Hamiltonian description of the second order scalar self-force

The two body body problem in general relativity is of great theoretical and observational interest, and can be studied in the post-Newtonian, post-Minkowskian and small mass ratio approximations, as well as with effective one body and fully numerical techniques. When gravitational wave dissipation is turned off, the motion is expected to form a Hamiltonian dynamical system. This had been established to various orders in the post-Newtonian and post-Minkowskian approximations. In a previous work, we showed that the motion of a spinning point particle under the conservative (time even) piece of the first-order self force is Hamiltonian in any stationary spacetime, and found an explicit expression for the Hamiltonian in terms of a Greens function.

Including second-order effects is more difficult, due divergences that arise from the point particle limit of the secondary. Furthermore, the separation between conservative and dissipative pieces of the dynamics is not clear, due to ambiguities in their construction. In this work we focus on the scalar self-force up to second order as a toy model of the gravitational case. We explain a new method for obtaining a finite second order scalar self force for point particles and propose a prescription for selecting its conservative piece and the Hamiltonian function that determines the conservative dynamics.