Perturbation theory and stability analysis for string-corrected black holes in arbitrary dimensions

Transcription

1 CPHT-RR SPHT-T05/50 axiv:hep-th/ v Aug 006 Petubation theoy and stability analysis fo sting-coected black holes in abitay dimensions Filipe Moua Cente de Physique Théoique, École Polytéchnique F98 Palaiseau Cedex, Fance and Sevice de Physique Théoique, Ome des Meisies, CEA/Saclay F99 Gif-su-Yvette Cedex, Fance fmoua@spht.saclay.cea.f Abstact We develop the petubation theoy fo R sting-coected black hole solutions in d dimensions. Afte having obtained the maste equation and the α -coected potential unde tensoial petubations of the metic, we study the stability of the Callan, Myes and Pey solution unde these petubations. Pesent addess: Instituto Supeio Técnico, Depatamento de Matemática, Av. Rovisco Pais, Lisboa, Potugal.

2 Sting theoy low enegy effective actions have thee diffeent types of contibutions, with diffeent oigins. The classical tems come fom the expansion in α wold-sheet loops. The quantum tems depend on the sting coupling constant g s = e φ ; they can be petubative coming fom space-time loops and non-petubative. In this wok we conside only the classical α coections, neglecting any kind of sting quantum coection. Both the bosonic and the heteotic sting theoies have coections aleady at the fist ode in α, which ae at most quadatic in the Riemann tenso. In these coections we neglect the Ricci tems, which would only contibute in a highe ode in α ; we ae only consideing an effective action which is petubative in α. All these theoies also have antisymmetic tensos in thei massless specta, which can always be consistently set to zeo. That will be the case in the α -coected black hole solution we use in this wok. Although these theoies lie espectively in 6 o 0 space-time dimensions, we will conside in this aticle black hole space-times in geneic d-dimensions. This way we take, as ou effective action in the Einstein fame, g R 4 κ d µ φ µ φ + e 4 d φ λ Rµνρσ R µνρσ d d x + femion tems, with λ = α, α, 0 fo bosonic, heteotic and supestings, espectively. 4 The coected bosonic equations of motion fo the dilaton and the gaviton ae, to this ode, φ λ 4 e 4 d φ R ρσλτ R ρσλτ = 0, R µν + λe 4 d R φ µρστ Rν ρστ d g µνr ρσλτ R ρσλτ = 0. 3 We ae inteested in studying the behaviou of a sting-coected black hole solution unde petubations. We will be studying these petubations in geneic d spacetime dimensions, taking as backgound metic d s = f d t + f d + d Ω d 4 with dω d = γ ij θ dθ i dθ j, γ ij = g ij / being the metic of a d -sphee S d. One can in geneal conside petubations to the metic and any othe physical field of the system unde consideation. Geneal tensos of ank at most on the d - sphee can be uniquely decomposed in thei scala, vectoial and fo d > 4 tensoial components. In this wok we only conside tensoial in S d petubations to the metic, given by h µν = δg µν as we will show, we can consistently set the tensoial petubation to the dilaton to 0. These petubations ae woked out in 3, whee it is shown that they can be witten as with T ij satisfying h ij = y a H T y a T ij θ i, h i = h it = 0, h = h t = h tt = 0 5 γ kl D k D l + k T Tij = 0, D i T ij = 0, g ij T ij = 0. 6

3 D i is the S d covaiant deivative; T ij ae the eigentensos of the S d laplacian; on the same efeence 3, it is also shown that the eigenvalues ae given by k T = l l + d 3, l =, 3, 4... We actually need the vaiation of the components of the Riemann tenso. Using the components of h µν given in 5 and the Palatini equation, one gets δr ijkl = f H T + f H T g il T jk g ik T jl g jl T ik + g jk T il + H T D i D l T jk D i D k T jl D j D l T ik + D j D k T il 7 δr itjt = t H T + ff H T + ff H T T ij 8 δr ij = f f H T f f H T H T H T T ij 9 δr abcd = 0. 0 Using the explicit fom of the Riemann tenso and the vaiations 5 and 7-0, one can petub the field equations and 3. Fom, we ae able to show that we can consistently set the dilaton petubation δφ = 0. By petubing 3 we ae able to detemine the equation fo H T, which is given by λ f t H T f H T + d f ff + λ 44 d f f + 4f 3 f + f f H T + k T f + 4λ f ff + d f d 3f + λ f + d 4d f + d 3f f d We now wite the equation above in the fom of a maste equation Φ H T = 0. Φ t =: V TΦ. Fo that, fist we wite the petubation equation in tems of the totoise coodinate, defined by d /d = /f. As caefully explained in 5, following the pocedue intoduced in 4 we deive ou maste function and potential: Φ = H f T exp + d + 4 d 4λ f 4 λf λf f 3 f f 4λf d V T f = + λ f λ f + 4λ k T + + ff + 4λ f d 4 4 k T + f f + d 3f + 4λ f f f f f + d 3λ + d 4λ λf + d 8 ff λ d f f f ff + d 4f. 3 3

4 This is the potential fo tenso petubations of any kind of R -coected black holes in d dimensions, in tems of which the petubation equation is witten as a maste equation like. To study the stability of a solution, we use the S-defomation appoach fist intoduced in 3 and developed in 4. Afte having obtained the potential V T f, we assume that its solutions ae of the fom Φ, t = e iωt φ, such that Φ/ t = iωφ. The maste equation is then witten in the Schödinge fom AΦ = ω Φ, and a solution to the field equation is then stable if the opeato A is positive definite with espect to the following inne poduct: + φ, Aφ = φ d + + V φ d d = dφ + V φ d d. Defining D = d d fh T d Φ Φ d HT and afte some algebaic ticks 5, we ae left with with Q f = λ f φ, Aφ = k T + f λ d f + Dφ d + + 4λ f + Q f φ d, + d 6 f + λ f All that is necessay to guaantee the stability is to check the positivity of Q. f We consideed the R -coected black hole solution of the type of 4 studied in. Its only fee paamete is µ, which is elated to the classical ADM black hole mass though m cl = d A d µ, A κ n being the aea of the unit n sphee. Fo the classical Schwazschild-Myes-Pey solution, we have f = µ. In d 3 ode to intoduce the α -coections to this solution, we choose a coodinate system in which the position of the hoizon, given by = µ d 3 =: H, is not changed. Accoding to f is given, in this coodinate system, by H d 3 f = λ d 3d 4 4 H d 5 d d H d d 3 d 3. 5 This solution has as fee paametes the invese sting slope λ, the black hole mass paamete µ o, equivalently, the hoizon adius H and the spacetime dimension d. Since λ is a petubative paamete, we should take it small say λ, fo the potential to make sense. Fo small values of λ, fo each value of d between 5 and 0, and fo a wide ange of values of µ, we have studied numeically and made plots of Q f as it is given by 4, and we always found positive values. Fom this numeical study we conclude that this solution is stable unde tenso petubations fo evey elevant spacetime dimension, fo evey value of the black hole mass. Given the potential fo the metic tensoial petubations, we have also obtained an analytical poof of the stability and computed the absoption coss section of the black hole given by 5. Fo moe details see 5. 3 H

5 Acknowledgements This wok has been suppoted by a Chateaubiand scholaship fom EGIDE and by fellowship BPD/4064/003 fom Fundação paa a Ciência e a Tecnologia, and is pat of a joint poject with Ricado Schiappa 5. Refeences C. G. Callan, R. C. Myes and M. J. Pey, Black Holes in Sting Theoy, Nucl. Phys. B H. Kodama, A. Ishibashi and O. Seto, Bane Wold Cosmology: Gauge Invaiant Fomalism Fo Petubation, Phys. Rev. D , hep-th/ A. Ishibashi and H. Kodama, Stability of Highe Dimensional Schwazschild Black Holes, Pog. Theo. Phys , hep-th/ G. Dotti and R. J. Gleise, Linea stability of Einstein-Gauss-Bonnet static spacetimes. pat. I: Tenso petubations, Phys. Rev. D , axiv:g-qc/ F. Moua and R. Schiappa, Highe-deivative coected black holes: Petubative stability and absoption coss-section in heteotic sting theoy, axiv:hep-th/