More on the Classical Double Copy in Three Spacetime Dimensions

TL;DR

This paper explores the classical double copy in three-dimensional gravity, presenting new solutions and interpretations, and discusses implications for the validity of the double copy in 3D with or without a cosmological constant.

Contribution

It provides an alternative derivation of a 3D double copy solution, introduces a stationary version, and analyzes the nature of the source, clarifying the role of scalar fields and fluids.

Findings

A new stationary solution without ghost fields.

The scalar field interpretation does not hold for the stationary case.

Implications for the classical double copy in 3D are discussed.

Abstract

It is well-known that General Relativity (GR) in three spacetime dimensions (3D) has no well-defined Newtonian limit. Recently, a static solution mimicking the behaviour of the expected Newtonian potential has been found in arXiv:1904.11001 by studying the classical double copy of a point charge in gauge theory. This is the first example where the vacuum solution in the gauge theory leads to a non-vacuum solution on the gravity side. The resulting energy-momentum tensor was attributed to a free scalar ghost field; however, alternatively, the source can be seen as one resulting from a spacelike perfect fluid. In this paper, we first give an alternative derivation of the solution where there is no need to perform a generalized gauge transformation to obtain a quadratic Lagrangian without propagating ghost fields. Then, we present a stationary version of the solution and show that the scalar field interpretation of the source does not survive in this case, leaving the spacelike fluid as the only possibility. We give the gauge theory single copy of our solution and comment on the implications of our results on the validity of the classical double copy in 3D. The effect of the cosmological constant is also discussed.