Notes on moving mirrors

TL;DR

This paper extends the Davies-Fulling model of a moving mirror by incorporating partial reflection and a dynamical self-interaction approach, enabling regularized flux calculations in scenarios like black hole evaporation and uniform acceleration.

Contribution

It introduces a frequency-dependent reflection model and a self-interaction based dynamical model, broadening the theoretical framework for analyzing moving mirror phenomena.

Findings

Derived Bogoliubov coefficients for partially reflecting mirrors.

Provided exact solutions for the dynamical self-interaction model.

Demonstrated regularized flux calculations in complex scenarios.

Abstract

The Davies-Fulling (DF) model describes the scattering of a massless field by a non-inertial mirror in two dimensions. In this paper, we generalize this model in two different ways. First, we consider partially reflecting mirrors. We show that the Bogoliubov coefficients relating inertial modes can be expressed in terms of the frequency dependent reflection factor which is specified in the rest frame of the mirror and the transformation from the inertial modes to the modes at rest with respect to the mirror. In this perspective, the DF model is simply the limiting case when this factor is unity for all frequencies. In the second part, we introduce an alternative model which is based on self-interactions described by an action principle. When the coupling is constant, this model can be solved exactly and gives rise to a partially reflecting mirror. The usefulness of this dynamical model lies in the possibility of switching off the coupling between the mirror and the field. This allows to obtain regularized expressions for the fluxes in situations where they are singular when using the DF model. Two examples are considered. The first concerns the flux induced by the disappearance of the reflection condition, a situation which bears some analogies with the end of the evaporation of a black hole. The second case concerns the flux emitted by a uniformly accelerated mirror.