The classical double copy in maximally symmetric spacetimes

Abstract

The classical double copy procedure relates classical asymptotically-flat gravitational field solutions to Yang-Mills and scalar field solutions living in Minkowski space. In this paper we extend this correspondence to maximally symmetric curved spacetimes. We consider asymptotically (A)dS spacetimes in Kerr-Schild form and construct the corresponding single and zeroth copies. In order to clarify the interpretation of these copies, we study several examples including (A)dS-Schwarzschild, (A)dS-Kerr, black strings, black branes, and waves, paying particular attention to the source terms. We find that the single and zeroth copies of stationary solutions satisfy different equations than those of wave solutions. We also consider how to obtain Einstein-Maxwell solutions using this procedure. Finally, we derive the classical single and zeroth copy of the BTZ black hole.

Kerr-Schild double copy and complex worldlines

“…Through the double copy relationship, in which gravity amplitudes are closely tied to the squares of gauge theory amplitudes, it has become possible to compute gravity amplitudes which would otherwise be prohibitively complicated [1][2][3]. A natural question is whether a similar relationship holds between exact solutions of gauge and gravity theories, in which solutions to general relativity can be generated from solutions to a gauge theory.Indeed, using the Kerr-Schild transformation, a simple and precise relationship can be drawn between gauge fields and spacetime metrics [4][5][6][7][8][9][10][11]. It is striking that many exact solutions to the Einstein equations can be presented in Kerr-Schild form as the double copies of gauge theory solutions, including all four-dimensional black hole spacetimes.…”

“…Double copy presentations have been given for accelerating black holes [11] and a pp-wave known as the vortex [7], neither of which are stationary. Additionally, the double copy appears to work quite generally on a maximally symmetric background [9]. We will present a covariant version of the double copy, which applies in all these cases.…”

“…Following [9], we can write the metric for a rotating black hole in (A)dS space using the background spacetime The single copy gauge field is A µ = gq κM φk µ . The corresponding field strength is We can also determine the gauge source using (2.18).…”

“…µν = gq(r 2 − a 2 cos 2 θ) 4π(r 2 + a 2 cos 2 θ) Note that only the radial electric field survives in the limit a → 0. This field strength satisfies the Maxwell equations in (A)dS background with source[9] j µ = q 4πa 2 sec 3 θδ(r) 2, 0, 0, −…”

The classical double copy of a point charge

“…This is possible for a class of vacuum solutions in gravity, that of stationary Kerr-Schild spacetimes [5], where much work has been done in this context; see e.g. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The basic example is the relation of the Schwarzschild and Coulomb solutions.…”