Asymptotic quasinormal modes of a coupled scalar field in the Gibbons-Maeda dilaton spacetime

The study of quasinormal modes in a black hole background spacetime has a long history [1]-[19]. It is shown that their frequencies and damping times are entirely fixed by the black hole parameters and independent of the initial perturbations. Therefore, it is certain that quasinormal modes carry the characteristic information about a black hole and it can provide a new and direct way for astrophysicists to search black hole in the universe. Recently, a great deal of efforts[20]-[31] has been devoted to the study of the asymptotic quasinormal modes because that Hod’s conjecture[20] shows that there maybe exist a connection between the asymptotic quasinormal frequencies and quantum gravity. In terms of Bohr’s correspondence principle, the transition frequencies at large quantum numbers (n→ ∞) should equal to classical oscillation frequencies. Hod [20] generalized Bohr’s correspondence principle to the black hole physics and regarded the real parts of the asymptotic quasinormal frequencies (n → ∞) as the characteristic transition frequency for the black hole. Moreover, Hod observed that the real parts of highly damped quasinormal frequencies in Schwarzschild black hole can be expressed as ωR = TH ln 3, which is derived numerically by Nollert[21] and later confirmed analytically by Motl [22][23] and Andersson [24]. Together with the first law of black hole thermodynamics, Hod obtained the value of the fundamental area unit in the quantization of black hole horizon area. Following Hod’s works, Dreyer[25] found that the asymptotic quasinormal modes can fix the Barbero-Immirzi parameter which is introduced as an indefinite factor by Immirzi[26] to obtain the right form of the black hole entropy in the loop quantum gravity. Furthermore, Dreyer obtained that the basic gauge group in the loop quantum gravity should be SO(3) rather than SU(2). These exciting new results imply that Hod’s conjecture maybe create a new way to probe the quantum properties of black hole. However, the question whether Hod’s conjecture applies to more general black holes still remain open. In their deduction, it is obvious that the factor ln 3 in the quasinormal frequencies plays an essential and important role. In other word, whether Hod’s conjecture is valid depends on whether the factor ln 3 appears in the asymptotic quasinormal frequencies or not. Recently, we [27] probed the asymptotic quasinormal modes of a massless scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime and find that the frequency spectra formula satisfies Hod’s conjecture. For the non-flat spacetime, Cardoso and Yoshida [28][29]found the asymptotic quasinormal frequencies in the Schwarzschild de Sitter and Anti-de Sitter spacetimes depend on the cosmological constant. Only in the case that the cosmological constant vanishes, the real parts of the asymptotic quasinormal frequencies returns TH ln 3. For the Reissner-Nordström black hole, L. Motl and A. Neitzke[23] found the asymptotic quasinormal frequencies satisfy e + 2 + 3e−βIω = 0, (1)