http://arxiv.org/abs/1410.3909
We consider lattice Universes with spatial topologies $T\times T\times T$, $\; T\times T\times R\; $ and $\; T\times R\times R$. In the Newtonian limit of General Relativity, we solve the Poisson equation for the gravitational potential in the enumerated models. In the case of point-like massive sources in the $T\times T\times T$ model, we demonstrate that the gravitational potential has no definite values on the edges joining identical masses in neighboring cells, i.e. at points where masses are absent. Clearly, this is a nonphysical result. The only way to avoid this problem and get a regular solution at any point of the cell is the smearing of these masses over some region. In the cases of $\; T\times T\times R\; $ and $\; T\times R\times R$ topologies, there is no way to get any physically reasonable and nontrivial solution. The only solutions we can get here are the ones which reduce these topologies to the $T\times T\times T$ one.
M. Brilenkov, M. Eingorn and A. Zhuk
Thu, 16 Oct 14
17/55
Comments: 16 pages, 1 figure