Cartesian Kerr-Schild variation on the Newman-Janis ansatz

TL;DR

This paper introduces a simplified two-step variation of the Newman-Janis trick that directly uses the Kerr-Schild Cartesian metric, aiming to make the procedure easier to understand and work with.

Contribution

It presents a novel, streamlined two-step version of the Newman-Janis trick utilizing Kerr-Schild Cartesian coordinates, simplifying the original four-step process.

Findings

The two-step method reproduces the Kerr spacetime from Schwarzschild.

The approach simplifies the original Newman-Janis procedure.

It leverages the interplay between oblate spheroidal and Cartesian coordinates.

Abstract

The Newman-Janis trick is a procedure, (not even really an ansatz), for obtaining the Kerr spacetime from the Schwarzschild spacetime. This 50 year old trick continues to generate heated discussion and debate even to this day. Most of the debate focusses on whether the Newman-Janis procedure can be upgraded to the status of an algorithm, or even an inspired ansatz, or is it just a random trick of no deep physical significance. (That the Newman-Janis procedure very quickly led to the discovery of the Kerr-Newman spacetime is a point very much in its favour.) In the current article we will not answer these deeper questions, we shall instead present a much simpler alternative variation on the theme of the Newman--Janis trick that might be easier to work with. We shall present a 2-step version of the Newman-Janis trick that works directly with the Kerr-Schild "Cartesian" metric presentation of the Kerr spacetime. That is, we show how the original 4-step Newman-Janis procedure can, (using the interplay between oblate spheroidal and Cartesian coordinates), be reduced to a considerably cleaner 2-step process.