Higher Derivative Corrected Black Holes: Perturbative Stability and Absorption Cross Section in Heterotic String Theory

Transcription

1 Pepint typeset in JHEP style - HYPER VERSION CPHT-RR SPHT-T05/51 ITFA CERN-PH-TH/ hep-th/ Highe Deivative Coected Black Holes: Petubative Stability and Absoption Coss Section in Heteotic Sting Theoy Filipe Moua a,b,c and Ricado Schiappa d a Cente de Physique Théoique, École Polytéchnique, F9118 Palaiseau Cedex, Fance b Sevice de Physique Théoique, CEA/Saclay, F91191 Gif su Yvette Cedex, Fance c Instituut voo Theoetische Fysica, Univesiteit van Amstedam, Valckeniestaat 65, 1018 XE Amstedam, The Nethelands d Theoy Division, Physics Depatment, CERN, CH 111 Genève 3, Switzeland fmoua@spht.saclay.cea.f, icados@mail.cen.ch Abstact: This wok addesses spheically symmetic, static black holes in highe deivative stingy gavity. We focus on the cuvatue squaed coection to the Einstein Hilbet action, pesent in both heteotic and bosonic sting theoy. The sting theoy low enegy effective action necessaily descibes both a gaviton and a dilaton, and we concentate on the Callan Myes Pey solution in d dimensions, descibing stingy coections to the Schwazschild geomety. We develop the petubation theoy fo the highe deivative coected action, along the guidelines of the Ishibashi Kodama famewok, focusing on tenso type gavitational petubations. The potential obtained allows us to addess the petubative stability of the black hole solution, whee we pove stability in any dimension. The equation descibing gavitational petubations to the Callan Myes Pey geomety also allows fo a study of geybody factos and quasinomal fequencies. We addess gavitational scatteing at low fequencies, computing coections aising fom the cuvatue squaed tem in the stingy action. We find that the absoption coss section eceives α coections, even though it is still popotional to the aea of the black hole event hoizon. We also suggest an expession fo the absoption coss section which could be valid to all odes in α. Keywods: Stingy Cuvatue Coections, Black Holes in d Dimensions, Stability, Hawking Emission. Pesent addess.

2 Contents 1. Intoduction and Summay. Highe Deivative Coections in Sting Theoy 4.1 α Coections to Heteotic and Bosonic Stings 6 3. Gavitational Petubations to the R Coected Field Equations Geneal Setup of the Petubation Theoy 7 3. Petubation of a Spheically Symmetic Static Solution On the Poof of Petubative Stability 1 4. The Callan Myes Pey Black Hole Solution Potential fo Classically Non Dilatonic Solutions Poof of Petubative Stability Scatteing Theoy in the Callan Myes Pey Geomety Absoption Coss Section Quasinomal Fequencies 6. Conclusions and Futue Diections 3 1

3 1. Intoduction and Summay The scatteing of gavitons in sting theoy is adically distinct fom the scatteing of gavitons within geneal elativity. This is not only clealy tue at the quantum loop level afte all, geneal elativity is not a enomalizable theoy, while sting theoy is expected to be finite, but is also tue at the classical level: gavitational wave scatteing in classical sting theoy and in geneal elativity is totally diffeent. An immediate consequence of this fact is that a Ricci flat manifold can not be a solution to the sting theoy equations of motion [1]; the evolution of gaviton scatteing must be descibed in some othe way. One obvious way is to emain within the sting theoetic famewok and compute scatteing amplitudes using the standad stingy machiney. While this appoach is cetainly of geat inteest in a wide ange of applications, it is pehaps not the most adequate one if one is inteested in studying fully non petubative gavitational solutions, such as black holes. Anothe way to descibe gavitational sting physics is to include highe ode deivative coections in the standad Einstein Hilbet EH action and petubatively constuct a low enegy effective action descibing the sting physics. Of couse that, in this appoach, only an action including an infinite set of highe deivative coections could fully epoduce gaviton scatteing in sting theoy. Nevetheless, a given low enegy effective action with highe deivative stingy coections will epoduce sting scatteing amplitudes up to, and including, tems of the ode of the momenta aised to the numbe of deivatives in the highest ode coection. In this set up, the intinsic sting physics esides pecisely in these highe deivative coections and thei detemination is cucial if one wants to obtain a complete undestanding of stingy phenomena. This is the famewok in which we shall wok. The highe deivative coections to the pue EH action natually appea in divese powes of the cuvatue tenso, as well as powes of its contactions. Fo instance, in the type II supesting the fist coection appeas as a combination of quatic powes of the cuvatue tenso [1,, 3, 4, 5] and the esulting low enegy effective action coectly epoduces stingy gaviton scatteing up to eighth ode in the momenta. This natually modifies the equations of motion and is thus simple to undestand why Ricci flat manifolds may no longe be solutions. Howeve, this also seems to aise a puzzle on what concens supesting compactifications on Calabi Yau CY manifolds [1]. Indeed, CY spaces ae Ricci flat. The point, of couse, is that CY manifolds ae also Kähle [1]: the discussion above concening scatteing amplitudes no longe applies as thee ae no scatteing pocesses descibed by Kähle manifolds these manifolds cannot have Loentzian signatue due to the complex stuctue. Thus, Ricci flat Kähle solutions seem to be in good shape to descibe sting compactification. The eal stoy tuns out to be a little moe subtle: the equations of motion of type II sting theoy, including the quatic cuvatue coections, ae actually not satisfied by Ricci flat Kähle metics [, 3, 4]. The solution to this poblem is that while the exact solution to the metic ends up not being Ricci flat, it is still the case that it can always be elated to the initial Ricci flat metic by a non local field edefinition 1 [6, 7]. A detailed and ecent discussion concening some of these issues can be found in [8], whee the authos have explicitly computed quatic cuvatue coections to the oiginal Ricci flat Kähle metic fo non compact CY spaces, choosing a paticula subtaction scheme in which the CY manifold emains Kähle, but is no longe Ricci flat. Ou inteest in the pesent wok esides not in highe deivative coections to sting theoy compactifications but in highe deivative coections to sting theoetic black hole solutions. As such, we shall concentate on the simple quadatic cuvatue coections pesent in both heteotic and bosonic sting theoy [9, 10, 11]. One impotant aspect to have in mind when studying highe deivative coections 1 If one wishes to use sigma model language, the field edefinitions of the metic ae associated to the choice of subtaction scheme used to compute the beta functions. It tuns out that it is always possible to choose a subtaction scheme whee Ricci flat Kähle metics will have vanishing beta function to all odes in petubation theoy [6].

4 within a sting theoetic context, is the ole of the dilaton field. In fact, in sting theoy, one can not discad the dilaton when including highe deivative coections to the low enegy effective action [9]: the cuvatue coections act as a souce in the dilaton field equations, so that if the Riemann tenso has any non vanishing components so will the dilaton field be non tivial thoughout spacetime. In this way, solutions to the highe deivative coected field equations will necessaily descibe both gaviton and dilaton fields. Anothe point to have in mind is the exact fom of the quadatic cuvatue coection [10]. While sigma model consideations natually yield the squae of the Riemann tenso as the sole coection to include in the low enegy effective action [9, 11], a Lagangian which only includes a Riemann tenso squaed coection will contain ghost paticles in its spectum [10]. This is eally just a consequence of woking with a low enegy effective action and one can simply esolve this annoyance with a field edefinition which amounts to eplacing the Riemann tenso squaed tem in the effective action by the Gauss Bonnet GB combination also involving Ricci tenso squaed and Ricci scala squaed [10]. As ou inteest esides with black hole solutions, this is in fact a point we need not woy about. Black hole solutions in cuvatue coected gavity have a long histoy and we efe the eade to [1] fo a eview and a complete list of efeences. The effective theoy we will focus upon in this wok is the cuvatue squaed coection to the EH action, plus dilaton field [9, 11], pesent in both heteotic and bosonic sting theoy, and the paticula black hole solution we shall addess is the Callan Myes Pey CMP solution to this low enegy effective action [13], descibing a static, spheically symmetic black hole. This solution descibes stingy coections to the well known d dimensional Schwazschild solution see, e.g., [14]. The physics of the stingy coections is quite inteesting [13]: the stength of the dilaton coupling i.e., the sting coupling constant deceases as one appoaches the black hole, so that sting inteactions become weake in the vicinity of the black hole. Futhemoe, the black hole mass inceases due to the highe cuvatue coections, while the black hole tempeatue deceases. It is paticulaly inteesting to note that the expession fo the tempeatue yields both maximum and zeo tempeatue fo definite values of the size of the black hole the size of the event hoizon [13]. This could have pofound implications in the undestanding of the full Hawking evapoation pocess. As to the entopy of the black hole, it inceases with espect to the oiginal Schwazschild esult. It is impotant to notice that because the low enegy effective action is constucted petubatively in highe cuvatue coections, also the solutions to this effective action must be found petubatively in the specific dimensionless expansion paamete. In paticula, any solutions one may find will not be valid in egions of stong cuvatue and one cannot obtain stingy infomation concening the esolution of singulaities in geneal elativity. Still, thee is a ich vaiety of phenomena which may be studied within this famewok. Let us be moe concete about the specific poblems we wish to addess in this pape. Having obtained a black hole solution to the cuvatue coected low enegy effective action, it is impotant to know whethe this solution is stable o not. The analysis of the linea stability of fou dimensional black hole solutions in geneal elativity was fist addess a long time ago [15, 16]. The pocedue amounts to studying the lineaised Einstein equations in the given backgound and poceeds with a decomposition of the gavitational petubation in tenso spheical hamonics fo spheically symmetic backgounds in ode to obtain a adial equation descibing the popagation of linea petubations. But it was not until ecently that the black hole stability poblem in geneal elativity was addessed within a d dimensional setting [17, 18, 19, 0, 1]. A set of equations descibing linea gavitational petubations to static, spheically symmetic black holes in any spacetime dimension d > 3 was deived in [18, 0]. These equations ae of Schödinge type and will be denoted as the Ishibashi Kodama IK maste equations. The d dimensional petubations come in thee types: tenso, vecto and scala type petubations. This nomenclatue efes to the tensoial behavio on the sphee S d of each gauge invaiant type of petubation. The IK maste 3

5 equations wee used in [19, 0, 1] to study the stability of d dimensional black holes in geneal elativity. In the pesent pape, we apply the IK famewok to the stingy cuvatue squaed coected action, focusing on tenso type gavitational petubations. This is done in section 3. Tenso type petubations only exist in dimension d > 4 and they ae the simplest of the thee types of petubations. We obtain an equation fo the gauge invaiant petubation which descibes sting theoetic coections to the oiginal IK maste equation. This equation is still of Schödinge type; the coections appea as coections to the tenso potential. We then use this equation to pove stability of the CMP black hole solution, in any spacetime dimension d > 4. This is done in section 4. A full poof of stability would still equie analysis of vecto and scala type petubations. Howeve, it was also agued in [17] that it is expected that only tenso modes pobe the S d base manifold sufficiently enough in ode to poduce instabilities. Thus, ou esults point towads full stability of the stingy coected black hole solution. We should futhe notice that the equation we obtain, descibing gavitational petubations, can be applied to any othe static, spheically symmetic solutions of the cuvatue coected Einstein equations. Thee has been some othe wok in the liteatue along analogous guidelines [, 3, 4, 5]. Howeve, these woks have instead concentated on GB coections to the EH action; in paticula they have focused upon the solutions descibed in [6, 7]. The solution in [6] is non dilatonic and is futhemoe an exact solution to the action consisting of EH and GB tems. Lacking a dilaton and not taking into account the petubative natue of the stingy low enegy effective action, the solution in [6], although inteesting on its own, is thus not applicable in a sting theoy context. The stability of this solution was ecently studied in [, 3, 4] using the IK famewok. Studying tenso type gavitational petubations, it was shown in these papes that static, spheically symmetic black hole solutions of the EH plus GB system ae stable fo d > 4 and d 6 [, 3]. The case of vecto and scala type gavitational petubations tuns out to be moe complex, as while vecto petubations ae stable, scala petubations do lead to instabilities of spheically symmetic EH GB black holes [4]. It should be noted that the famewok in [, 3, 4] is quite close to the one we deploy in the pesent pape, in spite of analyzing a distinct physical situation. The solution descibed in [7] studies the EH and GB system with the inclusion of the dilaton field. This action may be egaded as a [complicated] field edefinition of the action we use in the pesent pape. The authos constuct a solution which has dilatonic chage and is thus distinct, even unde field edefinitions, of the CMP solution, but they fail to povide a full analytical expession to thei final esult. In spite of this, numeical wok has been done [5] studying the ange of stability of this paticula solution. One futhe application of the petubation equation we obtain in this pape concens the calculation of geybody factos and quasinomal fequencies, in the CMP backgound geomety. This is equied data in the study of Hawking emission specta and quasinomal inging, and could be of futue inteest as gavitational wave astonomy becomes an expeimental eality. In section 5 we addess the investigation of gavitational scatteing at low fequencies, obtaining coections to the emission specta due to the highe ode deivative coections in the low enegy effective action. We use this esult in ode to compute the absoption coss section at low fequency which, in spite of the fact that it does eceive α coections, tuns out to be popotional to the aea of the black hole event hoizon, just like it is in the EH case [9]. The expession we obtain suggests a natual genealization which could be valid to all odes in α.. Highe Deivative Coections in Sting Theoy Sting theoy low enegy effective actions have thee diffeent types of contibutions, each with a diffeent oigin. Thee ae classical tems, which come fom the expansion in α wold sheet loops. These epesent Futhe studies on the stability of non dilatonic solutions include, e.g., [8]. 4

6 the finite size of stings as compaed to point paticles. Then thee ae the quantum tems, which depend on the sting coupling constant g s = e φ, and these can eithe be petubative aising fom spacetime loops o non petubative. In this aticle we shall only conside the classical α coections, neglecting any kind of stingy quantum coections. Fo this to be possible, one must clealy have g s 1. In this section we begin by eviewing the pecise fom of the leading highe deivative α coections to bosonic, heteotic and supesting theoies. We will caefully wite down the low enegy effective actions, and espective equations of motion, fo both gaviton and dilaton fields. Although these sting theoies also contain antisymmetic tenso fields in thei massless specta, these tenso fields can always be consistently set to zeo. This shall be the case in the α coected black hole solution we will use in this wok. Also, although bosonic o supesymmetic sting theoy lives, espectively, in 6 o in 10 spacetime dimensions, in this aticle we shall always conside black hole spacetimes in geneic d dimensions. One can think of this as having compactified sting theoy on a flat tous, leaving uncompactified d spacetime dimensions. A low enegy sting theoy effective action is witten, in geneal, and in the sting fame, as 3 1 κ d d x g e φ R + 4 µ φ µ φ + zy R + femions..1 Hee, Y R is a scala polynomial in the Riemann tenso epesenting the highe deivative stingy coections to the metic tenso field, and z is, up to a numeical facto, the suitable powe of the invese sting tension α fo Y R. The dilaton field is φ, and we shall not make the femionic tems explicit. The dilaton and gaviton field equations which follow fom the above effective action ae, espectively, φ φ R + 1 zy R = 0, 4. δy R R µν + µ ν φ + z δg µν = 0..3 Fo most puposes it is moe convenient to wite the stingy effective action.1 in the Einstein fame, athe than in the sting fame which is the fame aising natually fom sigma model consideations. In ode to achieve that, one needs to pefom a edefinition of the metic by a confomal tansfomation involving the dilaton, which will also obviously affect the Riemann tenso [30]: 4 g µν exp d φ g µν,.4 ρσ R ρσ µν R µν = R ρσ µν δ [ρ [µ ν] σ] φ..5 The low enegy effective action thus eads, in the same ode in α and up to total deivatives, 1 κ d d x g R 4 d µ φ µ φ + z e 4 d 1+wφ Y R + femions..6 Hee w is the confomal weight of Y R, with the convention that w g µν = +1 and w g µν = 1. The coected equations of motion fo the dilaton and gaviton fields ae, in the Einstein fame, 3 Ou conventions ae Γ σ µν = 1 gσρ µg νρ + νg ρµ ρg µν fo the connection coefficients, R ρ σµν = µγ ρ νσ + Γ ρ µλ Γλ νσ νγ ρ µσ Γ ρ νλ Γλ µσ fo the Riemann tenso, R µν = R ρ µρν fo the Ricci tenso, and R = R µ µ fo the Ricci scala. 5

7 φ z e 4 d 1+wφ Y R = 0,.7 R µν + z e 4 d 1+wφ δy R δg µν + 1 d Y Rg µν 1 d g ρσ δy R µνg δg ρσ = 0..8 In both equations above we have eliminated cetain tems involving deivatives of φ, which would only contibute at highe odes in ou petubative paamete z..1 α Coections to Heteotic and Bosonic Stings Both heteotic and bosonic sting theoies have highe deivative coections aleady at fist ode in α, and these coections ae at most quadatic in the Riemann tenso. Fo these theoies, the gavitational coection Y R in.6 is given, to fist ode in α, by Y R = R 4R µν R µν + R µνρσ R µνρσ, the [fou dimensional] GB combination, in ode to avoid ghosts as discussed in [10]. Since ou focus will be on non petubative solutions of the classical theoy, and not on quantization, we shall not be concened with this issue. Futhemoe, we ae only consideing a low enegy effective action which is petubative in α ; in this way one can neglect the Ricci tems in Y R, which fom.8 would only contibute at a highe ode in α, and take z Y R λ Rµνρσ R µνρσ,.9 as in [13]. Hee w Y R = and λ = α, α 4 and 0, fo bosonic, heteotic and type II stings, espectively. With this choice of deivative coections, the coected equations of motion fo the dilaton and gaviton fields ae, to this ode, φ λ 4 e 4 d φ R ρσλτ R ρσλτ = 0,.10 R µν + λ e 4 d R φ µρστ R ρστ 1 ν d g µνr ρσλτ R ρσλτ = These ae the field equations we shall focus upon, in the following. 3. Gavitational Petubations to the R Coected Field Equations Having set up the low enegy effective theoy we wish to study, we poceed with the constuction of the IK maste equations fo the esulting field equations. This will allow us to addess stability, unde gavitational petubations, of the R coected field equations. But fist, let us begin with a comment of [1] concening the geneal stability poblem in highe deivative effective actions. If one wites the highe cuvatue field equations of motion as R µν = λj µν, with J µν some highe deivative contibution and λ the [dimensionfull] petubative paamete associated to the highe cuvatue tem in the action, one can then constuct solutions petubatively as g µν = g µν 0 + λg µν 1 + Oλ. The petubative equations of motion to solve follow as R µν [g 0 ] = 0 and g 0 g 1 µν = J µν[g 0 ], 3.1 6

8 whee g 0 is the second ode diffeential opeato that aises fom the lineaisation of the Ricci tenso about the unpetubed metic g 0. The fist equation is simply the Einstein equation of motion, while the second amounts to solving fo a lineaised petubation in the chosen backgound, including a souce tem. If one now consides a futhe petubation to the coected black hole backgound, one needs to be caeful in oganizing the new distubance within the petubative λ expansion. One should expand as g µν = ĝ µν + h µν = ĝ 0 µν + h 0 µν + λĝ 1 µν + λh 1 µν + Oλ, 3. whee ĝ µν = ĝ 0 µν + λĝ 1 µν is the backgound metic satisfying the pevious equations of motion. The petubation to the backgound metic will theefoe satisfy ĝ0 h 0 µν = 0 and ĝ0 h 1 µν = J µν[ĝ 0, h 0 ], 3.3 whee J µν [ĝ 0, h 0 ] is the lineaisation of J µν [ĝ 0 + h 0 ]. What is inteesting to obseve [1] is that petubations to the backgound metic and hence, the causal stuctue ae completely detemined by the oiginal backgound metic. This also guaantees that the black hole solution does emain a black hole solution. The stability poblem should now amount to analyzing whethe the pevious equations allow fo unaway solutions. In the following we shall make these aguments fully pecise, following [18, 0]. 3.1 Geneal Setup of the Petubation Theoy We ae inteested in studying the behavio, unde gavitational petubations, of sting coected black hole solutions to the field equations discussed in the pevious section. The analysis of these petubations shall be caied out in a geneic spacetime dimension d. As such, the most suitable tool to cay out such study is the d dimensional petubation theoy famewok developed by Ishibashi and Kodama, and which is itself a d dimensional genealization of the oiginal fou dimensional fomalism of Regge and Wheele [15] and Zeilli [16]. The IK fomalism was fist set up in [31] and then futhe developed in a seies of aticles, whee it was applied to the study of gavitational petubations of maximally symmetic neutal [18] and chaged [0] black holes in d dimensions. As set up, the IK famewok applies to geneic spacetimes of the fom M d = N d n K n, with coodinates {x µ } = { y a, θ i}. In hee K n is a manifold with constant sectional cuvatue K, which descibes the geomety of the black hole event hoizon. The metic in the total space M d is then witten as g = g ab y dy a dy b + y γ ij θ dθ i dθ j. 3.4 Fo ou puposes, we shall take n = d and the K n manifold, descibing the geomety of the hoizon, will be a d sphee thus, with K = 1. Also, coodinates will be {y a } = {t, } with {, θ i} being the usual spheical coodinates so that y = and γ ij θ dθ i dθ j = dω d. One can conside petubations eithe to the metic field o to any othe physical field of the system unde consideation. The most geneal petubation will obviously involve petubations to all the fields pesent in the low enegy effective action. Beginning with petubations to the metic tenso field, these ae given by h µν = δg µν and h µν = δg µν. 3.5 Fom this vaiation one can easily compute the vaiation of the Riemann tenso, accoding to the Palatini equation 7

9 Indeed, fom δr ρ σµν = µ δγ ρ νσ ν δγ ρ µσ. 3.6 δγ ρ µν = 1 µh ν ρ + ν h µ ρ ρ h µν 3.7 one immediately deives δr ρσµν = 1 R µνρ λ h λσ R µνσ λ h λρ µ ρ h νσ + µ σ h νρ ν σ h µρ + ν ρ h µσ. 3.8 Geneal tensos, of ank at most equal to two, can be uniquely decomposed in tenso, vecto and scala components, accoding to thei tensoial behavio on the d sphee [18]. In paticula, this is also tue fo the petubations to the metic tenso field, and this is the whole basis fo addessing petubation theoy within the IK famewok [31, 18] it should be futhe noted that metic petubations of tenso type only exist fo dimensions d > 4, unlike petubations of vecto and scala type, which exist fo dimension d 4. That this is a vey convenient decomposition immediately follows fom the fact that the geomety of the black hole event hoizon is pecisely that of a d dimensional sphee. In this wok we shall only conside tenso type gavitational petubations to the metic field, fo α coected R black holes in sting theoy as we will show late, one can consistently set tenso type petubations to the dilaton field to zeo. These petubations wee woked out in detail in [18], 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 ia = 0, h ab = 0, 3.9 γ kl D k D l + k T T ij = 0, D i T ij = 0, g ij T ij = Hee, D i is the covaiant deivative on the d dimensional sphee, associated with the metic γ ij. Thus, the T ij ae the eigentensos of the Laplacian on the sphee S d, whose eigenvalues ae given by k T + = l l + d 3, whee l =, 3, 4,.... It should be futhe noticed that the expansion coefficient H T is gauge invaiant by itself. This is athe impotant: when dealing with linea petubations to a system with gauge invaiance one might always woy that final esults could be an atifact of the paticula gauge one chooses to wok with. Of couse the simplest way out of this is to wok with gauge invaiant vaiables, and this is pecisely implemented in the IK famewok [31, 18, 0]. Obseve that so fa we have only made the choice of backgound metic we wish to petub, and no choice of equations of motion plays any ole. Thus, the IK gauge invaiant vaiables ae also valid fo highe deivative theoies as long as diffeomophisms keep implementing gauge tansfomations. 3. Petubation of a Spheically Symmetic Static Solution Let us now focus on a vacuum type static, spheically symmetic backgound metic fo the emainde of this section. Such a metic is clealy of the type 3.4, and is given by 4 4 If one wee to conside the most geneal static, spheically symmetic metic, then the g component of the metic would equal a geneic function g. It is the vacuum field equations which imply g = f 1. This last condition is peseved by the solutions we shall consideed in this aticle, even in the pesence of the dilaton and highe deivative coections. 8

10 g = f dt dt + f 1 d d + dω d The nonzeo components of the Riemann tenso fo this metic ae R tt = 1 f, R itjt = 1 ff g ij, f R ij = 1 f g ij, R ijkl = 1 1 f g ik g jl g il g jk. 3.1 Petubations to the Einstein field equations without any α coections ae descibed by an equation which is obtained by fist petubing the Ricci tenso R ij. If one collects the above expessions fo h µν and the covaiant deivatives, and eplaces them on the Palatini equation pesented above, one obtains δr ij = f t H T Tij f H T Tij f H T T ij f H T T ij + d f H T T ij + + d 4 H T T ij + 6 d fh T T ij + k T H T T ij, 3.13 δr ia = 0, δr ab = 0, δr = By futhe using the so called totoise coodinate x, which is defined via dx = d f, it was shown in [18] that the equation fo the petubations, δr ij = 0, can be witten as a maste equation fo the gauge invaiant maste vaiable Φ d H T. This maste equation is of Schödinge type: Φ x + V TxΦ = Φ t Hee, V T is the potential fo tenso type gavitational petubations, and is given in its most geneal fom by [18] l l + d 3 d d 4 f V T = f d f Simila maste equations of Schödinge type can be obtained fo the vecto type and scala type gavitational petubations. Howeve, thei espective potentials ae moe complicated [18, 0]. Let us stess that the above maste equation is valid fo any static, spheically symmetic solution to the Einstein field equations of the fom 3.4, but so fa without any α coections. Ou goal in the following is to detemine which kind of maste equation one will obtain when consideing petubations to the solutions of the α coected field equations. It is expected that the potential above will get coected. In ode to detemine an equation fo H T with inclusion of the α coections fom.6, one needs to petub the α coected field equations.10 and.11. In ode to do so, and fo both of them, one fist needs to obtain the vaiation of the Riemann tenso unde geneic petubations of the metic, which 9

11 is given by 3.8. Replacing h µν by the expessions given in 3.9 and futhe using the expessions fo the components of the Riemann tenso as in 3.1, one obtains f δr ijkl = 1 HT + 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, 3.17 δr itjt = t H T + 1 ff H T + ff H T T ij, 3.18 δr ij = f f H T 1 f f H T H T H T T ij, 3.19 δr abcd = 0. By petubing.10 and.11 one futhe gets 3.0 δ φ λ 4 e 4 d φ δ R ρσλτ R ρσλτ + λ d e 4 d φ R ρσλτ R ρσλτ δφ = 0, 3.1 δr ij + λ e 4 d [δ φ R iρστ R ρστ 1 j d R ρσλτ R ρσλτ h ij 1 ] d g ij δ R ρσλτ R ρσλτ + 4 d R ij δφ = Using the explicit fom of the Riemann tenso 3.1, alongside with the vaiations 3.9 and , one can staightfowadly compute most of the tems in 3.1 and 3.. What emains to be analyzed is the equation descibing the dilaton petubation, δφ. In this famewok, we have δ φ = g ab δ a b φ h ij i j φ + g ij δ i j φ. 3.3 Fom 3.7 one can easily show that both δγ c ab = 0 and gij δγ a ij = 0. Assuming as it is the pesent case, due to the spheical symmety of the backgound that the dilaton does not depend on the angula coodinates, i.e., that k φ = 0, we ae left with the following δ φ = g ab a b δφ g ab Γ c ab cδφ + g ij i j δφ g ij Γ k ij kδφ g ij Γ a ij aδφ. 3.4 Using this esult in 3.1, and afte showing that it tuns out that δ R ρσλτ R ρσλτ = 0 in ou situation, one concludes that the equation descibing petubations to the dilaton field is a homogeneous diffeential equation fo δφ. Thus, thee is no immediate inconsistency in setting δφ = 0. Because futhemoe the dilaton is a scala field, which does not admit tenso type petubations, we ae led to conclude that it is indeed the case that δφ = 0 5. This is the infomation one obtains fom 3.1. Collecting the seveal expessions, the esult fo 3. finally becomes 5 Let us point out that ou deivation of this esult is only valid fo tenso type petubations to the action we ae consideing. Let us also point out that we do not expect this to hold fo othe types of gavitational petubations, in paticula this will most cetainly not happen fo scala type petubations. Rathe, one will find coupled equations between petubations of the metic field and of the dilaton field. 10

12 [ 1 λ f [ f d f ] t H T f H T + f + λ + f [ l l + d 3 + λ 4l l + d 3 1 f f 1 f d 4 ff f ] H T + f + d 3 1 f + d f f d ] H T = As it stands, the above equation fo the petubations looks athe complicated, specially if compaed to 3.15 and Indeed, this expession does not even have the Schödinge fom usually associated to the maste equations fo gavitational petubations. So, we would now like to e wite the equation above in the fom of a Schödinge like maste equation, as in In ode to achieve so, one fist wites the petubation equation in tems of the totoise coodinate, x. Then, and following a pocedue athe simila to the one in [3], one fist consides a geneal equation of the fom Futhe defining t H T F H T + P H T + Q H T = k = 1 exp F d P F, Φ = kh T, 3.7 it is easy to see that 3.6 may then be witten as a Schödinge type equation, Φ x Φ Q t = + F 4 F F P + P 4F + P F Φ V [f] Φ, 3.8 F whee Φ is the gauge invaiant maste vaiable fo the gavitational petubation in the pesent famewok, including α coections. In ou case, fom 3.5 it immediately follows F = f, f P = 1 λ f Q = f 1 λ f + λ [ d f + f + λ [ l l + d 3 f ] f 1 f d 4 ff f, f + d l l + d 3 1 f 1 f + d 3 f d ]. 3.9 While the algebaic manipulations above have poduced non polynomial tems in λ, it is impotant to ecall the physics of the poblem at hand. Indeed, in the pesent context, any black hole solution is built petubatively in λ and a solution, chaacteized by a function f, will only be valid in egions whee 11

13 λ, i.e., any petubative solution is only valid fo black holes whose event hoizon is much bigge than the sting length. Woking to fist ode in the petubative paamete λ thoughout then equies the above fomulae to obey the same citeia. A simple powe seies expansion yields F = f, P = f Q = f [d f + f + λd 4 f 1 f [ l l + d 3 f f + d λ l l + d 3 1 f + f ], + f d 3 1 f + 1 f + f 4 f f d ] Fom 3.8 and 3.30 one finally obtains l l + d 3 V T [f] = f + d d 4 f 4 + d 3 1 f + d 6 f + + d 4 f + λ f [ l l + d 3 d 4 d 5 f d 3 1 f f f + f + 3d 3 4d 13f 4 f ] + d 4 ff f d This is the potential fo tenso type gavitational petubations of any kind of static, spheically symmetic R sting coected black hole in d dimensions. This is also one of the main esults in this pape. 3.3 On the Poof of Petubative Stability In ode to study the stability of a solution, we shall use the so called S defomation appoach, fist intoduced in [19] and late futhe developed in [, 3, 4]. Let us biefly eview this technique in the following fo moe details we efe the eade to the oiginal discussion in [19]. Afte having obtained the potential 3.31 fo the maste equation 3.15, one assumes that its solutions ae of the fom Φx, t = e iωt φx, such that Φ t = iωφ. In this way the maste equation may be witten in Schödinge fom, as ] [ d dx + V x φx Aφx = ω φx. 3.3 A given solution of the gavitational field equations will then be petubatively stable if and only if the opeato A, defined above, has no negative eigenvalues fo x R [19]. Technically, A needs to be a positive self adjoint opeato in the Hilbet space of squae integable functions of x. Consideing a set of smooth functions {φx}, on the ange of x, the above condition is equivalent to the positivity fo any given φ of the following inne poduct [19] 1

14 φ Aφ = + ] φ x [ d dx + V x φx dx = + [ dφ dx + V x φ ] dx, 3.33 whee, in the last step, we have made an integation by pats. Afte some futhe algeba, and anothe integation by pats, one finds that the inne poduct φ Aφ may be e witten as φ Aφ = + [ Dφ + Ṽ x φ ] dx, 3.34 whee we have defined D = d ds dx + S and Ṽ x = V x + f d S, with S a completely abitay function. The stability is then guaanteed by the positivity of Ṽ x, fo x R, whateve function S is chosen [19]. If one now follows [], and takes into consideation the fom of the potential V x given in 3.8, the most convenient choice fo S, i.e., the one which leads to the simplest analysis, is S = F k Indeed, with this choice of S, we simply obtain Ṽ x = Q and ae left with φ Aφ = + Dφ dx + dk d Qx φ dx Thus, all that is necessay in ode to guaantee the stability of a given solution is to check the positivity of Q, fo x R, with Q given by This is paticulaly inteesting if one ecalls the definition of the totoise coodinate, dx = d f. It is simple to ealize that the second tem of the expession above becomes + Q φ d, 3.37 f whee is the adius of the black hole event hoizon. Because f is a positive function fo >, petubative stability of a given black hole solution, with espect to tenso type gavitational petubations, then follows if and only if one can pove that Q is a positive function fo, i.e., is a positive function outside of the black hole event hoizon. We shall come back to this question in the following. 4. The Callan Myes Pey Black Hole Solution The petubation theoy we have developed in the pevious section is valid fo any static, spheically symmetic solution to the field equations.10 and.11, i.e., fo any backgound metic of the type Fo the emainde of this wok we shall focus on a paticula black hole solution and study both petubation and scatteing theoies associated to this paticula solution, within the famewok we have developed. But fist, let us make some geneic emaks concening black hole solutions in sting theoy. As noticed in [3, 33, 34], the pesence of highe ode cuvatue tems in the low enegy effective action affects the enegy momentum tenso in such a way that its time component, T t t, epesenting the local enegy density in the Einstein case, will not be necessaily positive definite anymoe. It so happens that this assumption, of positive definiteness, is a necessay condition fo the no hai conjectue. Anothe necessay condition fo this conjectue, which again is not necessaily veified by the α coected solutions, is the elation T t t = T θ θ between time and angula components of the enegy momentum tenso a elation 13

15 which was checked to be valid in the case of spheically symmetic solutions to the Einstein theoy. As these conditions ae no longe veified fo highe deivative coected black hole solutions, it may thus be possible to cicumvent the no hai conjectue and find seveal static, spheically symmetic black hole solutions to the cuvatue coected field equations. These new black hole solutions may have eithe pimay o seconday hai outside the hoizon. In the fist case, of pimay hai, besides intoducing new fields outside of the hoizon, the solutions also intoduce new paametes e.g., dilatonic chages. In the second case, of seconday hai, thee ae no new paametes and only new fields ae intoduced, the chages of which being eventually expessed in tems of the oiginal paametes. One of the new fields which must always be pesent in any sting theoetic black hole solution is the dilaton field. As we have said befoe, a constant dilaton solution is only possible at the classical level, without any stingy coections, as the cuvatue coection tems act as a souce fo the dilaton field equation. Since we wok in a sting theoetic famewok one could also expect that these black hole solutions should include, besides the gaviton and the dilaton which we have alluded to above, couplings to diffeent antisymmetic tenso fields depending on the bosonic massless spectum of each theoy. But, in this case, it tuns out that one can always consistently set the antisymmetic tenso fields to zeo, without esticting the dilaton and gaviton tems though the field equations. The same holds tue fo any femionic fields aising in heteotic sting theoy. Still, consideing black hole solutions with both highe deivative coections and couplings to antisymmetic tenso fields would be inteesting on its own, but, to ou knowledge, these solutions have only been studied in fou dimensions, in [3, 33, 34]. Solutions with dilatonic pimay hai i.e., an independent dilatonic chage wee studied in [7]. We shall instead conside the case of seconday hai. The black hole solution we will focus upon in this pape is a d dimensional R stingy coection to the Schwazschild solution, and is thus natually of the type of This highe deivative coected solution was fist obtained by Callan, Myes and Pey in [13] and is the mentioned CMP black hole solution. Its only fee backgound paamete is µ, a paamete which is elated to the classical ADM mass of the black hole though M cl = d A d κ µ, 4.1 whee A n is the aea of the unit n sphee. Fo the classical d dimensional Schwazschild solution, we have f = 1 µ. 4. d 3 In ode to intoduce α coections to this solution, one chooses a coodinate system in which the position of the hoizon, given by = µ 1 d 3 RH in the solution above, is not changed. Such a solution was woked out in [13], petubatively in λ, and in such a coodinate system f is given by f = 1 Rd 3 H d 3d 4 λ R 1 d 3 R 1 d 1 H H d 3 d 1 RH d Rd 3 H d 3 The α coected ADM mass fo this stingy black hole is given by M = d A d κ lim d 3 1 f = d 3d λ R H d Ad κ µ. 4.4 This epesents an ADM mass which is both inetial and gavitational, since in the CMP solution we ae consideing, and to fist ode in λ, g tt = 1 g = f. The dilaton, which is classically set to vanish, will 14

16 get α coections. This means it will be of the fom φ = λϕ, with ϕ a solution to the dilaton equations of motion without any explicit dependence on the petubative paamete λ. One futhe ealizes that the dilaton coections will only affect the field equations at an ode in λ which is highe than the one we ae consideing, and theefoe we will not need it. Its full explicit expession may be found in [13]. In ode to compute the black hole tempeatue [35], one fist Wick otates to Euclidean time t = iτ. The esulting manifold is smooth as long as τ is a peiodic vaiable, with a peiod β which is elated to the black hole tempeatue as T = 1 β. The pecise smoothness condition amounts to π = lim β f 1 df 1, 4.5 d fom which follows T = 1 4π f. In ou paticula case, this immediately implies T = d 3 d 1 d 4 λ π One leans that the stingy α coections actually educe the black hole tempeatue fo d > 4. This seems to suggest that black hole tempeatue may each a maximum value. Indeed, taking the above fomula liteally one finds a maximal tempeatue of T max 0.06 α fo finite black hole adius [13]. Unfotunately, the black hole adius at which this happens is of the ode of the sting scale and thus outside the validity ange of the petubative solution. Nevetheless, one may notice that this value of the maximal tempeatue is close to the Hagedon tempeatue in the fee sting spectum, T citical Futhemoe, the α tempeatue fomula yields zeo tempeatue at a adius smalle than the one fo maximal tempeatue [13]. Such a black hole could pesumably be egaded as a sting soliton. Let us finally add that if one computes the entopy of these stingy black holes, it follows that S = A H d d 3 λ R H R H, 4.7 so that entopy no longe equals one quate of the hoizon aea [13]. Indeed, the entopy of black holes is inceased in sting theoy see, e.g., [36, 37, 38, 39]. Having eviewed the black hole solution which we have decided to focus upon, in the following we shall analyze its stability in detail, using the machiney we have developed ealie. Then, in section 5, we shall futhe analyze scatteing in this spacetime geomety. 4.1 Potential fo Classically Non Dilatonic Solutions As descibed above, the CMP solution has vanishing dilaton field at the classical level i.e., at ode Oλ 0, while at fist ode in λ it has a non tivial dilaton field. It tuns out that in this case one may futhe simplify some of ou pevious esults concening the petubation theoy of spheically symmetic static solutions. Indeed, upon the futhe assumption that the dilaton field is of the type φ = λϕ, one may judiciously use the field equation fo R ij in.11 to fist ode in λ i.e., neglecting all the dilaton tems, which would only contibute at least to ode Oλ, and deive the elation λr abcd R abcd = λ [ f = g ij R ij + λr ijkl R ijkl = d d 3 1 f 1 + λ 1 f f ]

17 In the following, we shall use the elation above in ode to emove the f tem in 3.31 and futhe simplify ou next calculations. It is not too had to obtain: l l + d 3 d d 4 f V T [f] = f d f + + λ f [ l l + d 3 d 4 d 5 f + + d 4 f 1 f + f + f + 4d 3 5d 16f 4 f ] + d 4 ff. 4.9 This expession is valid fo any sting theoy coected, spheically symmetic, static solution, which has no dilaton field at the classical level as is the case of the CMP solution. The fist thing one notices is that the classical tem in the potential above, at ode Oλ 0, pecisely matches the IK potential fo tenso type gavitational petubations, as expessed in This is cetainly to be expected, as the IK potential was deived in pue EH gavity, whee thee is no dilaton. This also tells us that the CMP solution is, in this context, a vey natual stingy extension of the Schwazschild black hole. We shall use the potential above in the following studies of the CMP geomety. 4. Poof of Petubative Stability We ae now fully set to pove stability of the CMP solution. As explained in section 3, stability follows if and only if Q 0 fo. Using 4.8 one may e wite Q in 3.9 as f Q = 1 λ f l l + d 3 f [ l l + d 3 + 4λl l + d 3 1 f ] [ 1 + λ 4 1 f ] + f Because f > 0 fo > this is, in some sense, the definition of the event hoizon 6, it is clea that one will have Q 0 fo, in any spacetime dimension, as long as 1 f + f > in the vey same egion. But this tem is the stingy coection, so that the function f in it is only to be evaluated at the classical level, i.e., at ode Oλ 0. In this case it tivially follows fo the CMP solution 1 f + f µd 1 = d, 4.1 Oλ 0 which is positive fo any >. This poves stability of the CMP black hole solution in any spacetime dimension d > 4 whee tenso type gavitational petubations exist, fo positive values of the black hole mass and sting length. But this esult also poves a bit moe: all we used above was the geneic fom of the potential and the classical esult fo f. So, one futhe concludes that any spheically symmetic, static solution, which has no dilaton at ode Oλ 0, is stable unde tenso type gavitational petubations. 6 Keep in mind that in the pesent context the CMP solution satisfies f > 0 fo > as long as the black hole in consideation is lage, i.e., as long as one emains in the domain of validity of sting petubation theoy. 16

18 5. Scatteing Theoy in the Callan Myes Pey Geomety The equation descibing gavitational petubations to the CMP solution is also the equation which allows fo a study of scatteing in this spacetime geomety. Moe pecisely, it is the equation which allows fo a study of geybody factos and quasinomal fequencies, equied data in the study of the Hawking emission specta and quasinomal inging. In the following, we shall addess both these questions. 5.1 Absoption Coss Section Let us begin with the computation of the CMP black hole absoption coss section fo low fequency tenso type gavitational waves. Stating with [40], thee is a geat deal of liteatue on geybody factos and black hole absoption coss sections, computed at low enegies, and we efe the eade to [41] fo a eview and complete list of efeences. A classical esult in EH gavity is the fact that, fo any spheically symmetic black hole in abitay dimension, minimally coupled massless scala fields have an absoption coss section which is equal to the aea of the black hole hoizon [9]. This esult is quite spectacula as it points towads a univesality of the low fequency absoption coss sections of geneic black holes in EH gavity. In spite of this, not much wok has been done on tying to extend such esult away fom the EH ealm, with the inclusion of highe deivative coections. The only exception we ae awae of is [4], whee the computation of geybody factos in cuvatue squaed Lovelock gavity without a dilaton field is addessed; which is a diffeent context fom the one in this pape. Futhemoe, the authos focused moe on obtaining numeical esults fo a wide ange of fequencies, athe than on the analytical solution to the low fequency poblem. Hee, we shall addess the analytical computation of the low fequency absoption coss section fo the CMP black hole, following the standad analysis descibed in [40, 41]. As we have said, we will conside scatteing of tenso type gavitational waves, at low fequencies, ω 1. The low fequency equiement is necessay as we shall use the technique of matching solutions see [40, 41] fo details; it is pecisely when the wave length of the scatteed field is vey lage, as compaed to the adius of the black hole, that one can actually match solutions nea the event hoizon to solutions at asymptotic infinity [40, 41]. Also, we shall only focus on the leading contibution to the scatteing pocess, given by the s wave, whee l = 0 7. Let us begin nea the CMP black hole event hoizon. At the pecise location of the hoizon, the potential descibing tenso type gavitational petubations vanishes, and the maste equation educes to a simple fee field equation whose solutions ae eithe incoming o outgoing plane waves, in the totoise coodinate. Vey close to the event hoizon,, one has d d 3 d 1d 4 λ RH V T 1 d RH RH 3 x R H d 1d 4 λ RH 1 + log d 3 R H + O, + O. 5.1 The above equations tell us that as long as ω one will have V T ω and in this nea hoizon egion one may neglect the potential V T in the maste equation. One thus obtains, vey close to the event hoizon, 7 At fist this might seem puzzling as, fo tenso type petubations, one should conside l. Ou point of view in hee is to analyze the simplest possible example, and as such one consides the l = 0 case as a good fist appoximation in a patial wave expansion it is cetainly a leading tem with espect to all othe l > 0 tems. Natually, futue wok should conside the case fo geneic l. In spite of this, and as we shall find at the end of this section, the esult we obtain tuns out to be valid both as the fist appoximation to tenso type gavitational petubations and as the exact esult fo minimally coupled massless scala fields, thus also validating the appoximation we choose to make in hee, of setting l = 0. 17

19 d dx + ω kh T = The solutions to the above equation ae plane waves. As we ae inteested in studying the absoption coss section, we shall conside the geneal solution fo a puely incoming plane wave whee we have also evaluated k ir d+1 H H T x = A nea e iωx, 5.3 d + 1d λ R H + O, 5.4 which can be teated as a constant fo. One may still move slightly away fom the event hoizon, making use of the above expansion fo the totoise coodinate, 5.1. Indeed, if one uses 5.1 in 5.3, one may give one step futhe out fom the black hole event hoizon, while still maintaining the validity of the seies expansion [40, 41]. One obtains in this case H T A nea 1 + i R Hω d 1d d 3 λ R H log RH. 5.5 Anothe egion of spacetime which is simple to study is asymptotic infinity. The asymptotic egion of the CMP black hole is pecisely the same as the asymptotic egion of the Schwazschild spacetime, which is basically flat Minkowski spacetime. At asymptotic infinity, the potential descibing tenso type gavitational petubations vanishes, and the maste equation educes to a simple fee field equation whose solutions ae eithe incoming o outgoing plane waves, in the totoise coodinate. One may also solve the maste equation in the oiginal adial coodinate in tems of Bessel functions, obtaining H T = ω 3 d/ [ A J d 3/ ω + B N d 3/ ω ] see, e.g., [9, 40, 41]. At low fequencies, with ω 1, such solution becomes 1 H T A asymp d 3 Γ d 3 Γ d 3 + B asymp d 1 π ω d 3 + O ω. 5.6 In ode to compute the absoption coss section, one now needs to elate the coefficients A nea, A asymp and B asymp. This can be done via a standad technique to match nea hoizon to asymptotic solutions, and equies studying the maste equation in an intemediate egion, between the event hoizon and asymptotic infinity see [40, 41] fo a clea explanation of this pocedue. This is what we shall do in the following. As one moves away fom the black hole event hoizon, the potential begins to gow. Because we ae studying low fequency scatteing, eventually the potential will be much bigge than the scatteing fequency. This is the definition of the intemediate egion: it is the egion whee V T ω, but whee the low fequency constaint of ω 1 emains valid. This constaint, togethe with ω, thus defines the adial coodinate ange associated to the intemediate egion. To poceed, we need to solve the maste equation in this egion. Let us do this petubatively in λ, defining the expansion H T = H 0 + λh 1. It is not too had to see that H 0 satisfies [ f d f d d d 4f + f d d 4 + d f ] kh 0 = 0,