Quantum Field Theory of Black Hole Perturbations with Backreaction VI. Apparent Horizons, Quasi-Local Mass and Effective Classical Metrics

TL;DR

This paper develops a gauge-invariant, first-principle approach to black hole perturbations including backreaction, analyzing apparent horizons and deriving an effective classical metric to understand quantum effects like Hawking evaporation.

Contribution

It introduces a second-order, gauge-invariant framework for black hole perturbations with backreaction, and reconstructs an effective classical metric incorporating quantum corrections.

Findings

Explicit expression for apparent horizon shape to second order.

Effective classical metric derived from quantum expectation values.

Insights into horizon area decrease due to Hawking evaporation.

Abstract

In a recent series of papers we developed a first-principle and gauge invariant approach to black hole perturbation theory valid to any order. We included back reaction effects to tackle the situation of evaporating black holes and obtained an explicit expression for the dynamics of the reduced phase space to second order. The physics of evaporating black holes is in particular encoded by apparent horizons, an observer dependent generalisation of the event horizon. We determine the shape of the apparent horizon to second order in the perturbations. The area of the apparent horizon is an interesting observable which is expected to decrease in the quantum theory due to Hawking evaporation. We show how the full four dimensional metric can be reconstructed in terms of the reduced phase space variables. In the quantum theory, taking expectation values of this metric, we obtain an effective classical metric, whose causal structure can then be visualised in a quantum corrected Penrose diagram. We conclude with an outlook into the quantisation procedure in the reduced phase space formalism and the implications on the area of the apparent horizon.