Type D spacetimes and the Weyl double copy

Abstract

We study the double-copy relation between classical solutions in gauge theory and gravity, focusing on four-dimensional vacuum metrics of algebraic type D, a class that includes several important solutions. We present a double copy of curvatures that applies to all spacetimes of this type -the Weyl double copy -relating the curvature of the spacetime to an electromagnetic field strength. We show that the Weyl double copy is consistent with the previously known Kerr-Schild double copy, and in fact resolves certain ambiguities of the latter. The most interesting new example of the classical double copy presented here is that of the C-metric. This well-known solution, which represents a pair of uniformly accelerated black holes, is mapped to the Liénard-Wiechert potential for a pair of uniformly accelerated charges. We also present a new double-copy interpretation of the Eguchi-Hanson instanton. arXiv:1810.08183v2 [hep-th] 1 Apr 2019 -1 -an antisymmetric tensor. The focus of this article will be on the double copy in classical field theory. Moreover, we will restrict to pure Einstein gravity, so that the dilaton and the antisymmetric tensor are not present; we will comment on these fields in section 6.The KLT relations have the advantage that an underlying reason for the relation between gauge theory and gravity is clear: by joining two open strings, you get one closed string. But they have the disadvantage that the relations themselves become quite complicated for high multiplicity. More recently, Bern, Carrasco and Johansson (BCJ) discovered a new and simple form for the double copy [2] which also leads to a natural formulation of the double copy at loop level [3]. This form of the double copy has been extensively studied at tree level, leading to various proofs [4][5][6][7][8][9][10][11][12][13][14][15][16]. At loop level, the double copy has been extensively studied [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], but to date it is still a conjecture [27]. Nevertheless, the BCJ double copy is a powerful tool in the theory of scattering amplitudes, which has led to rich new insights into the structure of supergravity, e.g. [32][33][34][35][36][37][38][39][40]. A celebrated recent example is the detailed computation of the UV structure of maximal supergravity at five loops [41]. The double copy is reviewed in, for example, [42][43][44].The success of the double copy for scattering amplitudes motivated the investigation of its manifestation for solutions of the classical field equations, with early steps given in [45][46][47]. Since the principles of the double copy, as currently understood, are perturbative in nature, it is remarkable that exact relations between solutions can be found, as discovered in [48]. In the same way that solutions to the Maxwell equations provide a class of linear solutions to the Yang-Mills equations (with trivial colour dependence), there is a class of solutions that linearise the Einstein equations. Kerr-Schild metrics belong to this class, and so do certain multi-Kerr-Sch...

The classical double copy of a point charge

“…It should also be possible to extend the vacuum Weyl double copy [15], and a higher-dimensional version based on [77], to include the dilaton and B-field, using ideas from double field theory. In fact, this could potentially elucidate some aspects of the generalised curvatures of double field theory, alluded to in the appendix.…”

Kerr-Schild double copy and complex worldlines

“…Furthermore, it may be possible to extend the blueprint of Fig. 1 to the more general Weyl double copy, thus providing insight into all spacetimes of Petrov type D [16]. We leave all these questions to future work.…”

“…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.…”

“…(C. 15) We have assumed throughout that the single copy is formed with the unit timelike Killing vector µ and that · k = −1, so we have φ = κM qg A 0 = r s r r 2 + a 2 cos 2 θ , k µ = 1, 1, 0, − a r 2 + a 2 , (C. 16) exactly as in (B.10).…”

S-duality and the double copy

“…with coordinates (u, v, X, Y ) and constant λ. The single-copy (self-dual) Maxwell tensor is [46,59]…”