The residual gravity acceleration effect in the Poincare dodecahedral space

Details The residual gravity acceleration effect in the Poincare dodecahedral space The residual gravity acceleration effect in the Poincare dodecahedral space

2009-02-19

arxiv.org/abs/0902.3402v4

Astronomy & Astrophysics 502 (2009) 27

Astronomy and Astrophysics

Astrophysics

Cosmology and Extragalactic Astrophysics

11 pages, 5 figures, accepted in Astronomy & Astrophysics, computer algebra script available at http://adjani.astro.umk.pl/G

Scientific paper

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?

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