Kinematics in Metric-Affine Geometry

TL;DR

This paper extends the kinematic analysis of congruences to metric-affine geometry, deriving generalized evolution equations that include torsion and non-metricity effects without relying on a specific gravity theory.

Contribution

It provides the first derivation of generalized Raychaudhuri and Sachs equations in metric-affine geometry, incorporating torsion and non-metricity effects.

Findings

Torsion and non-metricity influence relative acceleration between curves.

Rotation of hypersurface orthogonal congruences is purely non-Riemannian.

Distances depend on the choice of photon trajectories in the presence of non-Riemannian features.

Abstract

In a given geometry, the kinematics of a congruence of curves is described by a set of three quantities called expansion, rotation, and shear. The equations governing the evolution of these quantities are referred to as kinematic equations. In this paper, the kinematics of congruence of curves in a metric-affine geometry are analysed. Without assuming an underlying theory of gravity, we derive a generalised form of the evolution equations for expansion, namely, Raychaudhuri equation (timelike congruences) and Sachs optical equation (null congruences). The evolution equations for rotation and shear of both timelike and null congruences are also derived. Generalising the deviation equation, we find that torsion and non-metricity contribute to a relative acceleration between neighbouring curves. We briefly discuss the interpretation of the expansion scalars and derive an equation governing angular diameter distances. The effects of torsion and non-metricity on the distances are found to be dependent on which curves are chosen as photon trajectories. We also show that the rotation of a hypersurface orthogonal congruence (timelike or null) is a purely non-Riemannian feature.