A kinetic theory of diffusion in general relativity with cosmological scalar field

TL;DR

This paper develops a relativistic diffusion model using a Fokker-Planck equation on spacetime, explores its inconsistency with Einstein's equations, and investigates cosmological implications of a scalar field affecting universe expansion.

Contribution

It introduces a new relativistic diffusion model coupled with scalar fields and analyzes its cosmological consequences in flat Robertson-Walker spacetime.

Findings

Diffusion model is incompatible with Einstein equations without additional matter.

A scalar field can resolve divergence issues in the energy-momentum tensor.

Depending on initial conditions, the universe may expand forever or end in a finite-time singularity.

Abstract

A new model to describe the dynamics of particles undergoing diffusion in general relativity is proposed. The evolution of the particle system is described by a Fokker-Planck equation without friction on the tangent bundle of spacetime. It is shown that the energy-momentum tensor for this matter model is not divergence-free, which makes it inconsistent to couple the Fokker-Planck equation to the Einstein equations. This problem can be solved by postulating the existence of additional matter fields in spacetime or by modifying the Einstein equations. The case of a cosmological scalar field term added to the left hand side of the Einstein equations is studied in some details. For the simplest cosmological model, namely the flat Robertson-Walker spacetime, it is shown that, depending on the initial value of the cosmological scalar field, which can be identified with the present observed value of the cosmological constant, either unlimited expansion or the formation of a singularity in finite time will occur in the future. Future collapse into a singularity also takes place for a suitable small but positive present value of the cosmological constant, in contrast to the standard diffusion-free scenario.