Geodesic deviation, Raychaudhuri equation, and tidal forces in modified gravity with an arbitrary curvature-matter coupling

TL;DR

This paper explores how modified gravity theories with arbitrary curvature-matter coupling affect geodesic deviation, Raychaudhuri equations, and tidal forces, revealing new influences on tidal motion and Roche limit calculations.

Contribution

It derives the modified geodesic deviation and Raychaudhuri equations considering extra forces from curvature-matter coupling, and applies these to analyze tidal forces and Roche limit modifications.

Findings

Tidal forces are affected by the gradient of the extra force and velocity-curvature coupling.

Modified equations reveal new influences on tidal motion in alternative gravity models.

Explicit expression for Roche limit in these models is derived.

Abstract

The geodesic deviation equation, describing the relative accelerations of nearby particles, and the Raychaudhury equation, giving the evolution of the kinematical quantities associated with deformations (expansion, shear and rotation) are considered in the framework of modified theories of gravity with an arbitrary curvature-matter coupling, by taking into account the effects of the extra force. As a physical application of the geodesic deviation equation the modifications of the tidal forces due to the supplementary curvature-matter coupling are obtained in the weak field approximation. The tidal motion of test particles is directly influenced not only by the gradient of the extra force, which is basically determined by the gradient of the Ricci scalar, but also by an explicit coupling between the velocity and the Riemann curvature tensor. As a specific example, the expression of the Roche limit (the orbital distance at which a satellite will begin to be tidally torn apart by the body it is orbiting) is also obtained for this class of models.