Gravitational waves and conformal time transformations

Recent interest in the “memory effect” prompted us to revisit relation of gravitational waves oscillators. 50 years ago Niederer (1973) found that an isotropic harmonic oscillator with a constant frequency can be mapped onto free particle. Later Takagi (1990) has shown “time-dependent scaling” extends versus particle correspondence time-dependent when scale factor satisfies Sturm–Liouville equation. More recently Gibbons (0000) pointed out time redefinition is conveniently studied terms Schwarzian derivative. The “Eisenhart-Duval lifts” conformal transformation between Bargmann spaces (Eisenhart, 1928; Duval et al., 1985; 1991; Cariglia 2016). These methods are extended spacetimes which not conformally flat and have profile, then applied geodesic motion plane wave.