On K\"ahler Structures of Taub-NUT and Kerr Spaces

TL;DR

This paper investigates the K"ahler properties of Taub-NUT and Kerr spaces, revealing that Taub-NUT is hyper-K"ahler and Kerr is conformally K"ahler, with implications for understanding their geometric structures in general relativity.

Contribution

It demonstrates that Euclidean Taub-NUT is hyper-K"ahler and Euclidean Kerr is conformally K"ahler, providing explicit constructions and conditions for these structures.

Findings

Taub-NUT space is hyper-K"ahler with explicit coframe

Kerr space is globally conformally K"ahler

Conformal scaling admits K"ahler structures via Lee-form or Weyl tensor

Abstract

In this paper, we study the K\"ahlerian nature of Taub-NUT and Kerr spaces which are gravitational instanton and black hole solutions in general relativity. We show that Euclidean Taub-NUT metric is hyper-K\"ahler with respect to the usual almost complex structures by employing an alternative explicit coframe, and Euclidean Kerr metric is globally conformally K\"ahler. We also show that conformally scaled Euclidean Kerr space admits a K\"ahler structure by applying a conformal scaling factor stemming from the Lee-form of the original metric or alternatively a factor coming from self-dual part of the Weyl tensor $(W^+)$.