Gravitational shock waves and scattering amplitudes

TL;DR

This work builds a covariant framework linking classical GR solutions to the classical parts of scattering amplitudes, deriving the gravitational Aichelburg–Sexl shock wave from an off-shell three-point amplitude and generalizing to $D$ dimensions. It establishes a classical double copy between gravitational and electromagnetic shock waves and reveals a spin–shift relation $x^{\mu}\to x^{\mu}-i a^{\mu}$ that maps spinless shock waves to gyratons, providing all-spin gyraton solutions. The approach naturally reproduces the distributional shock profile without singular coordinate transformations and explains the exactness at linear order via absence of classical higher-loop contributions for three-point functions. It also recovers the high-energy scattering angle $\theta$ in the massless limit, $\theta=\frac{4 G_N \sqrt{s}}{b}$, and extends to all orders in spin, matching ACV results and highlighting the power of amplitudes methods for classical GR phenomena.

Abstract

We study gravitational shock waves using scattering amplitude techniques. After first reviewing the derivation in General Relativity as an ultrarelativistic boost of a Schwarzschild solution, we provide an alternative derivation by exploiting a novel relation between scattering amplitudes and solutions to Einstein's field equations. We prove that gravitational shock waves arise from the classical part of a three point function with two massless scalars and a graviton. The region where radiation is localized has a distributional profile and it is now recovered in a natural way, thus bypassing the introduction of singular coordinate transformations as used in General Relativity. The computation is easily generalized to arbitrary dimensions and we show how the exactness of the classical solution follows from the absence of classical contributions at higher loops. A classical double copy between gravitational and electromagnetic shock waves is also provided and for a spinning source, using the exponential form of three point amplitudes, we infer a remarkable relation between gravitational shock waves and spinning ones, also known as gyratons. Using this property, we infer a family of exact solutions describing gravitational shock waves with spin. We then compute the phase shift of a particle in a background of shock waves finding agreement with an earlier computation by Amati, Ciafaloni and Veneziano for particles in the high energy limit. Applied to a gyraton, it provides a result for the scattering angle to all orders in spin.