Asymptotic Quasinormal Frequencies of Brane-Localized Black Hole

The asymptotic quasinormal frequencies of the brane-localized (4 + n)dimensional black hole are computed. Since the induced metric on the brane is not an exact vacuum solution of the Einstein equation defined on the brane, the real parts of the quasinormal frequencies ω do not approach to the wellknown value TH ln 3 but approach to TH ln kn, where kn is a number dependent on the extra dimensions. For the scalar field perturbation Re(ω/TH) = ln 3 is reproduced when n = 0. For n 6= 0, however, Re(ω/TH ) is smaller than ln 3. It is shown also that when n > 4, Im(ω/TH) vanishes in the scalar field perturbation. For the gravitational perturbation it is shown that Re(ω/TH) = ln 3 is reproduced when n = 0 and n = 4. For different n, however, Re(ω/TH) is smaller than ln 3. When n = ∞, for example, Re(ω/TH) approaches to ln(1 + 2 cos √ 5π) ≈ 0.906. Unlike the scalar field perturbation Im(ω/TH) does not vanish regradless of the number of extra dimensions. ∗Email:[email protected] 1 The stability problem of the black holes, when perturbed by the external fields, is a long-standing issue in the context of the general relativity [1–4]. It is well-known that all perturbations are radiated away, which is characterized by the quasinormal modes [5]. These quasinormal modes are defined as solutions of the perturbation wave equation, belonging to complex-characteristic frequencies and satisfying the boundary conditions for the purely outgoing waves at infinity and purely ingoing waves at the horizon, i.e. Ψ ∼ e as z → −∞ (1) Ψ ∼ e as z → ∞ where z is an appropriate “tortoise” coordinate and the time dependence of the fields is taken as e. In our notation the quasinormal frequencies ω should satisfy Im(ω) ≥ 0. As a consequence, the quasinormal modes diverge exponentially at both boundaries. This makes it extremely difficult to determine the quasinormal frequencies numerically. This is a main reason why only very few frequencies with moderate imaginary parts were known [6]. About two decades ago Leaver [7] found a possibility to compute the quasinormal frequencies without having to deal with the corresponding quasinormal modes numerically. Instead of the solutions he used the recursion relation which provides an infinite continued fraction. However, this method has a technically problem of convergence when Im(ω) >> Re(ω). This defect of the continued fraction method was mostly removed by Nollert [8] by computing the “remaining” infinite continued fraction. For the case of the 4d Schwarzschild black hole Nollert showed using his improved numerical method that the asymptotic quasinormal frequencies for the scalar field and gravitational perturbations become ω = n̄+ 1/2 2 i+ 0.0874247 (n̄ = 0, 1, 2, · · ·) (2) with, for simplicity, assuming rh = 1 where rH is an horizon radius. This was confirmed by Andersson [9] by the phase integral method, which is an improved WKB-type technique. 2 After few years Hod [10] claimed surprisingly that the numerical number in Eq.(2) is identified as 0.0874247 → ln 3 4π = TH ln 3 (3) where TH is an Hawking temperature. This identification and the Bohr’s correspondence principle naturally imply that the minimal quantum area is 4 ln 3, one of the values 4 ln k suggested in Ref. [11]. This is intriguing from the loop quantum gravity [12] point of view because it suggests that the gauge group should be SO(3) rather SU(2) [13]. Subsequently, the identification (3) was analytically shown in Ref. [14,15]. Especially in Ref. [15] the authors transformed the boundary condition of the quasinormal modes at the horizon into the monodromy in the complex plane of the radial coordinate. We will use this method to compute the asymptotic quasinormal frequencies of the brane-localized (4+ n)-dimensional Schwarzschild black hole. For the quasinormal modes of the other asymptotically flat black holes see Ref. [18–21] and references therein. Recently, much attention is paid to the higher-dimensional black holes. Besides its own theoretical interest the main motivation of it seems to be the emergence of the TeV scale gravity arising in the brane-world scenarios [22–25], which opens the possibility to make a tiny black holes factory in the future high-energy colliders such as LHC [26–30]. In this reason the absorption and emission problems of the higher-dimensional black holes were extensively explored recently [31–34]. The lower quasinormal frequencies for the brane-localized 5-dimensional rotating black holes were recently computed numerically [35]. In this letter we would like to go further for the study of the brane-world black holes by examining the asymptotic quasinormal frequencies of the brane-localized Schwarzschild black holes. Since the metric of the branelocalized Schwarzschild black hole is induced by the higher-dimensional bulk metric, it is not a vacuum solution of the Einstein equation defined on the brane. We will show that this fact 1See also Ref. [16,17] for asymptotically non-flat black holes 3 makes the real part of the asymptotic quasinormal frequencies not to be the well-known value TH ln 3 but to be TH ln kn, where TH is an Hawking temperature of the higher-dimensional black hole and kn is a number dependent on the extra dimensions. We start with the (4+n)-dimensional Schwarzschild black hole whose metric is given by [36,37] dsB = −h(r)dt + h(r)dr + rdΩn+2 (4) where h(r) = 1− ( rH r )n+1 (5) dΩn+2 = dθ 2 1 + sin 2 θ1 [