Quantization of entropy spectra of black holes

From the quasinormal modes of black holes, we obtain the quantizations of the entropy and horizon area of black holes via Bohr-Sommerfeld quantization, based on Bohr’s correspondence principle. For this, we identify the appropriate action variable of the classical system corresponding to a black hole. By considering the BTZ black holes in topologically massive gravity as well as Einstein gravity, it is found that the spectra of not the horizon areas but the entropies of black holes are equally spaced. We also propose that other characteristic modes of black holes, which are non-quasinormal modes or holographic quasinormal modes, can be used in quantization of entropy spectra just like quasinormal modes. From these modes, it is found that only the entropy spectrum of the warped AdS₃ black hole is equally spaced as well. Furthermore, by considering a scattering problem in a black hole, we propose that the total transmission modes and total reflection modes of black holes can be regarded as characteristic modes of black holes and result in the equally spaced entropy of the Kerr and Reissner-Nordström black holes. Finally, we conclude that there is a universal behavior that the entropy spectra of various black holes are equally spaced.