Affine Symmetry, Geodesics, and Homogeneous Spacetimes

TL;DR

This paper clarifies how affine symmetries relate to conservation laws in geodesic equations, providing detailed computations for homogeneous solutions in various Einstein field scenarios.

Contribution

It demonstrates that affine symmetries of Lorentzian manifolds lead to conservation laws via Noether's theorem, correcting previous claims and applying this to homogeneous Einstein solutions.

Findings

Derived all proper affine symmetries for homogeneous spacetimes.

Computed associated geodesic conservation laws for various energy-momentum contents.

Clarified the relationship between affine symmetries and conservation laws in Lorentzian manifolds.

Abstract

We show that the conservation laws for the geodesic equation which are associated to affine symmetries can be obtained from symmetries of the Lagrangian for affinely parametrized geodesics according to Noether's theorem, in contrast to claims found in the literature. In particular, using Aminova's classification of affine motions of Lorentzian manifolds, we show in detail how affine motions define generalized symmetries of the geodesic Lagrangian. We compute all infinitesimal proper affine symmetries and the corresponding geodesic conservation laws for all homogeneous solutions to the Einstein field equations in four spacetime dimensions with each of the following energy-momentum contents: vacuum, cosmological constant, perfect fluid, pure radiation, and homogeneous electromagnetic fields.