Non-Euclidean Newtonian cosmology

Abstract

We formulate and solve the problem of Newtonian cosmology under the assumption that the absolute space of Newton is non-Euclidean. In particular, we focus on the negatively-curved hyperbolic space, H 3 . We point out the inequivalence between the curvature term that arises in the Friedmann equation in Newtonian cosmology in Euclidean space and the role of curvature in the H 3 space. We find the generalisation of the inversesquare law and the solutions of the Newtonian cosmology that follow from it. We find the generalisations of the Euclidean Michell 'black hole' in H 3 and show that it leads to different maximum force and area results to those we have found in general relativity. We show how to add the counterpart of the cosmological constant to the gravitational potential in H 3 and explore the solutions and asymptotes of the cosmological models that result. We also discuss the problems of introducing compact topologies in Newtonian cosmologies with non-negative spatial curvature.

On non-Euclidean Newtonian theories and their cosmological backreaction

“…The authors in [2,15] use this theory to compute the gravitational field created by one mass point. For the purpose of this section we only give the solution in the case where the topology § is S 3 which can be found when solving equation (2) in spherical symmetry with ρ = 0.…”

“…To our knowledge, only one NEN theory has been propose to date, by [2,15]. As we will see in the following sections, this theory is unphysical and cannot be used to compute the backreaction.…”

“…In section 2, we introduce the strategy we use to define an NEN theory. Section 3 reviews the NEN theory of [2,15]. We give a precise mathematical definition of an NEN theory in section 4 based on the concept of Galilean manifolds.…”

“…However once we introduce a non-zero curvature in the spatial connection present in these equations, they are not equivalent anymore and therefore define two different NEN theories. In [2,15], the authors use the Poisson equation. We detail this approach in the next section and we will see that physical inconsistencies appear.…”

Worldlines and social networks

“…When Newton formulated his theory of gravity, he assumed time to be linear, with unchangeable rate of flow. He assumed space to the absolute, unchanging, and Euclidean: 'the divine sensorium' [18].…”