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

TL;DR

This paper investigates the non-relativistic limit of Einstein's equations in relation to the universe's topology, revealing limitations in non-Euclidean spaces and proposing a modified theory to accommodate arbitrary topologies.

Contribution

It introduces a modified Einstein equation with a reference connection, enabling non-relativistic limits in non-Euclidean topologies, unlike standard general relativity.

Findings

NR limit only possible in Euclidean topology in standard GR

Modified theory introduces a second reference connection

Allows non-relativistic limits in non-Euclidean topologies

Abstract

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 the Einstein equation is only possible in Euclidean topologies, i.e. for which the covering space is $\mathbb{E}^3$. We interpret this result as an inconsistency of general relativity in non-Euclidean topologies and propose a modification of that theory which allows for the limit to be performed in any topology. For this, a second reference non-dynamical connection is introduced in addition to the physical spacetime connection. The choice of reference connection is related to the covering space of the spacetime topology. Instead of featuring only the physical spacetime Ricci tensor, the modified Einstein equation features the difference between the physical and the reference Ricci tensors. This theory should be considered instead of general relativity if one wants to study a model universe with a non-Euclidean topology and admitting a non-relativistic limit.