r 1Dˆar Dˆ h= − DˆaXDˆbX+ − Dˆac +2F DˆaY 2Y DˆaF Dˆbc +2F DˆbY 2Y DˆbF − a (cid:20) 4 4 − − (cid:16) (cid:17) (cid:0) (cid:1) +X 1e 2ja DˆaF DˆbF +X 1e2ja DˆaY DˆbY +Dˆaj Dˆbj gˆ 2Dˆ zDˆ z , (22) − − − ab a b · − i (cid:2) (cid:3) X 2 X 2 r 1DˆazDˆ h=2 − DˆaXDˆbX+ − Dˆac +2F DˆaY 2Y DˆaF Dˆbc +2F DˆbY 2Y DˆbF − a (cid:20) 4 4 − − (cid:16) (cid:17) (cid:0) (cid:1) +X 1e 2ja DˆaF DˆbF +X 1e2ja DˆaY DˆbY +Dˆaj Dˆbj Dˆ r Dˆ z. (23) − − − a b i The equations (22) and (23) are decoupled from the group equations (17) - (21). Once the solution of the system equations (17) - (21) is known we can determine the function h. Therefore the problem of the classification of the 4D Einstein-Maxwell-dilaton black holes can be studied as 2-dimensional boundary value problem for the nonlinear partially differentialequationsystem(17)-(21)as theboundaryconditionsare specified below. At thisstageweintroducethestrictlypositive2 definitemetric dX2+(dc +2F dY 2Y dF )2 e 2ja dF 2+e2ja dY 2 dL2 =G dXAdXB = − + − +dj 2 (24) AB 4X2 X onthe5-dimensionalmanifoldN = (X,c ,F ,Y ,j ) R5;X >0 . { ∈ } Theequations(17)-(21)can beobtainedfromavariationalprinciplebasedonthefunc- tional I[XA]= d2xr gˆgˆabG (XC)¶ XA¶ XB. (25) AB a b ZMˆ p Furtherweconsiderthemapping X :Mˆ N (26) 7→ ofthe2-dimensionalRiemannianmanifoldMˆ ontothe5-dimensionalRiemannianmanifold N thelocalcoordinaterepresentationofwhich X :(r ,z) XA (27) 7→ satisfies the equations (17) - (21) derived from the functional (25). It is well known that X belongstotheclassoftheso-calledharmonicmaps. We shall close this section with important comments. The Riemannian manifold (N ,G ) is not a symmetric space in the general case for arbitrary dilaton coupling pa- AB rametera . This isseen fromthefact that fortheRiemannian curvaturetensorwehave (cid:209) R =0 (28) E ABCD 6 2TheaxialKillingfieldh isstrictlyspacelikeeverywhere,i.e.X =g(h ,h )>0exceptonthesymmetryaxis whereh vanishes.Inthepresentworkweexcludethepossibilityofclosedtimelikecurves. 6 as onecan check. Only in the cases a =0 and a 2 =3, (N ,G ) is a symmetricspace. The AB first case corresponds to the pure Einstein-Maxwell gravity for which it is well known that metric G possesses the maximal group of symmetries (isometries) which in turn insures AB the complete integrability of the stationary and axi-symmetric Einstein-Maxwell equations. The second case corresponding to 4D Kaluza-Klein gravity inherits its symmetries via the dimensional reduction of the 5D vacuum Einstein gravity which also possesses high degree ofsymmetry,especially forR U(1)2 groupofspacetimeisometries. × 4 Classification theorem Westartthissectionwithintermediateresultswhichwillbeusedinthecentralclassification theorem. Lemma: The manifold(N ,G )is geodesicallycompleteforanya . AB Proof: Let s g (s)bean affinely parameterized geodesic, 7→ g (s)=(X(s),c (s),F (s),Y (s),j (s)). (29) Then G(g˙,g˙)=C>0 is aconstantofmotion,i.e. 1 X˙ 2 c˙+2F Y˙ 2Y F˙ 2 e 2ja F˙2 e2ja Y˙2 G X˙AX˙B = + − + − + +j˙2 =C. (30) AB 4(cid:18)X(cid:19) (cid:0) 4X2 (cid:1) X X Henceit followsthatwehave 1 X˙ 2 C, j˙2 C, (31) 4(cid:18)X(cid:19) ≤ ≤ and F˙2 CXe2ja , Y˙2 CXe 2ja , c˙+2F Y˙ 2Y F˙ 2 4CX2. (32) − ≤ ≤ − ≤ (cid:0) (cid:1) Therefore any geodesic can be extended to arbitrary values of the affine parameter, i.e. the metric is geodesically complete. Even though there is an edge at X = 0 this edge is at an infinitedistancefrom anypointofthemanifold. Lemma: The manifold(N ,G ) is manifoldwith nonpositiveRiemann curvatureoper- AB atorfora dilatoncouplingparametera satisfying0 a 2 3. ≤ ≤ Proof: Let L 2T (N ) be the linear space of 2-forms on N . We regard the Riemann ∗ curvaturetensorRiemann= R as an operator ABCD { } Rˆ :L 2T (N ) L 2T (N ). (33) ∗ ∗ → 7 which is symmetric with respect to the naturally induced metric3 , on L 2T (N ). The ∗ nonpositivityofthecurvatureoperatorthenmeansthat Rˆ W ,W = W h,iRˆ W =R (W ,W ) 0 forallW L 2T (N ). h i h i ≤ ∗ Inord∈ertoshowthenon-positivityoftheRiemannianoperatorRˆ wefix an orthonormal basis w i w j in L 2T (N ) where w i is an orthonormal basis in T N explicitly given ∗ ∗ { ∧ } { } by dX dc +2F dY 2Y dF w 1 =w X = , w 2 =w c = − , 2X 2X (34) e 2ja dF e2ja dY w 3 =w F = − , w 4 =w Y = , w 5 =w j =dj . √X √X InthisorthonormalbasisthematrixrepresentingtheRiemannoperatorissymmetricand isexplicitlygivenby 4 0 0 0 0 0 0 2 0 0  −0 1 0 0 0 1 0 0 a 0  0 −0 1 0 1 0 0 0 0 a  − − −   0 0 0 0 0 0 0 0 0 0  Rˆ = 00 10 −01 00 −01 01 00 00 0a −0a . (35)  − −   0 0 0 0 0 0 0 0 0 0     2 0 0 0 0 0 0 a 2 4 0 0   −   0 a 0 0 0 a 0 0 a 2 0   − −   0 0 a 0 a 0 0 0 0 a 2   − − −  Theeigenvaluesofthissymmetricmatrixare a 2 √16+a 4 a 2 √16+a 4 l = 4+ + , l = 4+ , (36) 1 2 − 2 2 − 2 − 2 l =l = 2 a 2, l =l =l =l =l =l =0. 3 4 5 6 7 8 9 10 − − Obviously, for 0 a 2 3 all the eigenvalues are nonpositive and therefore the Riemann ≤ ≤ operator is nonpositive. As an immediate consequence we also have that N is a manifold withnonpositivesectionalcurvaturefor0 a 2 3 . ≤ ≤ Before going to the formulation of the classification theorem let us give the definitions oftheangularmomentumJ,electricchargeQandmagneticchargeP. Theyarerespectively givenbytheintegrals 3For separable forms x y and u w the metric , is defined by x y,u w = G(x,u)G(y,w) ∧ ∧ h i h ∧ ∧ i − G(x,w)G(y,u). 8 1 J = ⋆dh , 16p ZS¥2 1 Q= e 2ja ⋆F, (37) 4p ZS¥2 − 1 P= F, 4p ZS¥2 whereS2 isthesphereat infinity. ¥ Classification(Uniqueness)Theorem: Therecanbeatmostonlyonestationary,asymptot- ically flat black hole spacetime satisfying the Einstein-Maxwell-dilaton field equations and the technical assumptionsstated in Sec. 2 for a given set of parameters l(I ),J,Q,P and H { } fora dilatoncouplingparametera satisfying0 a 2 3. ≤ ≤ Remark: In the theorem we assume that lim¥ j = j ¥ =0. When j ¥ = 0 it is easy to seethatthesetofparameters shouldbeexpandedto l(IH),J,Q,P,j ¥ 6 { } Proof: Consider two solutions (M,g,F,j ) and (M˜,g˜,F˜,j˜) as in the statement of the theo- rem. We use the same "tilde" notation to distinguish any quantities associated with the two solutions. Since l(I ) = l(I˜ ), we can identify the orbit spaces Mˆ and Mˆ˜. Moreover we H H canidentify M and M˜ as manifoldswithR U(1)-actionsincetheycan beuniquely hh ii hh ii × reconstructed from the orbit space. We may therefore assume that M = M˜ and that x =x˜,h =h˜. Wemay alsoassumethat r =r˜ andz=z˜. hh ii hh ii As a consequence of these identifications, (g,F,j ) and (g˜,F˜,j˜) may be considered as being defined on the same manifold. Moreover, on the base of these identifications, we can combinethetwosolutionsintoasingleidentityplayingakeyroleintheproofaswewillsee below. Furtherweconsidertwoharmonicsmaps(26)X :Mˆ N andX˜ :Mˆ N andasmooth 7→ 7→ homotopy T :Mˆ [0,1] N (38) × 7→ sothat T (t =0)=X , T (t =1)=X˜ (39) where 0 t 1 is the homotopy parameter. Rephrasing in local terms we consider two ≤ ≤ solutionsXA(r ,z)and X˜A(r ,z)and asmoothhomotopyT :Mˆ [0,1] N such that × 7→ T A(r ,z;t =0)=XA(r ,z), T A(r ,z;t =1)=X˜A(r ,z) (40) foreach point(r ,z) Mˆ. Asafurtherrequirementweimposethatthecurves[0,1] N be ∈ 7→ geodesicwhich inlocal coordinatesmeansthat 9 dSA +G A SBSC =0 (41) dt BC where dT A SA = (42) dt isthetangentvectoralongthecurvesandG A arecomponentsoftheLevi-Civitaconnection BC onN . The existence and uniqueness of the geodesic homotopy follow from the Hadamard- Cartan theorem [22] since the Riemannian manifold (N ,G ) is geodesically complete, AB simplyconnected andwithnonpositivesectionalcurvature. Thelengthofthegeodesics willbedenoted byS, i.e. 1 S= dt G SASB. (43) AB Z 0 Letus alsonotethatthetangentvectorsatisfies thet -independentnormalizationcondition SAS =S2. (44) A Nowweare at positiontowritetheBuntingidentity[6],[16] 1 Dˆ r SDˆaS =r dt (cid:209)ˆaS (cid:209)ˆ SA R SA(cid:209)ˆ T BSC(cid:209)ˆaT D . (45) a A a ABCD a Z0 (cid:16) − (cid:17) (cid:0) (cid:1) Here(cid:209)ˆ istheinducedconnectionalongtheharmonicmap,i.e. a (cid:209)ˆ VA =¶ T C(cid:209) VA =¶ VA+¶ T CG A VB. (46) a a C a a CB Since r 0, N has positivedefinite metric and nonpositivesectional curvature for 0 a 2 3we c≥oncludethat ≤ ≤ Dˆ r SDˆaS 0. (47) a ≥ (cid:0) (cid:1) ApplyingtheGausstheorem weobtain Dˆ r SDˆaS gˆd2x= r SDˆaSdS . (48) a a ZMˆ Z¶ Mˆ ¥ (cid:0) (cid:1)p ∪ We shall show that the boundary integral on the right hand side of (48) vanishes. To do thiswehavetoconsidertheboundaryconditions. In theasymptoticregion(r ¥ )wehave → thestandard boundaryconditions 10