On the Quantum Levels of Isolated Spherically Symmetric Gravitational Systems

TL;DR

This paper connects the canonical quantum theory of Schwarzschild black holes with Bekenstein and Mukhanov's discrete mass spectrum, deriving a quantized mass spectrum using boundary conditions and Euclidean time periodicity.

Contribution

It relates two different quantum descriptions of black holes by deriving a discrete mass spectrum from boundary conditions and Euclidean time periodicity.

Findings

Derived a discrete mass spectrum M_n=(1/2)√n m_P for black holes.

Linked plane wave quantum states to Bekenstein-Mukhanov quantization.

Established a connection between continuous and discrete quantum descriptions.

Abstract

The known canonical quantum theory of a spherically symmetric pure (Schwarzschild) gravitational system describes isolated black holes by plane waves exp(-iMc^2\tau/\hbar) with respect to their continuous masses M and the proper time \tau of obsevers at spatial infinity. On the other hand Bekenstein and Mukhanov postulated discrete mass levels for such black holes in the spirit of the Bohr-Sommerfeld quantisation in atomic physics. The two approaches can be related by postulating periodic boundary conditions in time for the plane waves and by identifying the period \Delta in real time with the period \Delta_H= 8\pi GM/c^3 in Euclidean time. This yields the mass spectrum M_n=(1/2)\sqrt{n}m_P, n=1,2,... .