Integrability of geodesics of totally geodesic metrics

TL;DR

This paper investigates the integrability of geodesic equations in specific Lorentzian metrics related to Einstein and Einstein-Maxwell equations, identifying conditions under which these equations are solvable.

Contribution

It provides a detailed analysis of geodesic integrability in metrics derived from Weyl tensor eigenspaces, extending the study to Einstein-Maxwell coupled models.

Findings

Identified cases of integrable geodesic equations.

Analyzed the impact of electromagnetic coupling on geodesic integrability.

Extended previous models to include electromagnetic fields.

Abstract

Analysis of the geodesics in the space of signature $(1,3)$ that splits in two-dimensional distributions resulting from the Weyl tensor eignespaces - hyperbolic and elliptic ones - described in [V. Lychagin, V. Yumaguzhin, \emph{Differential invariants and exact solutions of the Einstein equations}, Anal.Math.Phys. 1664-235X 1-9 (2016)] are presented. Cases when geodesic equations are integrable are identified. Similar analysis is performed for the same model coupled to Electromagnetism described in [V. Lychagin, V. Yumaguzhi, \emph{Differential invariants and exact solutions of the Einstein-Maxwell equation}, Anal.Math.Phys. 1, 19--29, (2017)].