A relativistic model of the topological acceleration effect

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 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 $M\sim 10^{1}4M_{\odot }$ and $L\sim 10$ to $20\hGpc$, this linear expression is accurate to $\pm 10$ 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 $x$.

12 pages, 2 figures, 1 table; v3: N-body simulation discussion removed, references added, minor errors corrected

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Cited by in corpus (4)

• Virialisation-induced curvature as a physical explanation for dark energy
• Solving the Einstein constraints in periodic spaces with a multigrid approach
• Gauss-Bonnet-Chern approach to the averaged Universe
• Topological implications of inhomogeneity