We consider lattice Universes with spatial topologies T × T × T, T × T × R, and T × R × 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 × T × 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 × T × R and T × R × 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 × T × T one.
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机译：我们考虑具有空间拓扑T×T×T，T×T×R和T×R×R的晶格宇宙。在广义相对论的牛顿极限中，我们在枚举模型中求解了泊松方程的重力势。在T×T×T模型中的点状块状源的情况下，我们证明了在连接相邻单元中相同质量的直线上，即在没有质量的点上，重力势没有确定的值。显然，这是非物理的结果，因为在这种情况下无法确定宇宙物体的动力学。避免此问题并在电池的任何位置获得常规解决方案的唯一方法是在某些区域上涂抹这些物质。因此，在N体模拟中对引力体的涂抹不仅是一种技术方法，而且是一种物理证实的过程。对于T×T×R和 T × R × R 拓扑，无法获得任何物理上合理且平凡的解决方案。我们在这里可以获得的唯一解决方案是将这些拓扑简化为 T × T ×× T 一个的解决方案。
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