https://doi.org/10.1140/epjc/s10052-024-13545-4
Regular Article
Topologically modified Einstein equation: a solution with singularities on 
1
Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziądzka 5, 87-100, Toruń, Poland
2
ENS de Lyon, CRAL UMR5574, Université Claude Bernard Lyon 1, CNRS, 69007, Lyon, France
3
Departament de Física, Universitat de les Illes Balears, IAC3-IEEC, Crta. Valldemossa km 7.5, 07122, Palma, Spain
Received:
31
August
2024
Accepted:
27
October
2024
Published online:
22
November
2024
Vigneron (Found Phys 54:15, https://doi.org/10.1007/s10701-023-00749-z, 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.
© The Author(s) 2024
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