m-Mode Regularization Scheme for the Self Force in Kerr Spacetime

Abstract

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.