Self-force via m-mode regularization and 2+1D evolution. III. Gravitational field on Schwarzschild spacetime

This is the third in a series of papers aimed at developing a practical time-domain method for self-force calculations in Kerr spacetime. The key elements of the method are (i) removal of a singular part of the perturbation field with a suitable analytic "puncture," (ii) decomposition of the perturbation equations in azimuthal (m-)modes, taking advantage of the axial symmetry of the Kerr background, (iii) numerical evolution of the individual m- modes in 2 + 1 dimensions with a finite difference scheme, and (iv) reconstruction of the local self-force from the mode sum. Here we report a first implementation of the method to compute the gravitational self-force. We work in the Lorenz gauge, solving directly for the metric perturbation in 2 + 1 dimensions, for the case of circular geodesic orbits. The modes m 0, 1 contain nonradiative pieces, whose time-domain evolution is hampered by certain gauge instabilities. We study this problem in detail and propose ways around it. In the current work we use the Schwarzschild geometry as a platform for development; in a forthcoming paper-the fourth in the series-we apply our method to the gravitational self-force in Kerr geometry.