Considerations on the Unruh Effect: Causality and Regularization

TL;DR

This paper addresses causality issues in Unruh effect calculations by deriving a time-dependent detector response that depends only on past trajectories, and proposes a new regularization method for physical consistency.

Contribution

It introduces a causally consistent response function for non-stationary detectors and proposes a novel regularization approach to improve physical accuracy.

Findings

Standard regularization leads to unphysical results.

A new regularization method aligns with causality.

Finite detector extension ensures consistent responses.

Abstract

This article is motivated by the observation, that calculations of the Unruh effect based on idealized particle detectors are usually made in a way that involves integrations along the {\em entire} detector trajectory up to the infinitely remote {\em future}. We derive an expression which allows time-dependence of the detector response in the case of a non-stationary trajectory and conforms more explicitely to the principle of causality, namely that the response at a given instant of time depends only on the detectors {\em past} movements. On trying to reproduce the thermal Unruh spectrum we are led to an unphysical result, which we trace down to the use of the standard regularization $t\to t-i\eps$ of the correlation function. By consistently employing a rigid detector of finite extension, we are led to a different regularization which works fine with our causal response function.