Relativistic Momentum and Manifestly Covariant Equipartition Theorem Revisited

Recently the discussion about the right relativistic generalization of thermodynamics has been revived. In particular the case of temperature has been investigated by alluding to a form of relativistic equipartition theorem. Now from the kinetic theory point of view a covariant equipartition involves necessarily the relativistic momentum of the system, which is given by an integral of the energy-momentum tensor over a spacelike hypersurface. Some authors have even proposed to trade the spacelike hypersurfaces entering in there by lightlike ones to accommodate Lorentz covariance. In this work we argue that a well defined momentum for a diluted gas can be given by making use of the velocity of the gas as whole and thereby selecting a hypersurface; this being in direct analogy with the case of an extended classical electron model and which turned out to solve the Abraham-Lorentz controversy codified in the wrong non-relativistic limit. We also discuss the effect of such choices on the equipartition theorem calculated through the covariant form of the Juettner distribution function.