Large mass expansion of quasi-normal modes in AdS5

We calculate analytically the asymptotic form of quasi-normal frequencies for massive scalar perturbations of large black holes in AdS5. We solve the wave equation, which reduces to a Heun equation, perturbatively and calculate the wave function as an expansion in 1/m, where m is the mass of the mode. The zeroth-order results agree with expressions derived by numerical analysis. We also calculate the first-order corrections to frequencies explicitly and show that they are of order 1/m. Research supported by the US Department of Energy under grant DE-FG05-91ER40627. 1 Quasi-normal modes of black holes in asymptotically AdS space-times have received a lot of attention recently [1–17] and should shed some light on the AdS/CFT correspondence. They are derived as complex eigenvalues corresponding to a wave equation which is solved subject to the conditions that the flux be ingoing at the horizon and the wave-function vanish at the boundary of AdS space. The wave equation reduces to a Hypergeometric equation in three dimensions (AdS3) and can therefore be solved exactly [7, 13]. In higher dimensions, an analytic solution is not readily available, as the wave equation develops unphysical singularities. It has been possible to obtain numerical values for quasi-normal frequencies [2, 18–20]. More recently, we proposed an analytic method of calculating massless quasi-normal modes of large AdS black holes. We applied the method to the calculation of low-lying frequencies [21] as well as in the asymptotic regime [22] in AdS5. The method was based on a perturbative expansion of the wave equation, which reduced to the Heun equation [23], in the dimensionless parameter ω/TH , where ω is the frequency of the mode and TH is the (high) Hawking temperature of the black hole. This is a non-trivial expansion, for the dependence of the wave-function on ω/TH changes as one moves from the asymptotic boundary of AdS space to the horizon of the black hole. We showed that our results were in agreement with numerical results [2, 18]. Here we extend the discussion of [21, 22] to the case of massive scalar modes. The quasinormal frequencies determine the poles of the retarded correlation functions of dual operators in finite-temperature N = 4 SU(N) SYM theory in the large-N , large ’t Hooft coupling limit [24]. They have also been recently studied in [25] where they were shown to arise in complexified geodesics. We shall derive a systematic analytic expansion of the frequencies in powers of 1/m, where m is the mass of the mode. Numerical results indicate [24] that in the limit of large mass, ωn TH ∼ 2π(±1− i)(n + h+ − 32) , n = 1, 2, . . . (1) where h± = 1 ± √ 1 +m2R2. Our results confirm analytically the above asymptotic result of numerical analysis. Our method is an extension of the discussion in [22] where the numerical result (1) was confirmed analytically in the massless case (m = 0). We also calculate the firstorder correction and show that it is of order 1/h+. A similar procedure was followed in [21]. The metric of a five-dimensional AdS black hole may be written as ds = −f(r) dt + dr 2 f(r) + rdΩ3 , f(r) = r R2 + 1− ω4M r2 (2)