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. This effect distinguishes different comoving 3-manifolds, potentially providing a theoretical justification for the Poincaré dodecahedral space observational hypothesis and a dynamical test for cosmic topology. 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 T\(^1\) (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 \(GM/c^2 \ll x \ll L/2\), has a residual acceleration away from the massive object of \(4\zeta(3) G(M/L^3)\,x\), where \(\zeta(3)\) is Apéry's constant. For \(M \sim 10^14 M_\odot\) and \(L \sim 10\) to \(20\hGpc\), this linear expression is accurate to \(\pm10%\) over \(3\hMpc \ltapprox x \ltapprox 2\hGpc\). 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 \(x\).