The Memory Effect for Plane Gravitational Waves

TL;DR

The paper analyzes gravitational memory for exact plane gravitational waves in vacuum, focusing on how freely falling detectors respond and how soft gravitons are encoded in the spacetime geometry. Using Brinkmann and Baldwin-Jeffery-Rosen coordinates, it studies linearly polarized sandwich waves with profile ${\mathcal{A}}(U)$, deriving the geodesic equations $d^2 \mathbf{X}/dU^2 - \frac{1}{2}\mathrm{diag}({\mathcal{A}},- {\mathcal{A}})\,\mathbf{X}=0$ and showing that transverse momentum is conserved in BJR coordinates while trajectories in Brinkmann coordinates are $\mathbf{X}(U)=P(u)\mathbf{x}^0$, with caustics arising at singular $P(u)$. The analysis confirms Bondi-Pirani's theorem: after the wave, detectors acquire a constant but nonzero asymptotic relative velocity (the velocity memory), in tension with Zel'dovich-Polnarev's original claim. The work also connects post-burst flat spacetime to soft gravitons via a gauge-like transformation with $\ddot P=0$, framing memory effects in terms of Bargmann-space dynamics and offering observational relevance through Doppler tracking, while emphasizing the geometric and Carroll/Carrollgroup structures at play.

Abstract

We give an account of the gravitational memory effect in the presence of the exact plane wave solution of Einstein's vacuum equations. This allows an elementary but exact description of the soft gravitons and how their presence may be detected by observing the motion of freely falling particles. The theorem of Bondi and Pirani on caustics (for which we present a new proof) implies that the asymptotic relative velocity is constant but not zero, in contradiction with the permanent displacement claimed by Zel'dovich and Polnarev. A non-vanishing asymptotic relative velocity might be used to detect gravitational waves through the "velocity memory effect", considered by Braginsky, Thorne, Grishchuk, and Polnarev.

Figures (4)

• Figure 1: Geodesics in Brinkmann coordinates for particles initially at rest for profile ${\mathcal{A}}(U)$ in (\ref{['Kd3']}), modelling gravitational collapse. The trajectories are focused where the effective potential is attractive, and are pushed apart where it is repulsive. In the flat afterzone, the acceleration vanishes.
• Figure 2: The numerical solution of eqn. (\ref{['chiSL']}) for the "collapse profile" (\ref{['Kd3']}) indicates that the coordinates become singular when $u_1 = 0.593342$ and when $u_2 = 1.97472$.
• Figure 3: The analytic (heavy line) and numerical (dashed line) solutions overlap perfectly in the afterzone $u\geq u_0=4$.
• Figure 4: Tissot diagram for the "collapse profile" (\ref{['Kd3']}). The initial circle is deformed to an ellipse which at $u=u_1$ degenerates to a line segment, and so on.