Particle motion under the conservative piece of the first-order gravitational self-force is Hamiltonian

The two 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, motion is expected to form a Hamiltonian dynamical system. This has been established to various orders in the post-Newtonian and post-Minkowskian approximations, but not yet in the small mass-ratio regime beyond the leading order of geodesic motion. We show that the motion under the conservative (time even) piece of the first-order self-force is Hamiltonian in any stationary spacetime, and find an explicit expression for the Hamiltonian in terms of a Green's function. In Kerr, this result extends previous results of Fujita et. al. who derived a Hamiltonian description valid only for non-resonant orbits. As applications, we derive a simple necessary condition that the self-force must satisfy for the motion to be integrable and clarify the domain of validity of the first law of binary black hole mechanics in the small mass-ratio regime.

 

 

 

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