Kerr-Schild Double Copy of the Coulomb Solution in Three Dimensions

TL;DR

This paper demonstrates that in three dimensions, the Kerr-Schild double copy of the Coulomb solution corresponds to a static Einstein-Maxwell black hole with a correct Newtonian limit, extending the double copy concept to lower dimensions.

Contribution

It shows that the static Einstein-Maxwell black hole in three dimensions is the double copy of the Coulomb solution, including cases with a negative cosmological constant, improving previous scalar field approaches.

Findings

The Einstein-Maxwell black hole in 3D has the correct Newtonian limit.

The charged BTZ black hole's single copy is a Coulomb solution with a linearly increasing electric field.

The approach extends the Kerr-Schild double copy to three-dimensional black holes.

Abstract

While the Kerr-Schild double copy of the Coulomb solution in dimensions higher than three is the Schwarzschild black hole, it is known that it should be a non-vacuum solution in three dimensions. We show that the static black hole solution of Einstein-Maxwell theory (with one ghost sign in the action) is the double copy with the correct Newtonian limit, which provides an improvement over the previous construction with a free scalar field that does not vanish at infinity. By considering a negative cosmological constant, we also study the charged Ba\~nados-Teitelboim-Zanelli black hole and find that the single copy gauge field is the Coulomb solution modified by a term which describes an electric field linearly increasing with the radial coordinate, which is the usual behaviour of the Schwarzschild-AdS black hole in higher dimensions when written around a flat background metric.