Abstract:
We generalize Moncrief's 3+1 formalism for gauge invariant perturbation in Schwarzschild background spacetime to include a point mass in the action. Variation of the action with respect to the metric perturbation gives the master equations for Zerilli-Moncrief function with source term. Variation of the action with respect the point mass's dynamical variables gives Hamilton equations or equations of motion for the point mass. It describes the test mass geodesic motion under the perturbed Schwarzschild background spacetime. In this formalism both the gravitational metric perturbation and point mass motion evolve self-consistently, and one can in principal apply any gauge choice. However, the field perturbation at test mass's location is generically divergent due to the point mass assumption. In order to regularize the field, we take Detweiler's approach to decompose the metric perturbation as singular piece and regular piece. The singular piece has analytical expression in the local Thornr-hartle-Zhang (THZ) frame of the test mass and it does not exert any force on the test mass. We subtract it from the total metric perturbation and obtain a master equation for the regular gauge invariant quantity with effective source. In addition, we also obtained a set of first order differential equations to evolve the regular gauge conditions. |