Gravitational Waves and Conformal Time Transformations

TL;DR

The paper develops a framework to map oscillator dynamics to free motion in gravitational-wave spacetimes using conformal time redefinitions and the Eisenhart–Duval (Bargmann) lift. It extends the Niederer correspondence to anisotropic, time-dependent oscillators by incorporating the Schwarzian derivative and a $\theta$-time formulation, enabling mappings for non-conformally-flat plane waves. The authors apply the method to linearly and circularly polarized waves, showing how Lukash and CPP spacetimes are related by conformal Bargmann transformations and yielding explicit constant-frequency descriptions. They discuss quantum aspects via exact propagators for quadratic systems and Maslov corrections, indicating a path to semiclassical and quantum analyses. Overall, the work broadens classical GW geodesic analysis and opens avenues for quantum extensions in curved backgrounds.

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.

Figures (4)

• Figure 1: (i) By \ref{['J0Y0eq']} the Takagi scale factor $a(t)\equiv a_{+}(t)$ is a combination of Bessel functions of the first and second kind with $\theta=2e^{-t/2}$ in its argument. $\textcolor{red}{a(t)}$ (shown in red) is regular and behaves as $\textcolor{red}{a(t) \approx \mathop{\rm const.}\nolimits}$ for large $t$. $\textcolor{blue}{a(t)}$ (shown in blue) has instead a zero, and $\textcolor{blue}{a(t) \approx -t}$ for large $t$. (ii) The fake time $\tau(t)$ can be found by numerical integration of $a^{-2}(t)$ in \ref{['Tredef']}. $\tau$ (shown in red) is regular and $\textcolor{red}{\tau \propto t}$ for large $t$. $\tau$ (shown in blue) is singular where the $a(t)$ is zero and $\tau$$\propto -{t}^{-1}$ for large $t$.
• Figure 2: (i) Scale factors $a_{-}(t)$ alias particle trajectories and (ii) fake times $\tau_-(t)$ for the repulsive potential \ref{['repfreq']}, cf. \ref{['reptau']}.
• Figure 3: Motions in redefined coordinates $(\zeta_{\pm}, \theta)$ for (i) the attractive \ref{['thetat']} and resp. (ii) the repulsive \ref{['repfreq']} frequencies. For large values of $\theta$ the perturbation term falls off and we are left approximately with harmonic motion : $\zeta_+(\theta)$ is approximately a shifted sinus/cosinus, and $\zeta_-(\theta)$ is approximately an exponential. (Caveat: the scales of (i) and (ii) are different .)
• Figure 4: Motion in the transverse plane for the choices (i) $\zeta_{+} \sim Y_0$ and for (ii) $\zeta_{+} \sim J_0$, completed, in both cases, with $\zeta_- \sim K_0$. For large $\theta$$\zeta_-(\theta)$ falls of exponentially and the $\zeta_+$ component moves approximately along a shifted sinus curve. Two more figures could be produced by other pairings of the curves taken from fig.\ref{['zetafigs']}.