The Kerr-Schild Double Copy in Lifshitz Spacetime

TL;DR

This paper extends the Kerr-Schild double copy framework to Lifshitz spacetimes, revealing how background curvature influences gauge theory sources and demonstrating solutions with curved backgrounds and matter couplings.

Contribution

It provides a general formulation of the Kerr-Schild double copy without background assumptions and explores its application to Lifshitz black holes with matter fields.

Findings

Curvature terms affect gauge theory sources in the double copy.

Lifshitz black holes exhibit constant charge densities influenced by background curvature.

Certain matter couplings can cancel non-localized source terms.

Abstract

The Kerr-Schild double copy is a map between exact solutions of general relativity and Maxwell's theory, where the nonlinear nature of general relativity is circumvented by considering solutions in the Kerr-Schild form. In this paper, we give a general formulation, where no simplifying assumption about the background metric is made, and show that the gauge theory source is affected by a curvature term that characterizes the deviation of the background spacetime from a constant curvature spacetime. We demonstrate this effect explicitly by studying gravitational solutions with non-zero cosmological constant. We show that, when the background is flat, the constant charge density filling all space in the gauge theory that has been observed in previous works is a consequence of this curvature term. As an example of a solution with a curved background, we study the Lifshitz black hole with two different matter couplings. The curvature of the background, i.e., the Lifshitz spacetime, again yields a constant charge density; however, unlike the previous examples, it is canceled by the contribution from the matter fields. For one of the matter couplings, there remains no additional non-localized source term, providing an example for a non-vacuum gravity solution corresponding to a vacuum gauge theory solution in arbitrary dimensions.