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.
Highlights
Copy structure when the background metric is curved, called Type-A and Type-B double copies
The principal result of our analysis is that the deviation of the background metric from a constant curvature spacetime, which is characterized by the deviation tensor defined in (2.13), affects nontrivially the gauge theory source as described in (2.17) for an arbitrary Killing vector and in (2.28) for the time-like Killing vector
We studied metrics that can be written in the KS form around a maximally symmetric background and presented the differences that arise due to the deviation if the Minkowski spacetime from a constant-curvature spacetime, which are a constant charge density in the source and correspondingly, electric and magnetic fields that linearly increase with the radial coordinate r
Summary
We give a general formulation of the KS double copy in curved spacetime. The principal result of our analysis is that the deviation of the background metric from a constant curvature spacetime, which is characterized by the deviation tensor defined in (2.13), affects nontrivially the gauge theory source as described in (2.17) for an arbitrary Killing vector and in (2.28) for the time-like Killing vector. This has been observed as a constant charge distribution filling all space when the background is taken to be flat. We will review some previously studied examples through our general formalism
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