First-Order Perturbative Hamiltonian Equations of Motion for a Point Particle Orbiting a Schwarzschild Black Hole

Abstract

We formulate a spherical harmonically decomposed 1+1 scheme to self-consistently evolve the trajectory of a point particle and its gravitational metric perturbation to a Schwarzschild background spacetime. Following the work of Moncrief, we write down an action for perturbations in space-time geometry, combine that with the action for a point-particle, and then obtain Hamiltonian equations of motion for metric perturbations, the particle's coordinates, as well as their canonical momenta. Hamiltonian equations for the metric-perturbation and their conjugate momenta reduce to Zerilli-Moncrief and Regge-Wheeler master equations with source terms, which are gauge invariant, plus auxiliary equations that specify gauge. Hamiltonian equations for the particle, on the other hand, now include effect of metric perturbations - with these new terms derived from the same interaction Hamiltonian that had lead to those well-known source terms. In this way, space-time geometry and particle motion can be evolved in a self-consistent manner, in principle in any gauge. However, the point-particle nature of our source requires regularization, and we outline how the Detweiler-Whiting approach can be applied. In this approach, a singular field can be obtained using Hadamard decomposition of the Green's function and the regular field, which needs to be evolved numerically, is the result of subtracting the singular field from the total metric perturbation. In principle, any gauge that has the singular-regular field decomposition is suitable for our self-consistent scheme. In reality, however, this freedom is only possible if our singular field has a high enough level of smoothness. In the case of Lorenz gauge, for each l and m, we have 2 wave equations to evolve gauge invariant quantities and 8 first order differential equations to fix the gauge and determine the metric components.