Kerr-Schild double copy of the Coulomb solution in three dimensions

Gökhan Alkaç,Mehmet Kemal Gümüş,Mehmet Ali Olpak

doi:10.1103/physrevd.104.044034

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.

Highlights

After its discovery in the context of scattering amplitudes [1,2], the double copy idea has found applications in the classical domain as perturbative and exact relations between gravity and gauge theories
We show that the static black hole solution of Einstein-Maxwell theory 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 study the charged Bañados-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 behavior of the Schwarzschild-AdS black hole in higher dimensions when written around a flat background metric

Summary

INTRODUCTION

After its discovery in the context of scattering amplitudes [1,2], the double copy idea has found applications in the classical domain as perturbative and exact relations between gravity and gauge theories. For metrics of the KS form [62], the Ricci tensor with mixed indices is linear in the perturbation for a flat [63] and curved [64,65,66] background metric Due to this crucial property, a map to solutions of Maxwell’s theory defined on the background spacetime, which has a linear nature, can be established. The ghost sign can be removed by a certain generalized gauge transformation; it is still somewhat unsatisfactory to not have a reasonable behavior of the matter field at infinity. When the cosmological constant is zero, we will show that a charged black hole solution with the correct Newtonian limit, which gives rise to the Coulomb solution as its single copy, can be obtained when a ghost sign is used in the action.

Solution with the correct Newtonian potential

Gauge theory single copy

SCALAR-VECTOR DUALITY AND ITS CONSEQUENCES

The most general static solution of Kerr-Schild form

CONCLUSIONS