The classical double copy of a point charge

Abstract

The classical double copy relates solutions to the equations of motion in gauge theory and in gravity. In this paper, we present two double-copy formalisms for relating the Coulomb solution in gauge theory to the two-parameter Janis-Newman-Winicour solution in gravity. The latter is a static, spherically symmetric, asymptotically flat solution that generically includes a dilaton field, but also admits the Schwarzschild solution as a special case. We first present the classical double copy as a perturbative construction, similar to its formulation for scattering amplitudes, and then present it as an exact map, with a novel generalisation of the Kerr-Schild double copy motivated by double field theory. The latter formalism exhibits the relation between the Kerr-Schild classical double copy and the string theory origin of the double copy for scattering amplitudes.

Weyl doubling

“…Does the derivative corrected Gibbon-Hawking expression have a conventional double copy interpretation in terms of Maxwell fields in flat space? It was shown in [79] that doubled geometry clarifies the double copy construction for the point charge, relating this to the JNW solution [125]. As we noted in section 6, the Kerr-Schild form in double field theory involves more than one gauge field -for example, two Maxwell fields in the case where there is a Kalb-Ramond field as well as the metric.…”

The convolutional double copy: a case study with a point

“…It would not be correct, for example, to conclude that the double copy of the point charge is only a special case of JNW: this would directly contradict the results of refs. [4,27]. Instead the problem can be traced back to the fact that we have demanded that the double copy dictionary of eqs.…”

“…Thus, we see that the full two-parameter JNW solution is obtained, commensurate with refs. [22,27]. Furthermore, the convolutional double copy provides a key insight into where these two parameters come from, namely the ability to source the ghosts and gauge fields (whilst keeping the Lorenz and de Donder gauges) independently.…”

“…Since its inception, a number of approaches have tried to extend its remit to classical solutions, exact or otherwise. Examples include the use of Kerr-Schild coordinates [4][5][6][7][8][9][10][11][12][13][14], spinorial methods [15,16], worldline methods [17][18][19][20][21], perturbative diagrammatic reasoning [22][23][24][25], and double field theory [26,27]. In this paper, we will focus on another approach, first introduced in ref.…”

“…This point of view was corroborated by the recent study of ref. [27], using double field theory. We will see that the BRST convolutional double copy provides a highly useful complementary analysis of this situation.…”