Linearized perturbations of the Kerr spacetime and outer boundary conditions in numerical relativity

Abstract

This paper discusses linearized vacuum gravitational perturbations of the Kerr spacetime in a neighbourhood of future null infinity . Unlike earlier discussion of perturbations of the Kerr spacetime we avoid the use of spheroidal harmonics and harmonic time dependence. Instead we develop the theory in terms of Hertz potentials and spherical harmonics with coupling between modes. The ‘master equation’ is a single complex scalar wave equation which, in the Minkowski limit, reduces to the Euler–Poisson–Darboux equation. We solve this by Picard iteration making extensive use of the flat spacetime Riemann–Green function. As an application we consider the problem of outer boundary conditions for numerical relativity and generalize earlier results of Buchman and Sarbach (2006 Class. Quantum Grav. 23 6709–44, 2007 Class. Quantum Grav. 24 S307–26) for the Schwarzschild case.