Scaling and conformal symmetries for plane gravitational waves

TL;DR

This work shows that exact plane gravitational waves possess a rich symmetry structure: their isometries form a five-parameter Carroll group, while a universal homothety provides a conformal transformation that preserves geodesic trajectories up to scaling. By lifting the spacetime to a Bargmann space, conformal vectors yield conserved quantities for null geodesics and project to non-relativistic conserved charges, with chrono-projective transformations central to this link. A key result is that all conformal Killing vectors of vacuum pp-waves are chrono-projective with constant chrono-projective factor, simplifying the classification of symmetries and associated charges. The paper develops the Bargmann and BJR coordinate formalisms to derive explicit charges and presents several concrete examples (Minkowski, Brdička, CPP, and expansion/screw cases) demonstrating how conformal and chrono-projective symmetries generate nontrivial conserved quantities and novel trajectory-aligning transformations. Overall, the chrono-projective framework clarifies the deep connections between gravitational wave symmetries, null geodesic conservation laws, and the structure of underlying non-relativistic dynamics.

Abstract

The isometries of an exact plane gravitational wave are symmetries for both massive and massless particles. Their conformal extensions are in fact chrono-projective transformations introduced earlier by Duval et al are symmetries for massless particles. Homotheties are universal chrono-projective symmetries for any profile. Chrono-projective transformations also generate new conserved quantities for the underlying non-relativistic systems in the Bargmann framework. Homotheties play a similar role for the lightlike "vertical" coordinate as isometries play for the transverse coordinates.

Figures (3)

• Figure 1: The $U$-translation in Brinkmann coordinates (\ref{['Utr']}) becomes, in BJR coordinates, "screwed".
• Figure 2: For the circularly polarized periodic profile (\ref{['Bprofile']}) with ${{\mathcal{A}_{+}}}(U) = \cos(U),\, {\cal A}_{\times}(U) = \sin(U)$ the homothety (\ref{['homothety0']}) takes the trajectory with initial condition $(U_0,{\bm X}_0,V_0)$ [in magenta] into that with initial condition $(U_0,\chi\,{\bm X}_0,\chi^2\,V_0)$ [in green].
• Figure 3: Dropping the $V$-coordinate and unfolding the transverse CPP trajectory by adding $U$ yields spirals. The screw-transformation (\ref{['CPPscrew']}) (in blue) carries the trajectory in magenta into another trajectory (in green).