Lie and Noether symmetries of geodesic equations and collineations

TL;DR

This paper explores the Lie and Noether symmetries of geodesic equations in Riemannian spaces, linking them to geometric collineations and applying results to key spacetime models.

Contribution

It provides explicit calculations of symmetries and first integrals for geodesic equations in various important spacetimes, connecting symmetry groups to geometric properties.

Findings

Geodesic equations admit three linear and two quadratic first integrals.

Explicit symmetry calculations for Einstein, Schwarzschild, and FRW spacetimes.

Symmetries are expressed in terms of the metric's collineation vectors.

Abstract

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.