Newton-Hooke spacetimes, Hpp-waves and the cosmological constant

TL;DR

The paper shows that the Newton-Hooke groups are the natural kinematical symmetries of non-relativistic cosmological models with a cosmological constant, emerging as a controlled c → ∞, Λ → 0 limit with Λ c^2 fixed. It realizes these symmetries geometrically by identifying Newton-Hooke spacetimes M4± as cosets and lifting them to 5D Bargmann spacetimes M5±, which are homogeneous pp-waves, yielding central (mass) extensions and a rich non-relativistic conformal structure. The authors establish a 13-parameter Bargmann conformal group acting on the cosmological equations and show its subsystems map time-dependent oscillator solutions and allow for rescalings of G and Λ, while connecting to the Dimitriev-Zel'dovich framework for structure formation. They also discuss Gao's Newton-Hooke–inspired matrix construction and highlight potential links to string theory, holography, and large-scale cosmology, providing a comprehensive geometric stage for non-relativistic cosmology with a cosmological constant.

Abstract

We show explicitly how the Newton-Hooke groups act as symmetries of the equations of motion of non-relativistic cosmological models with a cosmological constant. We give the action on the associated non-relativistic spacetimes and show how these may be obtained from a null reduction of 5-dimensional homogeneous pp-wave Lorentzian spacetimes. This allows us to realize the Newton-Hooke groups and their Bargmann type central extensions as subgroups of the isometry groups of the pp-wave spacetimes. The extended Schrodinger type conformal group is identified and its action on the equations of motion given. The non-relativistic conformal symmetries also have applications to time-dependent harmonic oscillators. Finally we comment on a possible application to Gao's generalization of the matrix model.