Taub-NUT Instanton as the Self-dual Analog of Kerr

TL;DR

This paper confirms that the self-dual Kerr-Schild spacetime is equivalent to the self-dual Taub-NUT instanton by using the Cartan-Karlhede algorithm, revealing their invariance and simplifying the analysis with complex conjugate null tetrads.

Contribution

It demonstrates the equivalence of these two metrics through invariants and introduces the first application of the Cartan-Karlhede algorithm with complex conjugate null tetrads.

Findings

Confirmed the conjecture of metric equivalence.

Applied the Cartan-Karlhede algorithm to complex null tetrads.

Provided the first example of this algorithm with such tetrads.

Abstract

It was recently conjectured that a certain vacuum Kerr-Schild spacetime, which may be regarded as a self-dual analog of the Kerr metric, is equivalent to the self-dual Taub-NUT instanton. We confirm this conjecture by applying the Cartan-Karlhede algorithm to each metric and showing that for suitable choices of null tetrad, the algorithm leads to the same invariants and linear isotropy groups for both, establishing their equivalence. While it is well-known that the Taub-NUT solution and its self-dual version admit a double Kerr-Schild form, the observation that the self-dual Taub-NUT instanton admits a single Kerr-Schild form has only been made very recently. The two metrics we compare may be regarded as either complex metrics with Lorentzian (1,3) signature or real metrics with Kleinian (2,2) signature; here we take the latter view. Significant simplifications occur when the null tetrads are chosen to consist of two pairs of complex conjugate null vectors rather than four real independent ones. As a bonus, our work provides the first example of applying the Cartan-Karlhede algorithm using a null tetrad of this type.