Memory Effect for Particle Scattering in Odd Spacetime Dimensions

TL;DR

This work extends gravitational memory analyses to odd spacetime dimensions by computing retarded fields from classical particle scattering for scalar, electromagnetic, and gravitational perturbations in Minkowski space. A key result is the absence of gravitational memory in all odd dimensions, with scalar and electromagnetic memory exhibiting a dimension-dependent pattern: infinite momentum memory at $d=3$, finite momentum memory at $d=4$, vanishing momentum memory but possible displacement memory at higher odd dimensions, and none for $d>6$ when sources are suitably smeared. The analysis employs the retarded Green's function in odd dimensions, slow-motion expansions, and memory diagnostics via geodesic deviation and momentum transfer, highlighting the crucial role of tail effects inside the light cone. The findings clarify how memory phenomena depend on spacetime dimensionality and the temporal smoothing of sources, informing expectations for radiation-memory behavior beyond standard four-dimensional spacetime.

Abstract

We investigate the gravitational memory effect for linearized perturbations off of Minkowski space in odd spacetime dimensions $d$ by examining the effects of gravitational radiation from classical point particle scattering. We also investigate analogous memory effects for electromagnetic and scalar radiation. We find that there is no gravitational memory effect in all odd dimensions. For scalar and electromagnetic fields, there is no memory effect for $d\geq 7$; for $d=3$ there is an infinite momentum memory effect, whereas for $d=5$ there is no momentum memory effect but the displacement of a test particle will grow unboundedly with time. Our results are further elucidated by analyzing the memory effect for any slowly moving source of compact spatial support in odd dimensions.

Figures (2)

• Figure 1: A spacetime diagram representative of the type of particle scattering we consider. Here we have five particles scattering at event $O$ and three particles emerge. The worldlines of the incoming and outgoing particles are timelike.
• Figure 2: The change in momentum of a scalar charge, initially at rest, as a function of retarded time in 5 (top left), 7 (top right), 9 (lower left) and 11 (lower right) dimensions after smearing out the particle interaction event with a Gaussian normalized in time. The test charge's momentum generically oscillates over the time scale of the interaction $(\tau)$ and then relaxes to its initial value. In this figure, $\tau$ is taken to be much smaller than the range of $U$ plotted.