The Problem of Time in the Quantum Theory of Black Holes

TL;DR

This paper addresses the problem of defining time in quantum black hole models by solving the Wheeler-DeWitt equation and introduces a novel inner product and notion of time that aligns with classical metrics.

Contribution

It presents an exact solution to the Wheeler-DeWitt equation for spherically symmetric gravity and proposes a new approach to defining time in quantum gravity.

Findings

Defined a positive definite inner product based on momentum constraints.

Introduced a natural notion of time via the Heisenberg equation.

Reproduced the classical metric's time-time component in the quantum framework.

Abstract

We discuss the problem of time in spherically symmetric pure Einstein gravity with the cosmological term by using an exact solution to the Wheeler-DeWitt equation. A positive definite inner product is defined, based on the momentum constraint rather than the hamiltonian constraint. A natural notion of time is introduced via the Heisenberg equation. This notion enables one to reproduce the time-time component of the classical metric. Non-Hermiticity of the hamiltonian is essential in the definition of time.