Role of symmetries in the Kerr-Schild derivation of the Kerr black hole

TL;DR

This paper demonstrates a simplified, rigorous derivation of the Kerr black hole solution by incorporating symmetry properties from the outset, leading to clearer insights into the solution's geometric and physical features.

Contribution

It introduces a symmetry-based approach to derive the Kerr solution using a stationary and axisymmetric Kerr-Schild ansatz, simplifying the integration of Einstein's equations.

Findings

Parameter $a$ linked to conserved angular momentum

Derivation reduces to a simple radial ODE

Explicit connection between symmetries and solution properties

Abstract

In this work we explore the consequences of considering from the very beginning the stationary and axisymmetric properties of the Kerr black hole as one attempts to derive this solution through the Kerr-Schild ansatz. The first consequence is kinematical and is based on a new stationary and axisymmetric version of the Kerr theorem that yields to the precise shear-free and geodesic null congruence of flat spacetime characterizing the Kerr solution. A straightforward advantage of this strategy is that now the parameter $a$ appears naturally as associated to the conserved angular momentum of the geodesics due to axisymmetry. The second consequence is dynamical and takes into account the circularity theorem. In fact, a stationary-axisymmetric Kerr-Schild ansatz is in general incompatible with the circularity property warranted by vacuum Einstein equations unless the remaining angular dependence in the Kerr-Schild profile appears fixed in a precise way. Thanks to these two ingredients, the integration of the Einstein equations reduces to a simple ordinary differential equation on the radial dependence, whose integration constant is precisely the mass $m$. This derivation of the Kerr solution is simple but rigorous, and it may be suitable for any textbook.