The Right Side of Einstein's Equation

Abstract: Recent developments in observational cosmology have led to attempts to make modifications on both sides of the Einstein equation to explain some of the puzzling new findings. What follows is an examination of the source of gravity that we usually find on the right hand side of Einstein's equation. The outcome is a modified version of the stress-energy tensor that is the source of the gravitational field. The derivation is based on the kinetic theory of a gas of identical particles with no internal structure. The presentation here is in two parts. In Part I, I describe the stress tensor that Xinzhong Chen and I have proposed for the matter tensor for a nonrelativistic gas with input from Hongling Rao and Jean-Luc Thiffeault. Our derivation of the equations of fluid dynamics is based on kinetic theory without recourse to the standard Chapman-Enskog approximation. In Part II, I present the analogous derivation of our form for the stress-energy tensor in the relativistic case. Then I exhibit its application to the usual isotropic cosmological model. The result of that, in addition to the Friedmann solution, is a second solution that arises from terms discarded in the usual Chapman-Enskog approximation. The new solution is a temporal analogue of a spatial shock wave. Just as the usual shock waves make transitions in properties within a mean free path, the new solution can change its properties appreciably in a mean flight time. Whereas the Friedmann solution is not dissipative, the new solution produces entropy at a rate that may be of cosmological interest. For the calculation of cosmic entropy production I use a formula derived in the ultrarelativistic limit in which particle masses are negligible. Independently of the cosmological aspects, the fluid dynamical equations that we derive are causal, even for the heat equation (or Fourier equation).