Scaling and conformal symmetries for plane gravitational waves

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. [Classical Quantum Gravity 3, 461 (1986); Classical Quantum Gravity 32(17), 175006 (2015)]} and 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.