$m$-mode regularization scheme for the self-force in Kerr spacetime

We present a new, simple method for calculating the scalar, electromagnetic, and gravitational self-forces acting on particles in orbit around a Kerr black hole. The standard “mode-sum regularization” approach for self-force calculations relies on a decomposition of the full (retarded) perturbation field into multipole modes, followed by the application of a certain mode-by-mode regularization procedure. In recent years several groups have developed numerical codes for calculating black hole perturbations directly in $2+1$ dimensions (i.e., decomposing the azimuthal dependence into $m$-modes, but refraining from a full multipole decomposition). Here we formulate a practical scheme for constructing the self-force directly from the $2+1$-dimensional $m$-modes. While the standard mode-sum method is serving well in calculations of the self-force in Schwarzschild geometry, the new scheme should allow a more efficient treatment of the Kerr problem.