An asymptotic framework for gravitational scattering

TL;DR

This work constructs a comprehensive, symmetry-driven framework for classical gravitational scattering in four-dimensional, asymptotically flat spacetimes by unifying the five asymptotic regions under a single BMS group. It develops Beig–Schmidt expansions, imposes physically motivated constraints to realize the BMS algebra, and derives explicit expressions for BMS charges at i^±, I^±, and i^0, demonstrating their global conservation via flux balance; it also provides new, frame-independent definitions of spin, scattering angle, and impact parameter. A central achievement is the detailed matching of charges across all infinities, including antipodal relations across spatial infinity, and the analysis of gravitational memory and its decomposition into soft and hard contributions. The framework offers a conceptually clean, background-free way to interpret gravitational scattering in terms of asymptotic symmetries, with potential implications for quantum gravity and S-matrix approaches, while also highlighting practical challenges in computing these charges in realistic spacetimes.

Abstract

Asymptotically flat spacetimes have been studied in five separate regions: future/past timelike infinity $i^\pm$, future/past null infinity $\mathcal{I}^\pm$, and spatial infinity $i^0$. We formulate assumptions and definitions such that the five infinities share a single Bondi-Metzner-Sachs (BMS) group of asymptotic symmetries and associated charges. We show how individual ingoing/outgoing massive bodies may be ascribed initial/final BMS charges and derive global conservation laws stating that the change in total charge is balanced by the corresponding radiative flux. This framework provides a foundation for the study of asymptotically flat spacetimes containing ingoing and outgoing massive bodies, i.e., for generalized gravitational scattering. Among the new implications are rigorous definitions for quantities like initial/final spin, scattering angle, and impact parameter in multi-body spacetimes, without the use of any preferred background structure.

Figures (4)

• Figure 1: $3\to2$ scattering
• Figure 2: Diagrams for asymptotically flat spacetime. Future timelike infinity $i^+$ (top left) is a unit hyperboloid obtained from rescaling the induced metric on late-time hyperboloids. Finite-sized massive bodies (red, blue) approach points of $i^+$ corresponding to their asymptotic velocities. Future null infinity $\mathcal{I}^+$ (top right) is a cylinder, the celestial sphere cross time, whose cross-sections (colors) represent the arrival times of light rays (correspondingly colored). In our treatment, only the angular metric is rescaled; $\mathcal{I}^+$ does not arise from the induced metric on any bulk surface. Spatial infinity $i^0$ (bottom right) is defined analogously to $i^+$ using suitably rescaled Lorentzian hyperboloids at large radius $\rho$. Past null infinity $\mathcal{I}^-$ and past timelike infinity $i^-$ are the time-reverses of their future counterparts. The five infinities are naturally represented on a "puzzle piece" diagram (bottom left).
• Figure 3: Overlap regions
• Figure 4: The five infinities can be represented as the boundaries of the region defined by $\vert t^2-r^2 \vert < \mathcal{T}$ and $r<\mathcal{R}$ for large constants $\mathcal{T}$ and $\mathcal{R}$. Here we plot this boundary for a sequence (a) $\mathcal{T}=0.25$, $\mathcal{R}=1$; (b) $\mathcal{T}=25$ and $\mathcal{R}=10$; and (c) $\mathcal{T}=2500$ and $\mathcal{R}=100$, showing the 4 overlapping boundary regions as bullets. The "puzzle piece" shape is invariant on a Minkowski spacetime diagram (top), but converges to a diamond on a conformal diagram (bottom). (Here $T=U+V$ and $R=V-U$ are the standard conformal coordinates constructed from $\tan U = t-r$ and $\tan V=t+r$.) The puzzle piece can adequately represent all asymptotic regions, while the conformal diagram does not resolve $i^+$, $i^-$ and $i^0$, nor any of the overlap regions.