Do Black Holes Exist in a Finite Universe Having the Topology of a Flat 3-Torus?

Abstract: Based on perturbation theory, we present the exact first-order solution to the Einstein equations for the exterior static gravitational field of an isolated non-rotating star in a spatially finite universe having the topology of a flat 3-torus. Since the method of images leads to a divergent Poincare' series, one needs a regularization which we achieve by using the Appell respectively the Epstein zeta function. The solution depends on a new positive constant which is completely fixed by the mass of the star and the spatial volume of the universe. The physical interpretation is that a stable or metastable equilibrium requires a topological dark energy which fills the whole universe with positive energy density and negative pressure. The properties of the gravitational field are discussed in detail. In particular, its anisotropy is made explicit by deriving an exact multipole expansion which shows that in this case Birkhoff's theorem does not hold. While the monopole describes the Newtonian potential, there is no dipole but always a non-vanishing quadrupole which leads to a repulsive force experienced by a planet at rest. Finally, we put forward the conjecture that black holes exist in a toroidal universe and that their gravitational field is in the weak-field limit well approximated by the first-order field.