Moving Mirrors, Black Holes, Hawking Radiation and All That

TL;DR

This paper presents a canonical quantization of a scalar field around a Schwarzschild black hole, deriving Hawking radiation through a non-static Hamiltonian approach and comparing it with moving mirror models to clarify approximations.

Contribution

It introduces a simplified, canonical quantization method in Lemaître coordinates for black hole backgrounds, emphasizing the time-dependent Hamiltonian and steady-state phenomena like Hawking radiation.

Findings

Derived Hawking radiation using a time-dependent Hamiltonian approach.

Highlighted the non-static nature of the quantization in black hole backgrounds.

Compared black hole radiation with moving mirror models to clarify approximations.

Abstract

In this talk I show how to canonically quantize a massless scalar field in the background of a Schwarzschild black hole in Lema\^itre coordinates and then present a simplified derivation of Hawking radiation based upon this procedure. The key result of quantization procedure is that the Hamiltonian of the system is explicitly time dependent and so problem is intrinsically non-static. From this it follows that, although a unitary time-development operator exists, it is not useful to talk about vacuum states; rather, one should focus attention on steady state phenomena such as the Hawking radiation. In order to clarify the approximations used to study this problem I begin by discussing the related problem of the massless scalar field theory calculated in the presence of a moving mirror.