Abstract
The double copy relates scattering amplitudes in gauge and gravity theories, and has also been extended to classical solutions. In this paper, we study solutions in three spacetime dimensions, where the double copy may be expected to be problematic due to the absence of propagating degrees of freedom for the graviton, and the lack of a Newtonian limit. In particular, we examine the double copy of a gauge theory point charge. This is a vacuum solution in gauge theory, but leads to a non-vacuum solution in gravity, which we show is consistent with previously derived constraints. Furthermore, we successfully interpret the non-trivial stress-energy tensor on the gravity side as arising from a dilaton profile, and the Newtonian description of a point charge emerges as expected in the appropriate limit. Thus, our results provide a non-trivial cross-check of the classical Kerr-Schild double copy.
Highlights
Where gdenotes a graph obtained from g by exchanging an odd number of vertices and the graphs gt and gu are obtained from the graph gs by picking an internal leg in gs, calling it the s-channel, and switching it to the t- or u-channel, leaving the rest of the graph unchanged
Both the color-kinematic duality and BCJ double copy described above are purely perturbative relations, but their seeming validity at multiple loops indicates that they may be the perturbative manifestations of some nonperturbative symmetry or duality between gauge and gravity theories, though it is as yet unknown how they may be made explicit at the Lagrangian level
This is in stark contrast to the Schwarzschild black hole in four dimensions, in which the point mass M on the gravity side is associated with a point charge in the gauge theory
Summary
To see the significance of the sign choice consider a Kerr-Schild spacetime sourced by a stationary pressureless fluid in four dimensions with a small energy density ρ, in which case the stress-energy tensor is Tμν = ρUμUν with U μ∂μ = (−g00)−1/2∂0. We will choose k0 = +1 so that we can identify the Kerr-Schild scalar field φ as the scalar potential on the gauge side. — given a Kerr-Schild graviton kμkνφ we can always choose k0 = ±1 by rescaling kμ → ±kμ/k0 and φ → (k0)2φ. These redefinitions leave our spacetime unchanged but alter the gauge field with which it is identified.
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