Dirac brackets for classical radiative observables

TL;DR

This paper develops a gauge-invariant, causal formalism for classical gravitational scattering with radiation, linking observables directly to amplitude matrix elements and computing spin and angular momentum changes up to second order in G.

Contribution

It introduces a new Dirac bracket-based approach combining KMOC formalism for classical gravitational scattering with radiation, enabling direct, gauge-invariant calculations of observables.

Findings

Derived compact, gauge-invariant expressions for scattering observables.

Calculated spin kick and angular momentum change up to second order in G.

Provided examples including impulse, waveform, and radiative fluxes.

Abstract

We introduce a new coherent state expansion of the exponential representation of the S-matrix for the classical gravitational two-body problem. By combining the Kosower-Maybee-O'Connell (KMOC) formalism with the Dirac bracket structure emerging in the classical limit, we derive compact and gauge-invariant expressions for scattering observables in the presence of radiation. This causal formulation bypasses the calculation of KMOC cuts and provides a direct link between observables and a minimal set of classical matrix elements extracted from amplitudes. We illustrate our method with several examples, including the impulse, spin kick, angular momentum, waveform and the related radiative fluxes. Finally, using our formalism we evaluate for the first time the spin kick and the change in angular momentum of each particle up to $\mathcal{O}(G^2 s_1^{j_1} s_2^{j_2})$ with $j_1+ j_2 \leq 11$.