Relativistic diffusing particles, entropy and varying cosmological constants

Abstract

We discuss Einstein equations with a non-conserved energy-momentum $T^{\mu \nu }$. The energy-momentum tensor ${\tilde{T}}^{\mu \nu }$ on the right hand side of Einstein equations is expressed in the form $8\pi G{T}^{\mu \nu }-g^{\mu \nu }\Lambda +{\Sigma }^{\mu \nu }$ where $G$ is the Newton constant, $\Lambda$ is the cosmological term and $T^{\mu \nu }$ is the energy-momentum of an ideal fluid. The remaining part ${\Sigma }^{\mu \nu }$ describes the viscosity and a heat flow. We discuss space-time-dependent Newton ``constant" $G$ and the cosmological term $\Lambda$. The covariant conservation law for ${\tilde{T}}^{\mu \nu }$ imposes a relation between the energy-momentum (non)conservation and the variation of $G$ and $\Lambda$. The covariant derivative ${\nabla }_{\mu }{T}^{\mu \nu }$ 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 $P_t=w_t{\rho }_t$ with $0<w_t\le \frac{1}{3}$. 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.

arXiv comment29 pages