First-order perturbative Hamiltonian equations of motion for a point particle orbiting a Schwarzschild black hole

We formulate a spherical harmonically decomposed 1þ 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 spacetime geometry, combine that with the action for a point particle moving through this spacetime, and then obtain Hamiltonian equations of motion for metric perturbations, the particle’s coordinates, as well as its canonical momenta. Hamiltonian equations for the metric perturbation and their conjugate momenta, for even and odd parities, reduce to Zerilli-Moncrief and Regge-Wheeler master equations with source terms, which are gauge invariant, plus auxiliary equations that specify the 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 lead to those well-known source terms. In this way, spacetime geometry and particle motion can be evolved in a self-consistent manner, in principle in any gauge. However, the pointparticle 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 analytically using the 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. For a singular field with a minimum quality, one can adopt the Lorenz gauge condition, which we have recast into our formalism: for each l and m, we have two wave equations to evolve odd and even parity gauge invariant quantities and eigh first-order differential equations to fix the Lorenz gauge and determine the metric components.