Generalized Galilean Algebras and Newtonian Gravity

TL;DR

This paper constructs and analyzes generalized Galilean algebras as non-relativistic limits of relativistic algebras, explores their contractions and expansions, and links these structures to Newtonian gravity and potential dark matter interpretations.

Contribution

It introduces new generalized Galilean algebras, derives their relations via contraction and expansion methods, and connects non-relativistic gravity modifications to dark matter effects.

Findings

Generalized Galilean algebras of types I and II are constructed.

The non-relativistic limit of Einstein--Chern--Simons gravity modifies the Poisson equation.

Dark Matter effects may be explained as a non-relativistic limit of Dark Energy.

Abstract

The non-relativistic versions of the generalized Poincar\'{e} algebras and generalized $AdS$-Lorentz algebras are obtained. This non-relativistic algebras are called, generalized Galilean algebras type I and type II and denoted by $\mathcal{G}\mathfrak{B}_{n}$ and $\mathcal{G}\mathfrak{L}_{_{n}}$ respectively. Using a generalized In\"{o}n\"{u}--Wigner contraction procedure we find that the generalized Galilean algebras type I can be obtained from the generalized Galilean algebras type II. The $S$-expansion procedure allows us to find the $\mathcal{G}\mathfrak{B}_{_{5}}$ algebra from the Newton--Hooke algebra with central extension. The procedure developed in Ref. \cite{newton} allow us to show that the non-relativistic limit of the five dimensional Einstein--Chern--Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.