Quantum Mechanics of a Black Hole

TL;DR

This paper develops a quantum model of Schwarzschild black holes based on discrete area spectra, deriving an algebra of observables, and explains Hawking radiation as a consequence of area-phase interactions influenced by initial states.

Contribution

It introduces an algebra of black hole observables and models Hawking radiation through an area-phase interaction, linking initial states to black hole evolution.

Findings

Reproduces semi-classical Hawking radiation power

Shows initial states determine black hole evolution

Models black hole entropy via area degeneracy

Abstract

Beginning with Bekenstein, many authors have considered a uniformly spaced discrete quantum spectrum for black hole horizon area. It is also believed that the huge degeneracy of these area levels corresponds to the notion of black hole entropy. Starting from these two assumptions we here infer the algebra of a Schwarzschild black hole's observables. This algebra then serves as motivation for introducing in the system's Hamiltonian an interaction term. The interaction contains the horizon area operator, which is a number operator, and its canonical conjugate, the phase operator. The Hawking radiation from a Schwarzschild black hole is seen to be a consequence of an area-phase interaction. Using this interaction we have reproduced the semi-classical result for the Hawking radiation power. Furthermore, we show that the initial state of the black hole determines the nature of its development. Thus, a state which is an area eigenstate describes a static eternal black hole, but a coherent state describes a radiating black hole. Hence, it is the observer's initial knowledge or uncertainty about the horizon area which determines the evolution.