Duality in Einstein's Gravity

TL;DR

This paper reveals an $SO(2)$ duality symmetry in vacuum Einstein gravity that transforms solutions like Kerr into Kerr-Taub-NUT, paralleling electric-magnetic duality in gauge theories.

Contribution

It introduces a novel duality symmetry in Einstein's vacuum equations and demonstrates its action on classical solutions such as Kerr and Kerr-Taub-NUT.

Findings

Vacuum Einstein equations are invariant under $SO(2)$ duality.

Duality maps Kerr solution to Kerr-Taub-NUT.

The duality operation is analogous to electric-magnetic duality.

Abstract

We show that the Einstein equations in the vacuum are invariant under an $SO(2)$ duality symmetry which rotates the curvature 2-form into its tangent space Hodge dual. Akin to electric-magnetic duality in gauge theory, the duality operation maps classical solutions into each other. As an example, we demonstrate that the Kerr solution is non-linearly mapped by duality into Kerr-Taub-NUT.