On black hole spectroscopy via adiabatic invariance

TL;DR

This paper derives a coordinate-independent method for black hole area quantization using adiabatic invariance and tunneling, confirming equally spaced spectra for Schwarzschild black holes.

Contribution

It introduces a covariant adiabatic invariant approach to black hole spectroscopy, ensuring coordinate invariance and consistent area quantization results.

Findings

Black hole horizon area is quantized with equal spacing.

The quantization is independent of coordinate choice.

The method applies to Schwarzschild black holes in different coordinates.

Abstract

In this paper, we obtain the black hole spectroscopy by combining the black hole property of adiabaticity and the oscillating velocity of the black hole horizon. This velocity is obtained in the tunneling framework. In particular, we declare, if requiring canonical invariance, the adiabatic invariant quantity should be of the covariant form $I_{\textrm{adia}}=\oint p_idq_i$. Using it, the horizon area of a Schwarzschild black hole is quantized independent of the choice of coordinates, with an equally spaced spectroscopy always given by $\Delta \mathcal{A}=8\pi l_p^2$ in the Schwarzschild and Painlev\'{e} coordinates.