Causality constraints on radiative transfer

TL;DR

This paper addresses causality issues in radiative transfer equations by incorporating finite photon speed, proving their stability and causality, and providing an analytical solution that respects physical constraints.

Contribution

It corrects the instability in Spiegel's formula by including transit time effects, ensuring the transport coefficients stay within the causality-preserving hydrohedron.

Findings

Radiative transfer equations are proven to be causal and stable.

Analytical solutions accounting for photon transit time are derived.

Transport coefficients remain within the causality-preserving hydrohedron.

Abstract

The standard formula, due to Spiegel, for the smoothing of temperature fluctuations by radiative transfer is unstable in relativity. This is due to the fact that Spiegel neglected the transit time of light, thereby allowing the transport coefficients to move outside the convex geometry compatible with causality (the "hydrohedron"). Here, we fix this pathology. First, we prove that the linearized radiative transfer equations are causal and covariantly stable by construction. Then, we repeat Spiegel's calculation accounting for the finite speed of photons. We find that the full transfer problem can be solved analytically. All the infinite (exact) transport coefficients arising from it fall inside the hydrohedron. Our analysis also accounts for isotropic scattering.