Approximate Noether Symmetries of the Geodesic Equations for the Charged-Kerr Spacetime and Rescaling of Energy

TL;DR

This paper investigates the symmetries of geodesic equations in charged-Kerr spacetime using approximate symmetry methods, revealing that energy rescaling depends on the radial coordinate and comparing it with Reissner-Nordström spacetime.

Contribution

It introduces a novel application of approximate symmetry methods to charged-Kerr spacetime and identifies energy rescaling factors related to the spacetime's parameters.

Findings

No non-trivial approximate symmetries found for Kerr spacetime.

Energy in charged-Kerr spacetime requires r-dependent rescaling.

Rescaling factor compared with Reissner-Nordström metric.

Abstract

Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no non-trivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is $r$-dependent. This rescaling factor is compared with that for the Reissner-Nordstr\"{o}m metric.