Angular momentum loss in gravitational scattering, radiation reaction, and the Bondi gauge ambiguity

Recently, Damour computed the radiation reaction on gravitational scattering as the (linear) response to the angular momentum loss which he found to be of <math altimg="si1.svg"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math> in the gravitational constant. This is a puzzle because any amplitude calculation would predict both radiated energy and radiated angular momentum to start only at <math altimg="si2.svg"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math>. Another puzzle is that the resultant radiation reaction, of <math altimg="si2.svg"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math>, is nevertheless correct and confirmed by a number of direct calculations. We ascribe these puzzles to the BMS ambiguity in defining angular momentum. The loss of angular momentum is to be counted out from the ADM value and, therefore, should be calculated in the so-called canonical gauge under the BMS transformations in which the remote-past limit of the Bondi angular momentum coincides with the ADM angular momentum. This calculation correctly gives the <math altimg="si2.svg"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math> radiative loss. On the other hand, we introduce a gauge in which the Bondi light cones tend asymptotically to those emanating from the center of mass world line. We find that the angular momentum loss in this gauge is precisely the one used by Damour for his radiation reaction result. We call this new gauge “intrinsic” and argue that, although the radiated angular momentum is to be computed in the canonical gauge, any mechanical calculation of gauge-dependent quantities – such as angular momentum – gives the result in the intrinsic gauge. Therefore, it is this gauge that should be used in the linear response formula. This solves the puzzles and establishes the correspondence between the intrinsic mechanical calculations and the Bondi formalism.