Quantum tunneling, adiabatic invariance and black hole spectroscopy

Eur. Phys. J. C (2017) 77:314 DOI 10.1140/epjc/s10052-017-4901-y Regular Article - Theoretical Physics Quantum tunneling, adiabatic invariance and black hole spectroscopy Guo-Ping Li1,a , Jin Pu1,2,b , Qing-Quan Jiang2,c , Xiao-Tao Zu1,d 1 2 School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China College of Physics and Space Science, China West Normal University, Nanchong 637009, China Received: 30 March 2017 / Accepted: 8 May 2017 / Published online: 17 May 2017 © The Author(s) 2017. This article is an open access publication Abstract In the tunneling framework, one of us, Jiang, together with Han has studied the black hole spectroscopy via adiabatic invariance, where the adiabatic invariant quantity has been intriguingly obtained by investigating the oscillating velocity of the black hole horizon. In this paper, we attempt to improve Jiang–Han’s proposal in two ways. Firstly, we once again examine the fact that, in different types (Schwarzschild and Painlevé) of coordinates as well as  in different gravity frames, the adiabatic invariant Iadia = pi dqi introduced by Jiang and Han is canonically invariant. Secondly, we attempt to confirm Jiang–Han’s proposal reasonably in more general gravity frames (including Einstein’s gravity, EGB gravity and HL gravity). Concurrently, for improving this proposal, we interestingly find in more general gravity theories that the entropy of the black hole is an adiabatic invariant action variable, but the horizon area is only an adiabatic invariant. In this sense, we emphasize the concept that the quantum of the black hole entropy is more natural than that of the horizon area. 1 Introduction The black hole as a special object with the strong gravitational field, has always been regarded as a test bed for any proposed scheme for a quantum-gravity theory. It therefore shows in principle that the exploration of the black hole entropy/area quantum has great significance, since it may provide a window on finding an effective way to quantize a gravitational field. However, there is as yet no complete quantum theory of gravity. Hence, reenforcing our understanding of these issues at a semiclassical level may be an appropriate juncture. In the context of a semiclassical notion, Bekenstein has reported that black hole entropy is always quantized in units of 2p [3],1 and the quantum of the horizon area  usually appears in the form A = 8π 2p (where  p = h̄G/c3 is the Planck length) [7]. This idea mainly comes from the remarkable fact that the black hole area in the nonextremal case is an adiabatic invariant [8–11]. Later on, various types of other semiclassical approaches have appeared to further study the quantization of the black hole entropy [12–52]. The most popular one among them was proposed by Kunstatter in 2002. It is pointed out that, for a given system (black hole) equipped with the energy E and the vibrational frequency ω(r ), the  action I = dE/ω(E) is an adiabatic invariant [12]. Then, by using the Bohr correspondence principle for the quasinormal frequency, the entropy spectrum of a d-dimensional spherically symmetric black hole is equally spaced with its quantum to be equal to the one given by Hod [13,14], as well as by Bekenstein and Mukhanov [15].2 However, in the Kunstatter’s treatment, if one employs the oscillating frequencies of the black hole only coming from the real part of the highly damped quasinormal frequencies, there is an illdefined (adiabatic invariant) integral with inclusion of a large logarithmic term when it is used with the case of the rotating black hole [24]. In 2008, Maggiore has suggested that the black hole can be treated as a damped harmonic oscillator, and correspondingly, it is relevant that the frequency should contain contributions from both real and imaginary parts of the complex quasinormal mode frequencies, rather than only from the real part [23]. 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 2p [25], which is exactly equal to the original result of Beken- a e-mail: b e-mail: 1 c e-mail: 2 d e-mail: This was also confirmed by several different approaches [4–6]. Here, a lot of work in this direction can be found in [15–22] and the references therein. 123 314 Page 2 of 7 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 [24–35,49–52]. Recently, the interesting notion has been found that, without the aid of quasinormal modes, one can quantize the entropy and horizon area of the black hole  in the tunneling mechanism [2].3 Here, the action I = pi dqi is an adiabatic invariant, which can be intriguing (...truncated)