On black hole spectroscopy via adiabatic invariance

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.