Area spectra of the rotating BTZ black hole from quasinormal modes

Abstract

Following Bekenstein's suggestion that the horizon area of a black hole should be quantized, the discrete spectrum of the horizon area has been investigated in various ways. By considering the quasinormal mode of a black hole, we obtain the transition frequency of the black hole, analogous to the case of a hydrogen atom, in the semiclassical limit. According to Bohr's correspondence principle, this transition frequency at large quantum number is equal to classical oscillation frequency. For the corresponding classical system of periodic motion with this oscillation frequency, an action variable is identified and quantized via Bohr-Sommerfeld quantization, from which the quantized spectrum of the horizon area is obtained. This method can be applied for black holes with discrete quasinormal modes. As an example, we apply the method for the both non-rotating and rotating BTZ black holes and obtain that the spectrum of the horizon area is equally spaced and independent of the cosmological constant for both cases.

Quantum tunneling, adiabatic invariance and black hole spectroscopy

“…Introducing a covariant form for the adiabatic invariant, Jiang and Han have combined the oscillating velocity of the black hole horizon to revisit the black hole spectroscopy of the Schwarzschild black hole in the tunneling picture [1]. It has been shown that the area spectrum of the Schwarzschild black hole is described with an equally spacing A = 8π 2 p , which is in agreement with Bekenstein's original proposal [7] and is fully in consistence with the previous findings by combining the black hole property of adiabaticity and the black hole quasinormal frequencies [24][25][26][27][28][29][30][31][32][33][34][35][49][50][51][52].…”

“…Following this new explanation and the black hole property of adiabaticity, it has been shown that the equally spaced area spectrum of a slowly rotating black hole can be expressed as the form A = 8π 2 p [25], which is exactly equal to the original result of Beken-stein [7]. Later, much further work has shown that the black hole spectroscopy in more general gravity frames can also be properly reproduced by combining the new explanation for quasinormal mode frequency and the black hole property of adiabaticity [24][25][26][27][28][29][30][31][32][33][34][35][49][50][51][52].…”

“…Moreover, Kwon and Nam pointed out that an action variable of the classical system is an adiabatic invariant, but an adiabatic invariant is not necessarily an action variable. The Bohr-Sommerfeld quantization rule can be applied only for the action variable of the classical system [26][27][28], rather than the adiabatic invariant. In this case, it was later showed by quasinormal modes that the area spectra of a rotating BTZ black hole is equally spaced [26][27][28].…”

Some properties of the de Sitter black holes in three dimensional spacetime

“…Later, by considering the asymptotic QNMs of a black hole, it was proposed that the transition frequency in quantum levels of black holes is relevant to the imaginary part of the QNMs in the semi-classical limit [47]. By identifying an action variable in a black hole system with this transition frequency, it was proposed that the following quantization condition is obtained via the Bohr-Sommerfeld quantization in the semi-classical limit [48]:…”

“…In particular, it was found in lots of literatures [29,48,49,50,51,52,53,54,55,56] that the entropies of black holes in diverse gravity theories are quantized with same spacing. Our results are also consistent with the universal property of equidistant entropy spectrum.…”