Late-time expansion in the semiclassical theory of the Hawking radiation

TL;DR

This paper analyzes the late-time back-reaction effects on Hawking radiation spectrum using semiclassical methods, revealing caustic phenomena and providing corrections to the standard Hawking spectrum.

Contribution

It introduces a detailed semiclassical framework for late-time back-reaction effects, including caustic analysis and an iterative solution for spectrum corrections.

Findings

Caustics generally occur in the mode boundary value problem.

No caustics occur for radii below a critical value $r_c$.

First-order correction to Hawking spectrum is proportional to $rac{ ext{frequency}}{ ext{mass}}$.

Abstract

We give a detailed treatment of the back-reaction effects on the Hawking spectrum in the late-time expansion within the semiclassical approach to the Hawking radiation. We find that the boundary value problem defining the action of the modes which are regular at the horizon admits in general the presence of caustics. We show that for radii less that a certain critical value $r_c$ no caustic occurs for all values of the wave number and time and we give a rigorous lower bound on such a critical value. We solve the exact system of non linear equations defining the motion, by an iterative procedure rigorously convergent at late times. The first two terms of such an expansion give the $O(\omega/M)$ correction to the Hawking spectrum.