Symmetry analysis of the charged squashed Kaluza-Klein black hole metric

TL;DR

This paper conducts a comprehensive symmetry and conservation law analysis of the charged squashed Kaluza-Klein black hole spacetime, identifying Lie symmetries, optimal systems, and Noether symmetries to understand its geometric and physical properties.

Contribution

It provides a complete group analysis of the spacetime metric, computes optimal systems for various dimensions, and derives associated conservation laws using Noether's theorem.

Findings

No n-dimensional optimal system for geodesic systems with n≥5.

Identified point symmetries and Noether symmetries of the spacetime.

Derived conservation laws for geodesic equations.

Abstract

In this paper, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza--Klein black hole spacetime in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space-time metric using Lie point symmetries are presented and then it the $n$-dimensional optimal system of this space-time metric, for $n=1,\ldots,4$, are computed. It is shown that there is not any $n$-dimensional optimal system of Lie symmetry subalgebra associated to the system of geodesic for $n\geq5$. Then the point symmetries of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found and then the conservation laws associated to the system of geodesic equations are calculated via Noether's theorem.