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

We show explicitly how the Newton–Hooke groups N±10 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 M±4 and show how these may be obtained from a null reduction of five-dimensional homogeneous pp-wave Lorentzian spacetimes M±5. This allows us to realize the Newton–Hooke groups and their Bargmann-type central extensions as subgroups of the isometry groups of M±5. The extended Schr?dinger-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 Gaoʹs generalization of the matrix model.