Gauge and averaging in gravitational self-force

A difficulty with previous treatments of the gravitational self-force is that an explicit formula for the force is available only in a particular gauge (Lorenz gauge), where the force in other gauges must be found through a transformation law once the Lorenz-gauge force is known. For a class of gauges satisfying a “parity condition” ensuring that the Hamiltonian center of mass of the particle is well-defined, I show that the gravitational self-force is always given by the angle average of the bare gravitational force. To derive this result I replace the computational strategy of previous work with a new approach, wherein the form of the force is first fixed up to a gauge-invariant piece by simple manipulations, and then that piece is determined by working in a gauge designed specifically to simplify the computation. This offers significant computational savings over the Lorenz gauge, since the Hadamard expansion is avoided entirely and the metric perturbation takes a very simple form. I also show that the rest mass of the particle does not evolve due to first-order self-force effects. Finally, I consider the “mode sum regularization” scheme for computing the self-force in black hole background spacetimes, and use the angle-average form of the force to show that the same mode-by-mode subtraction may be performed in all parity-regular gauges. It appears plausible that suitably modified versions of the Regge-Wheeler and radiation gauges (convenient to Schwarzschild and Kerr, respectively) are in this class.

Figure 1

A diagram illustrating the geometrical setup of the mode-sum regularization argument. The particle is at the pole of rotated Boyer-Lindquist coordinates. The mode decomposition is taken relative to the background coordinate sphere, while the self-force is given by an average over the local inertial sphere. For the change in bare force, a general theorem relates the mode sum at the particle/pole to the average over the polar circle, which agrees with the average over the local inertial sphere when the parity condition is satisfied.