Covariant kinematics and gravitational bounce in Finsler space-times

TL;DR

This paper explores covariant kinematics and gravitational bounce phenomena in Finsler space-times, revealing how generalized geometry affects congruence dynamics and potentially prevents singularities.

Contribution

It extends the covariant kinematic analysis of congruences to Finsler geometry, deriving a modified Raychaudhuri equation with color-curvature coupling and demonstrating a non-singular bounce.

Findings

Finsler geometry introduces new geometric entities affecting congruence evolution.

A modified Raychaudhuri equation with color-curvature coupling is derived.

A non-singular bounce model is developed within Finsler space-time.

Abstract

The similarity between Finsler and Riemann geometry is an intriguing starting point to extend general relativity. The lack of quadratic restriction over the line element (color) naturally generalize the Riemannian case and breaks the local symmetries of general relativity. In addition, the Finsler manifold is enriched with new geometric entities and all the classical identities are suitably extended. We investigate the covariant kinematics of a medium formed by a time-like congruence. After a brief view in the general case we impose particular geometric restrictions to get some analytic insight. Central role to our analysis plays the Lie derivative where even in case of irrotational Killing vectors the bundle still deforms. We demonstrate an example of an isotropic and exponentially expanding cross-section that finally deflates or forms a caustic. Furthermore, using the 1+3 covariant formalism we investigate the expansion dynamics of the congruence. For certain geometric restrictions we retrieve the Raychaudhuri equation where a color-curvature coupling is revealed. The condition to prevent the focusing of neighboring particles is given and is more likely to fulfilled in highly curved regions. Then, we introduce the Levi-Civita connection for the osculating Riemannian metric and develop a (spatially) isotropic and homogeneous dust-like model with a non-singular bounce.