Non-Relativistic Spacetimes with Cosmological Constant

TL;DR

This work develops a group-theoretic framework to study non-relativistic spacetimes with a cosmological constant by deriving Newton-Hooke kinematics as nonrelativistic limits of de Sitter spacetimes via Inonu-Wigner contractions. It shows that Newton-Hooke spacetimes are non-metric, curved homogeneous spaces endowed with a nontrivial invariant connection, contrasting with Galilean spacetime which has a flat connection and no invariant metric. The paper provides explicit constructions: the de Sitter spacetimes have a metric-compatible canonical connection and Maurer-Cartan curvature, while the Newton-Hooke limit preserves curvature through $R^{a0}{}_{b0}=rac{ u}{ au^2}\, ext{delta}^a{}_b$, and translations become noncommuting. The results highlight potential observable implications for nonrelativistic physics in cosmological contexts (e.g., virial-type analyses) and advocate viewing spacetime as a manifold equipped with a connection, consistent with gauge approaches to spacetime symmetries. This framework broadens the understanding of how a cosmological constant can influence nonrelativistic kinematics and cosmology.

Abstract

Recent data on supernovae favor high values of the cosmological constant. Spacetimes with a cosmological constant have non-relativistic kinematics quite different from Galilean kinematics. De Sitter spacetimes, vacuum solutions of Einstein's equations with a cosmological constant, reduce in the non-relativistic limit to Newton-Hooke spacetimes, which are non-metric homogeneous spacetimes with non-vanishing curvature. The whole non-relativistic kinematics would then be modified, with possible consequences to cosmology, and in particular to the missing-mass problem.