A relativistic model of the topological acceleration effect

It has previously been shown heuristically that the topology of the Universe affects gravity, in the sense that a test particle near a massive object in a multiply connected universe is subject to a topologically induced acceleration that opposes the local attraction to the massive object. It is necessary to check if this effect occurs in a fully relativistic solution of the Einstein equations that has a multiply connected spatial section. A Schwarzschild-like exact solution that is multiply connected in one spatial direction is checked for analytical and numerical consistency with the heuristic result. The T¹ (slab-space) heuristic result is found to be relativistically correct. For a fundamental domain size of L, a slow-moving, negligible-mass test particle lying at distance x along the axis from the object of mass M to its nearest multiple image, where GM/c² ≪ x ≪ L/2, has a residual acceleration away from the massive object of 4ζ(3)G(M/L³) x, where ζ(3) is Apéry's constant. For M ∼ 10¹⁴M_⊙ and L ∼ 10-20h^-1 Gpc, this linear expression is accurate to ±10% over 3\;{{h^{-1} Mpc}}\ \lower.6ex{\buildrel < \over \sim }\ x\ \lower.6ex{\buildrel<\over \sim }\ 2\;{{h^{-1} Gpc}}. Thus, at least in a simple example of a multiply connected universe, the topological acceleration effect is not an artefact of Newtonian-like reasoning, and its linear derivation is accurate over about three orders of magnitude in x.