Gauge and Averaging in Gravitational Self-force
by Gralla, Samuel E.
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. 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 gauges satisfying the parity condition. This helps provide a practical foundation for the computation of self-forces in the Kerr background.