Non-relativistic regime and topology I: topological term in the Einstein equation

We study the non-relativistic (NR) limit of relativistic spacetimes in relation with the topology of the Universe. We first show that the NR limit of Einstein's equation is only possible in Euclidean topologies, i.e. for which the covering space is $\mathbb{E}^3$. We propose and motivate a modification of that equation which allows for the limit to be performed in any topology. For this, a second reference non-dynamical metric is introduced in addition to the physical spacetime metric. The choice of reference metric is related to the covering space of the topology. Instead of featuring the physical spacetime Ricci tensor, the modified Einstein equation features the difference between the physical and the reference non-dynamical spacetime Ricci tensor. This equation should be considered instead of Einstein's equation if one wants to study a model universe with a non-Euclidean topology and admitting a non-relativistic limit.Comment: 43 pages, 5 figures, submitted to Classical and Quantum Gravity. The difference with the previous version is that we answer the question raised at the en