  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 $x$.
