On non-Euclidean Newtonian theories and their cosmological backreaction

Abstract

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 S 3 . ‡ Note that the addition of a spatial curvature is not done in order to take into account relativistic effects, as in [1], but only to allow for non-Euclidean topologies.

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

“…This suggests that extending Newton's theory to non-Euclidean topologies is possible, and therefore describing that aspect of cosmology with a Newtonian theory. Such a theory is called a non-Euclidean Newtonian theory (NEN theory), and was proposed using Galilean structures and a minimal modification of the Newton-Cartan equations by Vigneron [18]. More than being a theoretical curiosity, this theory is actually a powerful tool to study the effect of cosmic topology on the gravitational field and on structure formation [20], mainly because it is much easier to use than general relativity, while still encoding non-linearities in other topologies than the usual 3-torus.…”

“…Therefore any non-relativistic theory of gravitation should be retrieved with a nonrelativistic limit of the Einstein equation to be qualified as a physical theory. This holds in particular for the NEN theory of Vigneron [18] which has yet to be derived from such a limit, and therefore which remains still heuristic.…”

“…3 Because of property (1), Newton's theory, for which the spatial Ricci tensor is zero, is only defined for Euclidean topologies. Therefore a non-Euclidean Newtonian theory, such as the one defined in [18] for spherical and hyperbolic topologies, requires the spatial Ricci tensor to be non-zero. The validity of general relativity, Newton's theory and NEN theory in [18,20], as a function of Thurston topological class is summarised in Table I.…”

“…Table I. This table shows, as function of Thurston's topological class, which theory among general relativity, Newton's theory and NEN theory in [18,20] are defined. Only general relativity is defined for any possible spatial topology.…”

“…Only general relativity is defined for any possible spatial topology. The, still heuristic, NEN theory of Vigneron [18] is only defined for the Euclidean, spherical and hyperbolic cases (in the Euclidean case it is equivalent to Newton's theory). The goal of this paper is to determine if general relativity naturally defines a NEN theory, i.e.…”