On the Gravitational Back Reaction to Hawking Radiation

TL;DR

This paper demonstrates that adding a boundary term to the Einstein-Hilbert action is essential for accurately describing quantum transitions around black holes, linking black hole radiation to horizon area changes and the first law of black hole dynamics.

Contribution

It introduces the use of a boundary surface term in the action to compute gravitational corrections to Hawking radiation, establishing a direct connection to black hole thermodynamics.

Findings

Emission probability relates to horizon area change as e^{-ΔA/4}

Boundary term ensures a well-defined quantum action principle

Gravitational corrections modify Hawking radiation amplitudes

Abstract

We show that a surface term should be added to the Einstein-Hilbert action in order to properly describe quantum transitions occurring around a black hole. The introduction of this boundary term has been advocated by Teitelboim and collaborators and it has been used in the computation of the black hole entropy. Here, we use it to compute the gravitational corrections to the transition amplitudes giving rise to Hawking radiation. This surface term implies that the probability to emit a particle is given by $e^{- \Delta A/4}$ where $\Delta A$ is the change in the area of the black hole horizon induced by the emission. Its inclusion at the level of the amplitudes therefore relates quantum black hole radiation to the first law of black hole dynamics. In both cases indeed, the term expressing the change in area directly results from the same boundary term introduced for the same reason: to obtain a well defined action principle.