http://arxiv.org/abs/2201.02112
Constructing an extension of Newton’s theory which is defined on a non-Euclidean topology, called a non-Euclidean Newtonian theory, such that it can be retrieved by a non-relativistic limit of general relativity is an important step in the study of the backreaction problem in cosmology and might be a powerful tool to study the influence of global topology on structure formation. After giving a precise mathematical definition of such a theory, based on the concept of Galilean manifolds, we propose two such extensions, for spherical or hyperbolic classes of topologies, using a minimal modification of the Newton-Cartan equations. However as for now we do not seek to justify this modification from general relativity. The first proposition features a non-zero cosmological backreaction, but the presence of gravitomagnetism and the impossibility of performing N-body calculations makes this theory difficult to be interpreted as a Newtonian-like theory. The second proposition features no backreaction, N-body calculation is possible and no gravitomagnetism appears. In absence of a justification from general relativity, we argue that this non-Euclidean Newtonian theory should be the one to be considered, and could be used to study the influence of topology on structure formation via N-body simulations. For this purpose we give the mass point gravitational field in $\mathcal{S}^3$.
Q. Vigneron
Fri, 7 Jan 22
19/34
Comments: Submitted to Classical and Quantum Gravity