  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.
