Non-singular extension of the Kerr-NUT-(anti) de Sitter spacetimes

TL;DR

This paper develops a method to extend Kerr-NUT-(anti) de Sitter spacetimes into non-singular solutions, addressing a long-standing issue with conical singularities and providing a comprehensive global geometric analysis.

Contribution

It introduces a novel extension technique for Kerr-NUT-(anti) de Sitter spacetimes that removes singularities and broadens the class of non-singular solutions within Einstein's equations.

Findings

Derived non-singular extensions for all admissible parameters.

Analyzed the global structure and conformal geometry of the extended spacetimes.

Presented conditions for the existence of non-singular Killing horizons.

Abstract

In 1963 Ezra Ted Newman and his two students Louis A. Tamburino, and Theodore W. J. Unti, proposed a deformation of the Shwarzschild spacetime that made it twisting. In the cosmological context, an equivalent solution had been found earlier, in 1951, by Abraham Haskel Taub. The problem that these solutions have is a conical singularity along the symmetry axis at all distances from the origin. In 1969 Misner proposed a non-singular interpretation of Taub-NUT spacetimes. We extend and refine his method to include a broader family of solutions and completely solve the outstanding issue of a non-singular extension of the Kerr-NUT- (anti) de Sitter solutions to Einstein's equations. Our approach relies on an observation that in 2 dimensional algebra of Killing vector fields there exist 2 distinguished vector fields that may be used to define $U(1)$-principal bundle structure over the non-singular spaces of non-null orbits. For all admissible parameters we derive appropriate Killing vector fields and discuss limits to spacetimes with less parameters. The global structure of spacetime, together with non-singular conformal geometry of the infinities is presented and (possibly also projectively non-singular) Killing horizons is presented.