Gravitational Waves and Conformal Time Transformations

TL;DR

This paper explores the relationship between gravitational waves and oscillators through conformal time transformations, extending classical mappings to complex spacetimes and analyzing geodesic motion in gravitational wave backgrounds.

Contribution

It extends oscillator-free particle mappings to non-conformally flat spacetimes with time-dependent profiles, applying conformal transformations to gravitational wave geodesics.

Findings

Extended Niederer and Takagi mappings to complex spacetimes

Applied conformal transformations to analyze geodesics in gravitational waves

Connected oscillator dynamics with gravitational wave memory effects

Abstract

Recent interest in the "memory effect" prompted us to revisit the relation of gravitational aves and oscillators. 50 years ago Niederer [1] found that an isotropic harmonic oscillator with a constant frequency can be mapped onto a free particle. Later Takagi [2] has shown that "time-dependent scaling" extends the oscillator versus free particle correspondence to a time-dependent frequency when the scale factor satisfies a Sturm-Liouville equation. More recently Gibbons [3] pointed out that time redefinition is conveniently studied in terms of the Schwarzian derivative. The oscillator versus free particle correspondence "Eisenhart-Duval lifts" to a conformal transformation between Bargmann spaces [4-7]. These methods are extended to spacetimes which are not conformally flat and have a time-dependent profile, and can then be applied to the geodesic motion in a plane gravitational wave.