Gravitational waves and conformal time transformations

Abstract

Recent interest in the “memory effect” prompted us to revisit the relation of gravitational wavesto oscillators. 50 years ago Niederer (1973) found that an isotropic harmonic oscillator with a constant frequency can be mapped onto a free particle. Later Takagi (1990) 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 (0000) 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 (Eisenhart, 1928; Duval et al., 1985; Duval et al., 1991; Cariglia et al., 2016). 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.