Radiated Angular Momentum and Dissipative Effects in Classical Scattering

We present a new formula for the angular momentum $J^{\mu \nu }$ carried away by gravitational radiation in classical scattering. This formula, combined with the known expression for the radiated linear momentum $P^{\mu }$, completes the set of radiated Poincaré charges due to scattering. We parametrize $P^{\mu }$ and $J^{\mu \nu }$ by nonperturbative form factors and derive exact relations using the Poincaré algebra. There is a contribution to $J^{\mu \nu }$ due to static (zero-frequency) modes, which can be derived from Weinberg’s soft theorem. Using tools from scattering amplitudes and effective field theory, we calculate the radiated $J^{\mu \nu }$ due to the scattering of two spinless particles to third order in Newton’s constant $G$, but to all orders in velocity. Our form-factor analysis elucidates a novel relation found by Bini, Damour, and Geralico between energy and angular momentum loss at $\mathcal{O}(G^3)$. Our new results have several nontrivial implications for binary scattering at $\mathcal{O}(G^4)$. We give a procedure to bootstrap an effective radiation reaction force from the loss of Poincaré charges due to scattering.

Figure 1

The initial configuration of the binary system. The spatial momenta of the two particles are along the $x$ direction, and the impact vectors $b_{1,2}^{\mu }$ are along the $y$ direction.

Figure 2

Sample diagrams depicting contributions to ${\mathcal{T}}^{\mu \nu }(k)$. Straight and wavy lines denote matter and gravitons. Radiation can be emitted from any of the external matter legs. The top row illustrates relevant diagrams for the zero-frequency limit in Eq. (8). The solid and hallow blobs correspond to the $\mathcal{O}(G)$ and $\mathcal{O}(G^2)$ deflections. The bottom row depicts ${\mathcal{T}}^{\mu \nu }(k)$ at $\mathcal{O}(G)$ with general frequency.