Scaling and conformal symmetries for plane gravitational waves

Abstract

The isometries of an exact plane gravitational wave are symmetries for both massive and massless particles. Their conformal extensions which are in fact Chrono-Projective transformations introduced by Duval et al are symmetries for massless particles and play an important role for the Memory Effect. The generically 5-parameter isometry group extends under special circumstances to a 6 parameter group, further extended by homotheties, which are universal Chrono-Projective symmetries for any profile.

Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields

“…(The argument can be extended to electromagnetic backgrounds, provided the latter are also preserved). Similar results hold for homothetic fields whose conformal factor is a constant [1][2][3][4], or for massless geodesics [5][6][7].…”

“…ψ here is the so-called chrono-projective constant. This condition is satisfied, e.g., by N-type null fluids and by exact plane gravitational waves [7].…”

“…Firstly, we observe that the integral (III.5) can be computed explicitly for conformal vector fields which are "chrono-projective" [7,24], defined by the property…”

“…For massless geodesics the integrals of the motion associated with conformal fields are local cf. (II.6b) and [6,7]. Moreover, under some assumptions, these integrals project onto integrals of the motion for the underlying non-relativistic system.…”

“…In the previous section, we have noted that the conformal vector fields of N-type pp wave space-times satisfy the chrono-projectivity condition (III.9). Therefore, only their [7] and references therein. The v-coordinate can be expressed in a non-local form in terms of the action integral of the projected non-relativistic dynamics,…”