Gravitational Radiation from Massless Particle Collisions

TL;DR

Problem: compute classical gravitational bremsstrahlung in massless particle collisions at leading order in the deflection angle $\theta$. Approach: a simple, nonperturbative-in-$G$ framework using Aichelburg-Sexl shock waves and the Fraunhofer/Huygens setup yields an explicit two-dimensional integral form for the differential GW spectrum, valid in the massless limit. Findings: the massless limit exists and reproduces the zero-frequency limit; the spectrum features a low-frequency logarithmic region and a high-frequency scale-invariant tail with angular confinement $\rho \sim \theta(\omega E)^{-1/2}$, giving a total radiated energy $E^{GW}/E \sim (1/2\pi) \theta^2 \log \theta^{-2}$ under a UV cutoff near $|\mathbf{x}_{\min}| \sim E$. Significance: offers a transparent analytic handle on GW emission in trans-Planckian scattering, clarifies UV sensitivity and soft-graviton connections, and provides a rigorous benchmark for numerical and quantum-gravity approaches.

Abstract

We compute classical gravitational bremsstrahlung from the gravitational scattering of two massless particles at leading order in the (center of mass) deflection angle $θ\sim 4 G \sqrt{s}/b = 8 G E/b \ll 1$. The calculation, although non-perturbative in the gravitational constant, is surprisingly simple and yields explicit formulae --in terms of multidimensional integrals-- for the frequency and angular distribution of the radiation. In the range $ b^{-1} < ω< (GE)^{-1}$, the GW spectrum behaves like $ \log (1/GEω) d ω$, is confined to cones of angular sizes (around the deflected particle trajectories) ranging from $O(θ)$ to $O(1/ωb)$, and exactly reproduces, at its lower end, a well-known zero-frequency limit. At $ω> (GE)^{-1}$ the radiation is confined to cones of angular size of order $θ(GEω)^{-1/2}$ resulting in a scale-invariant ($dω/ω$) spectrum. The total efficiency in GW production is dominated by this "high frequency" region and is formally logarithmically divergent in the UV. If the spectrum is cutoff at the limit of validity of our approximations (where a conjectured bound on GW power is also saturated), the fraction of incoming energy radiated away turns out to be $\frac{1}{2 π} θ^2 \log θ^{-2}$ at leading logarithmic accuracy.