The residual gravity acceleration effect in the Poincare dodecahedral
  space
  In a flat space, the global topology of comoving space can induce a weak
acceleration effect similar to dark energy. Does a similar effect occur in the
case of the Poincare dodecahedral space S^3/I^*? Does the effect distinguish
the Poincare space from other well-proportioned spaces? The residual
acceleration effect in the Poincare space is studied here using a massive
particle and a nearby test particle of negligible mass, in S^3 embedded in R^4.
The weak limit gravitational attraction on a test particle at distance r is set
\propto [r_C \sin(r/r_C)]^{-2}, where r_C = curvature radius, in order to
satisfy Stokes' theorem. A finite particle horizon large enough to include the
adjacent topological images of the massive particle is assumed. The regular,
flat, 3-torus T^3 is re-examined, and two other well-proportioned spaces,
S^3/T^* and S^3/O^*, are also studied. The residual gravity effect occurs in
all four cases. In a perfectly regular 3-torus of side length L_a, and in
S^3/T^* and S^3/O^*, the highest order term in the residual acceleration is the
third order term in the Taylor expansion of r/L_a (3-torus), or r/r_C,
respectively. However, the Poincare dodecahedral space is unique among the four
spaces. The third order cancels, leaving the fifth order term \sim \pm 300
(r/r_C)^5 as the most significant. Not only are three of the four perfectly
regular well-proportioned spaces better balanced than most other multiply
connected spaces in terms of the residual acceleration effect by a factor of
about a million (setting r/L_a = r/r_C \sim 10^{-3}), but the fourth of these
spaces is about 10^4 times better balanced than the other three. This is the
Poincare dodecahedral space. Is this unique dynamical property of the Poincare
space a clue towards a theory of cosmic topology?
