A relativistic model of the topological acceleration effect

TL;DR

This paper demonstrates through a relativistic model that the topology of the universe induces a measurable acceleration effect, supporting the hypothesis that cosmic topology influences gravitational dynamics.

Contribution

It provides the first fully relativistic verification of the topological acceleration effect in a multiply connected universe, confirming heuristic and Newtonian results.

Findings

The topological acceleration effect is relativistically valid.

The effect's magnitude matches heuristic predictions within 10%.

The linear approximation holds over three orders of magnitude in distance.

Abstract

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 Poincar\'e 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 Ap\'ery'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 $x$.