Lattice Universe: examples and problems
  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 straight lines joining identical masses
in neighboring cells, i.e. at points where masses are absent. Clearly, this is
a nonphysical result since the dynamics of cosmic bodies is not determined in
such a case. 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.
Therefore, the smearing of gravitating bodies in $N$-body simulations is not
only a technical method but also a physically substantiated procedure. 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.
