The classical double copy in three spacetime dimensions

TL;DR

This work tests the classical Kerr-Schild double copy in three spacetime dimensions, where gravity lacks propagating gravitons and the Newtonian limit. Using the BTZ black hole and the double copy of a gauge-theory point charge, it shows that the gravity side must include a dilaton (and a two-form) to map gauge degrees of freedom, with the BTZ case yielding a constant gauge charge density and the point charge producing a nonvacuum, dilaton-supported spacetime. A Newtonian-like limit emerges when the double-copy source is treated appropriately, giving a logarithmic potential and defining $G_3=\kappa^2/(8\pi)$, while geodesic analyses reveal precession and bound-orbit structures. Overall, the results provide a nontrivial cross-check of the classical Kerr-Schild double copy in a setting with no graviton DOF and clarify how gauge-theory degrees of freedom map onto gravity plus a dilaton, offering insight into the broader applicability of the double copy.

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.

Figures (4)

• Figure 1: Effective GR and Newtonian potentials for $GM=1/2$ and different values of the angular momentum $L$. We can clearly observe that for $L<L_\text{min}$ the GR potential has no maximum, at $L=L_\text{min}$ it develops an inflection point, and for $L>L_\text{min}$ a maximum occurs and we can have stable orbits. On the other hand, the shape of the Newtonian potential stays unchanged and always allows stable orbits.
• Figure 2: Timelike geodesics obtained for the GR effective potential, where $GM=0.4$ and $L/r_0=3$.
• Figure 3: Timelike geodesics obtained for the GR effective potential, where $GM=1.8$ and $L/r_0=3$.
• Figure 4: Bounded orbits obtained via solving Eqns \ref{['Newtorb']} (displayed as the blue line) and \ref{['GRorb']} (displayed as the orange line) with $GM=.2$, $L/r_0=1$, and $E=1.1 V_{min}$.