Niederer's transformation, time-dependent oscillators and polarized gravitational waves

TL;DR

This paper explores the use of Niederer's transformation to relate time-dependent oscillators and pp-waves, revealing new solvable models of polarized gravitational waves with applications to electromagnetic interactions.

Contribution

It provides a geometric interpretation of Niederer's transformation via pp-waves and identifies new analytically solvable gravitational wave models with conformal symmetry.

Findings

Maximal conformal symmetry in gravitational waves allows analytical solutions.

Circularly polarized gravitational waves exhibit specific classical effects.

Additional integrals of motion are linked to conformal invariants.

Abstract

It is noted that the Niederer transformation can be used to find the explicit relation between time-dependent linear oscillators, including the most interesting case when one of them is harmonic. A geometric interpretation of this correspondence is provided by certain subclasses of pp-waves; in particular the ones strictly related to the proper conformal transformations. This observation allows us to show that the pulses of plane gravitational wave exhibiting the maximal conformal symmetry are analytically solvable. Particularly interesting is the circularly polarized family for which some aspects (such as the classical cross section, velocity memory effect and impulsive limit) are discussed in more detail. The role of the additional integrals of motion, associated with the conformal generators, is clarified by means of Ermakov-Lewis invariants. Possible applications to the description of interaction of electromagnetic beams with matter are also indicated.