Topologically modified Einstein equation: a solution with singularities on $\mathbb{S}^3$

Vigneron [Foundations of Physics, 54, 15, (2024)] recently proposed a modification of general relativity in which a non-dynamical term related to the spatial topology is introduced in the Einstein equation. The original motivation for this theory is to allow for the non-relativistic limit to exist in any physical topology. In the present paper, we derive a first inhomogeneous exact vacuum solution of this theory for a spherical topology, assuming staticity and spherical symmetry. The metric represents a black hole and a repulsive singularity at opposite poles of a 3-sphere. The solution is similar to the Schwarzschild metric, but the spacelike infinity is cut, and replaced by a repulsive singularity at finite distance, implying that the spacelike hypersurfaces have finite volume, and the total mass is zero. We discuss how this solution paves the way to massive, non-static solutions of this theory, more directly relevant for cosmology.