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 Poincare charges due to scattering. We parametrize $P^\mu$ and
$J^{\mu\nu}$ by non-perturbative form factors and derive exact relations using
the Poincare 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 Poincare charges due to scattering.

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