Memory and supertranslations on plane wave spacetimes: an on-shell perspective

TL;DR

The paper addresses how gravitational memory manifests for a particle moving through a plane-wave spacetime using on-shell amplitudes, correcting prior analyses that assumed weak memory. It derives the first all-memoried, tree-level waveform expressed through Synge's world function, with explicit tail terms and a careful treatment of memory via the memory matrix $c$ and caustics. A central result is an exact all-orders memory waveform on the plane-wave background, including the tail structure and the dependence on the BMS frame, with soft-dressing corresponding to a Veneziano-Vilkovisky supertranslation and a background waveshape rotation. These findings illuminate the interplay between memory, tail propagation, and boundary conditions in non-flat backgrounds, providing a controlled arena to study gravitational observables beyond flat spacetime and guiding future extensions to more general spacetimes.

Abstract

We revisit the computation of the classical gravitational waveform for a particle moving in a plane wave background using on-shell amplitudes. We emphasize the relationship between gravitational memory and the boundary conditions of external scattering states, which were neglected in previous works. We then provide the first tree-level expression for the waveform that captures all memory effects. The waveform is presented in terms of Synge's world function, with explicit tail terms, and a smooth weak memory limit. We also discuss the choice of BMS frame for the waveform on a plane wave background. In flat space, this corresponds to a choice of soft dressing of the initial state. We show that on a plane wave background, this dressing becomes a supertranslation of the waveform, in addition to a phase shift in the waveshape of the background.

Figures (2)

• Figure 1: The behaviour of the phase $\varphi_{k}^{\text{in}}(x)$ defined in \ref{['eq:asyphase']} on the future celestial sphere at $u = 0.6$, with parameters $k_1 = 0.5$, $k_2 = - 0.125$, $k_+ = 1$. The north pole corresponds to $\hat{x}^3 = 1$. The behaviour of this phase should be contrasted to the everywhere rapidly phase of the typical $e^{i k \cdot x}$, though this is compensated by the overall $1/r$ behaviour of the solution \ref{['asympScalarBehav']}
• Figure 2: An idealised depiction of the tail effect on a gravitational plane wave with $m =0$, projected onto the $(x^+, x^-)$ plane. The solid orange boundary represents points where Synge's world function $\sigma(x, y) = 0$ and is the past lightcone of the point $x$. On the other hand, the shaded orange region has $\sigma(x, y)<0$. Green's functions for gravitational radiation in plane waves have support in both regions Harte:2013dba, violating Huygen's principle. Support from the $\sigma(x, y)>0$ region is often known as the 'tail'. The blue line represents an example massive geodesic.