Relativistic diffusing particles, entropy and varying cosmological constants

Abstract

We discuss Einstein equations with a non-conserved energy-momentum $T^{μν}$. The energy-momentum tensor $\tilde{T}^{μν}$ on the right hand side of Einstein equations is expressed in the form $8πGT^{μν}-g^{μν}Λ+Σ^{μν}$ where $G$ is the Newton constant, $Λ$ is the cosmological term and $T^{μν}$ is the energy-momentum of an ideal fluid. The remaining part $Σ^{μν}$ describes the viscosity and a heat flow. We discuss space-time-dependent Newton ``constant&#34; $G$ and the cosmological term $Λ$. The covariant conservation law for $\tilde{T}^{μν}$ imposes a relation between the energy-momentum (non)conservation and the variation of $G$ and $Λ$. The covariant derivative $\nabla_μT^{μν}$ can be related to the first law of thermodynamics and expressed by the temperature $T$ and the entropy $S$. The energy-momentum non-conservation in our models is a consequence of a diffusion of its constituting particles. We describe a gas of diffusing massive particles in a homogeneous isotropic space-time as a system with a time-dependent linear barotropic equation of state $P_{t}=w_{t}ρ_{t}$ with $0<w_{t}\leq \frac{1}{3}$. The gas of massive diffusing particles in an expanding universe tends in a limit of infinite time to the gas of diffusing massless particles. We obtain results on the energy density, entropy and phase space dstribution in a soluble model of massless relativistic particles in a spatially flat expanding universe.