Towards the Raychaudhuri Equation Beyond General Relativity

TL;DR

This paper explores how modifications to General Relativity from theories like string theory, braneworld models, $f(R)$ gravity, and Loop Quantum Cosmology alter the Raychaudhuri equation, potentially weakening geodesic convergence and impacting singularity theorems.

Contribution

It computes leading corrections to the Raychaudhuri equation in various alternative gravity theories, revealing conditions that can weaken geodesic convergence.

Findings

Most models reinforce geodesic convergence

Some models introduce repulsive terms

Implications for singularity theorems

Abstract

In General Relativity, gravity is universally attractive, a feature embodied by the Raychaudhuri equation which requires that the expansion of a congruence of geodesics is always non-increasing, as long as matter obeys the strong or weak energy conditions. This behavior of geodesics is an important ingredient in general proofs of singularity theorems, which show that many spacetimes are singular in the sense of being geodesically incomplete and suggest that General Relativity is itself incomplete. It is possible that alternative theories of gravity, which reduce to General Relativity in some limit, can resolve these singularities, so it is of interest to consider how the behavior of geodesics is modified in these frameworks. We compute the leading corrections to the Raychaudhuri equation for the expansion due to models in string theory, braneworld gravity, $f(R)$ theories, and Loop Quantum Cosmology, for cosmological and black hole backgrounds, and show that while in most cases geodesic convergence is reinforced, in a few cases terms representing repulsion arise, weakening geodesic convergence and thereby the conclusions of the singularity theorems.