Instantaneous Temperatures \`a la Hadamard: Towards a generalized Stefan-Boltzmann law for curved spacetime

TL;DR

This paper introduces a method to define an instantaneous temperature for a scalar field in curved spacetime, generalizing the Stefan-Boltzmann law and incorporating effects of acceleration, curvature, and quantum field stress-energy.

Contribution

It develops a novel technique using Hadamard renormalization and derivative couplings to define local temperatures in curved spacetime, extending the Unruh effect.

Findings

Temperature includes local acceleration and curvature contributions

Method reproduces known results in Rindler, Minkowski, and de Sitter spacetimes

Generalizes Stefan-Boltzmann law to curved spacetime contexts

Abstract

In the celebrated Unruh effect, we learn that a uniformly accelerating detector in a Minkowski vacuum spacetime registers a constant temperature. Building on prior work, we present a technique based on derivative couplings of the two-point Wightman function and the Hadamard renormalization procedure to define an instantaneous temperature for a massive scalar field, non-minimally coupled to gravity. We find the temperature contains local contributions from the acceleration of the detector, the curvature of spacetime, and the renormalized stress-energy tensor of the field. Our result, which can be considered as a generalized Stefan-Boltzmann law for curved spacetimes, agrees with the familiar expressions found in 4D Rindler, thermal Minkowski, and de Sitter.