Lorentz Transformation of the Energy Spectrum of the Equilibrium State of Massive Free Fields

TL;DR

This paper derives how the energy spectrum of massive quantum fields transforms under Lorentz boosts, demonstrating the need for a four-vector temperature in relativistic thermodynamics beyond massless cases.

Contribution

It provides the Lorentz transformation of the energy spectral density for massive fields, extending the concept of temperature to a four-vector form in relativistic equilibrium.

Findings

Massless limit recovers black body radiation transformation

Massive fields require four-vector temperature description

Supports the non-scalar nature of temperature in relativistic systems

Abstract

In previous studies of relativistic thermodynamics, the temperature of a static system, as perceived by a moving observer, has traditionally been treated as a scalar. This assumption has also been extended to the research on the cosmic microwave background. However, the validity of this assumption is a consequence of the massless nature of photons. More generally, when an observer is in relative motion to a system, the thermal equilibrium state is characterized by a four-vector temperature. In this paper, we study the non-interacting massive Bosonic and Fermionic field systems. We derive the Lorentz transformation of the energy spectral density in the equilibrium state of these fields. In the massless limit for bosonic field, our results recover the transformation of black body radiation [G. W. Ford and R. F. O' Connell., Phys. Rev. E, 88, 044101(2013)], which corresponds to a scalar temperature with dipole anisotropy. For the massive fields, the moving equilibrium state cannot be characterized by a corresponding scalar temperature. This result shows the necessity of introducing four-vector temperature in relativistic thermodynamics.