The Raychaudhuri equation in spacetimes with torsion and non-metricity

TL;DR

This paper extends the Raychaudhuri equation to spacetimes with torsion and non-metricity, providing the most general form and exploring implications for astrophysics and cosmology.

Contribution

It develops a comprehensive 1+3 covariant framework for non-Riemannian spacetimes, deriving the general Raychaudhuri equation with applications to various geometric effects.

Findings

Most general expression of the Raychaudhuri equation for non-Riemannian spacetimes.

Identification of effects due to non-metricity on timelike congruences.

Recovery of torsion results from non-metricity analogues in symmetric spaces.

Abstract

We introduce and develop the 1+3 covariant approach to relativity and cosmology to spacetimes of arbitrary dimensions that have nonzero torsion and do not satisfy the metricity condition. Focusing on timelike observers, we identify and discuss the main differences between their kinematics and those of their counterparts living in standard Riemannian spacetimes. At the centre of our analysis lies the Raychaudhuri equation, which is the fundamental formula monitoring the convergence/divergence, namely the collapse/expansion, of timelike worldline congruences. To the best of our knowledge, we provide the most general expression so far of the Raychaudhuri equation, with applications to an extensive range of non-standard astrophysical and cosmological studies. Assuming that metricity holds, but allowing for nonzero torsion, we recover the results of analogous previous treatments. Focusing on non-metricity alone, we identify a host of effects that depend on the nature of the timelike congruence and on the type of the adopted non-metricity. We also demonstrate that in spaces of high symmetry one can recover the pure-torsion results from their pure non-metricity analogues, and vice-versa, via a simple ansatz between torsion and non-metricity.