From pp-waves to galilean spacetimes

Abstract

A. We exhibit all spatially isotropic homogeneous galilean spacetimes of dimension (n + 1) 4, including the novel torsional ones, as null reductions of homogeneous pp-wave spacetimes. We also show that the pp-waves are sourced by pure radiation fields and analyse their global properties.

Particle dynamics on torsional galilean spacetimes

“…The above shows however that the symmetric part 15 of the unique invariant connection (2.10) compatible with the invariant Newton-Cartan structure (2.13) coincides with the symmetric connection (4.3) compatible with the Newton-Cartan structure (4.10) and specified by the NC-doublet (4.11). The Newton-Cartan structure (4.10) is invariant under the subalgebra generated by the ξ P a , ξ B a and ξ J ab of (2.8), but this is not the case for the time translations: although L ξ H τ µ = 0 one finds that L ξ H h µν = −βh µν .…”

“…In section 4.3 we show how they also appear as the algebras of homotheties of the Lorentzian metrics (4.23), that are obtained through the Eisenhart lift of the damped harmonic oscillator. They are also the isometry Lie algebras of certain homogeneous pp-waves discussed in [15], as we explain in appendix D.…”

“…In this appendix we shortly recall that these descriptions are unique only up to a conformal redefinition. We refer to [34] for a complete discussion, here we simply mention how some key formulae of Section 4 behave under such a conformal redefinition and then focus on the case of interest in this paper, namely the damped harmonic motion describing a free particle on M (α,β) , and how some results in this paper are related to those in [15]. The starting observation is that the Type I NC geometries (τ µ , h µν ; τµ , hµν ) and (τ c µ , h Under such a conformal redefinition of the NC geometry essentially all results of Section 4 remain valid, but some of the parameters will depend on the choice of conformal prefactor.…”

“…Of course this is not the case for the Lorentzian metric (4. To make the equivalence to [15] explicit, make the coordinate transformation Upon the choice ψ = −β t this becomes exactly the metric described in Appendix A of [15].…”

“…In Appendix C we extend the discussion of the main text to galilean spacetimes in spacetime dimension ≤ 3, where particularly in dimension 3 there is a richer family of homogeneous galilean spacetimes. Finally, in Appendix D we comment on the freedom to conformally redefine the Eisenhart lift and use this to connect some of our results to those of [15].…”