On Nonrelativistic Isotropic and Homogeneous Universe

TL;DR

The paper develops a covariant, nonrelativistic cosmology based on Galilean covariance implemented on a five‑dimensional manifold, enabling a gravity description without a universal speed limit. It constructs a five‑dimensional isotropic, homogeneous metric with scale factor $a(x^4)$ and, via dimensional reduction to 3+1, derives Einstein-like equations with a fluid potential $V(\rho)=\lambda\rho$, analyzing two regimes: $\lambda=0$ (vacuum) and $\lambda=1$ (dust). The vacuum case yields a separable solution with an exponential–quadratic scale factor, while the dust case produces a non-interacting nonrelativistic fluid with density $\varrho(s,t)$; in both, the embedding introduces anisotropy in the reduced spacetime. In the flat-curvature limit ($K=0$), the model reduces to Milne’s Newtonian cosmology with zero pressure, offering an independent covariant nonrelativistic framework for cosmological dynamics. Overall, the work demonstrates that Galilean covariance provides a consistent, covariant setting for nonrelativistic cosmology with natural anisotropy arising from dimensional reduction, complementing Newtonian and relativistic approaches.

Abstract

This article deals with a nonrelativistic cosmological model based on Galilean covariance, formulated within a five-dimensional Galilean manifold. Within this framework, we construct an isotropic and homogeneous metric analogous to the Friedmann--Robertson--Walker metric but without a universal speed limit. Two distinct solutions of the Einstein-like field equations are obtained: (i) a vacuum configuration ($λ=0$) yielding an exponential--quadratic scale factor, and (ii) a dust-dominated universe ($λ=1$) described by a non-interacting nonrelativistic fluid. Upon dimensional reduction to $3+1$ spacetime through a specific embedding, the model naturally develops anisotropy in the scale factor and density, consistent with the near-zero spatial curvature inferred from Planck data. In the case of vanishing spatial curvature, the framework reproduces Milne's Newtonian cosmology because this condition leads to a vanishing pressure. This provides an independent nonrelativistic setting for cosmological dynamics within Galilean covariance.