A new exact rotating spacetime in vacuum: The Kerr--Levi-Civita Spacetime

TL;DR

This paper introduces a new exact vacuum solution in Einstein's theory, extending the Kerr metric with Levi-Civita features, which is free of singularities and exhibits unique geometric properties.

Contribution

It constructs a novel rotating vacuum spacetime by applying symmetry transformations to the Kerr metric, resulting in a regular, algebraically general solution with distinctive ergoregion behavior.

Findings

The spacetime is free of curvature singularities and closed timelike curves.

It has horizons at Kerr black hole radii and unique ergoregion structures.

The solution does not admit a Kerr-Schild form.

Abstract

We construct a new rotating solution of Einstein's theory in vacuum by exploiting the Lie point symmetries of the field equations in the complex potential formalism of Ernst. In particular, we perform a discrete symmetry transformation, known as inversion, of the gravitational potential associated with the Kerr metric. The resulting metric describes a rotating generalization of the Schwarzschild--Levi-Civita spacetime, and we refer to it as the Kerr--Levi-Civita metric. We study the key geometric features of this novel spacetime, which turns out to be free of curvature singularities, topological defects, and closed timelike curves. These attractive properties are also common to the extremal black hole and the super-spinning case. The solution is algebraically general (Petrov-type I), and its horizons lie at the horizon radii of the Kerr black hole. The ergoregions, however, are strongly influenced by the Levi-Civita-like asymptotic structure, producing an effect akin to the magnetized Kerr--Newman and swirling solutions. Interestingly, while its static counterpart permits a Kerr--Schild representation, the Kerr--Levi-Civita metric does not admit such a formulation.