Conformally related vacuum gravitational waves, and their symmetries

TL;DR

The paper develops a framework for conformally relating vacuum gravitational waves using a special Möbius time redefinition U → f(\tilde U), preserving vacuum conditions when the Schwarzian term vanishes, and shows how the transformed profile H transforms to tilde H via tilde H = f'(\tilde U) H[\cdot] + (1/4) S_{\tilde U}(f)\tilde X^2. It identifies two main outcomes: (i) interchanging globally defined GWs with a shifted or squeezed profile that forms (approximate) sandwich waves, and (ii) constructing a class of ${\rm O}(2,1)$-conformally invariant, anisotropic pp-waves that can remain invariant under SMCT; these include waves inspired by polar-molecule potentials. The study leverages a Sturm–Liouville–Killing correspondence to derive geodesics analytically and numerically for linearly and circularly polarized GWs, including damped rational-time cases and piecewise solutions. A molecular-physics analogy is developed via the Bargmann framework, mapping the gravitational problem to a non-relativistic particle in an anisotropic inverse-square potential, and the work further explores approximate sandwich waves with Gaussian envelopes and historical connections to Bohlin–Arnold duality. Overall, the results illuminate how conformal symmetries shape GW profiles, particle trajectories, and memory effects, with implications for both mathematical structure and physical interpretation.

Abstract

A special conformal transformation which carries a vacuum gravitational wave into another vacuum one is built by using Möbius-redefined time. It can either transform a globally defined vacuum wave into a vacuum sandwich wave, or carry the gravitational wave into itself. The first type, illustrated by linearly and circularly polarized vacuum plane gravitational waves, permutes the symmetries and the geodesics. Our pp waves with conformal O(2,1) symmetry of the second type, which seem to ahve escaped attention so far, are anisotropic generalizations of the familiar inverse-square profile. An example inspired by molecular physics, for which the particle can escape, or perform periodic motion, or fall into the singularity is studied in detail.

Figures (16)

• Figure 1: The conformal factor \ref{['N(U)']} determines the width and position of the wave \ref{['S-Gibbons']}: \ref{['Delta1']} is for parameters $\rho=1$ and $\delta=0$ and FIG. \ref{['Delta2']} is for $\rho=10$ and $\delta=1$, respectively.
• Figure 3: \ref{['LPP1']}: a particle in the LPP GW space-time \ref{['usual-LPP']} of Brdicka (drawn in steel blue) oscillates. It should be compared with what happens in the rational LPP GW \ref{['LPP-pulse']}, obtained by squeezing the wave as in \ref{['S-Gibbons']} and drawn in dark orchid in FIG. \ref{['LPP2']}, for which the particle initially in rest is shaken by the GW and then escapes with straightened-out velocity due to the damping factor $U^{-1}$ after the wave has passed.
• Figure 4: \ref{['LPP1anal']} shows analytically found geodesics for the LPP (Brdicka) \ref{['usual-LPP']}, and \ref{['LPP2anal']} for the rational LPP in \ref{['Xsol-rLPP']}-\ref{['Ysol-rLPP']}, metric respectively. These plots should be compared with the numerical ones in FIG. \ref{['LPP-plots']}.
• Figure 5: \ref{['CPP1']} : in the usual CPP GW (depicted in steel blue) the particle performs a "gear wheel - like" motion. \ref{['CPP2']} : in the rational CPP GW (in dark orchid) the particle which is at rest before the GW arrives escapes along an expanding screw after the GW has passed. For large $U$ its velocity becomes approximately constant due to the damping factor $\widetilde{U}^{-4}$ in \ref{['CPP-pulse-metric']}.
• Figure 6: The analytic rational CPP solution \ref{['Xsol-rLPPpiecewise']}-\ref{['Ysol-rLPPpiecewise']}, to be compared with the numerically found one in \ref{['CPP2']}.