  The classical double copy is a map between exact solutions of general
relativity and Maxwell's theory, where the nonlinear nature of general
relativity is circumvented by considering solutions in the Kerr-Schild form. In
this paper, we give a general formulation, where no simplifying assumption
about the background metric is made, and show that the gauge theory source is
affected by a curvature term that characterizes the deviation of the background
spacetime from a constant curvature spacetime. We demonstrate this effect
explicitly by studying gravitational solutions with non-zero cosmological
constant. We show that, when the background is flat, the constant charge
density filling all space in the gauge theory that has been observed in
previous works is a consequence of this curvature term. As an example of a
solution with a curved background, we study the Lifshitz black hole with two
different matter couplings. The curvature of the background, i.e., the Lifshitz
spacetime, again yields a constant charge density; however, unlike the previous
examples, it is canceled by the contribution from the matter fields. For one of
the matter couplings, there remains no additional non-localized source term,
providing an example for a non-vacuum gravity solution corresponding to a
vacuum gauge theory solution in arbitrary dimensions.
