Time parameterization and stationary distributions in a relativistic gas

TL;DR

This paper investigates how different time parameterizations affect the stationary velocity distribution in a relativistic gas, clarifying the distinction between Jüttner and modified Jüttner distributions and showing proper-time averaging does not alter the distribution.

Contribution

The study demonstrates that proper-time averaging results in the standard Jüttner distribution multiplied by a Lorentz factor, clarifying misconceptions about the modified Jüttner distribution.

Findings

Proper-time averaging introduces a Lorentz factor to the Jüttner distribution.

The modified Jüttner distribution is not obtained through proper-time averaging.

Results are consistent in both rest and moving frames.

Abstract

In this paper we consider the effect of different time parameterizations on the stationary velocity distribution function for a relativistic gas. We clarify the distinction between two such distributions, namely the J\"{u}ttner and the modified J\"{u}ttner distributions. Using a recently proposed model of a relativistic gas, we show that the obtained results for the proper-time averaging does not lead to modified J\"{u}ttner distribution (as recently conjectured), but introduces only a Lorentz factor $\gamma$ to the well-known J\"{u}ttner function which results from observer-time averaging. We obtain results for rest frame as well as moving frame in order to support our claim.