Time parametrization and stationary distributions in a relativistic gas

In this paper we consider the effect of different time parametrizations on the stationary velocity distribution function for a relativistic gas. We clarify the distinction between two such distributions, namely, the Jüttner and the Modified Jüttner 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 the Modified Jüttner distribution (as recently conjectured), but introduces a Lorentz factor γ(v) (i.e., energy) to the well-known Jüttner function which results from observer-time averaging. These two modifications (i.e., Modified Jüttner function or Jüttner divided by energy) are identical in the rest frame; however, their distinction comes to light when one considers a moving frame. We obtain results for rest frame as well as moving frame in order to support our claim. We finally conclude that reparametrizing time simply rescales stationary (Jüttner) distribution function according to a well-defined frame-independent relation.