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.

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That is, a diffeomorphism under which the metric pulls back to an (in general position dependent) positive multiple of itself. This should be distinguished from a “Weyl rescaling” of the metric by which the metric is replaced by a positive, in general position dependent, multiple of itself at the same point, i.e., in the same coordinate system.

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