Abstract
We show that the Wald Noether-charge entropy is canonically conjugate to the opening angle at the horizon. Using this canonical relation, we extend the Wheeler–DeWitt equation to a Schrödinger equation in the opening angle, following Carlip and Teitelboim. We solve the equation in the semiclassical approximation by using the correspondence principle and find that the solutions are minimal uncertainty wavefunctions with a continuous spectrum for the entropy and therefore also of the area of the black hole horizon. The fact that the opening angle fluctuates away from its classical value of 2π indicates that the quantum black hole is a superposition of horizonless states. The classical geometry with a horizon serves only to evaluate quantum expectation values in the strict classical limit.
Highlights
Quantum black holes have attracted continued interest over decades since the discovery of their unique thermodynamics [1, 2]
The Wheeler-DeWitt (WDW) equation [6], the standard equation that is used to study the quantum mechanics of black hole (BH)’s, has been extended by Carlip and Teitelboim [7] to a Schrodinger-like equation for the BH. Their reasoning was based on the canonical structure of Einstein gravity, making the key observation that the BH horizon should be considered as a boundary of spacetime in addition to the boundary at infinity
We identify the relationship between the Lie derivative of the area of a D − 2 hypersurface embedded in a D dimensional spacetime and the extrinsic curvature of the surface
Summary
Quantum black holes have attracted continued interest over decades since the discovery of their unique thermodynamics [1, 2]. We first show that the opening angle of the horizon is canonically conjugate to the Wald Noether charge entropy.
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