Stefan-Boltzmann law for massive photons

TL;DR

This paper extends the Stefan-Boltzmann law to massive photons, showing that their radiance depends on their velocity, not the speed of light, which alters the traditional understanding of blackbody radiation.

Contribution

It introduces a generalized Stefan-Boltzmann law accounting for photon mass and their variable speed, providing a more accurate description of radiance for massive photons.

Findings

Radiance is not proportional to energy density times the speed of light.

Massive photons travel at speeds less than light, affecting radiance calculations.

The generalized law modifies classical blackbody radiation formulas.

Abstract

This paper generalizes the Stefan-Boltzmann law to include massive photons. A crucial ingredient to obtain the correct formula for the radiance is to realize that a massive photon does not travel at the speed of (massless) light. It follows that, contrary to what could be expected, the radiance is not proportional to the energy density times the speed of light.