General relativistic dynamics of compact binaries at the third post-Newtonian order

The general relativistic corrections in the equations of motion and associated energy of a binary system of pointlike masses are derived at the third post-Newtonian (3PN) order. The derivation is based on a post-Newtonian expansion of the metric in harmonic coordinates at the 3PN approximation. The metric is parametrized by appropriate nonlinear potentials, which are evaluated in the case of two point particles using a Lorentzian version of a Hadamard regularization which has been defined in previous works. Distributional forms and distributional derivatives constructed from this regularization are employed systematically. The equations of motion of the particles are geodesiclike with respect to the regularized metric. Crucial contributions to the acceleration are associated with the nondistributivity of the Hadamard regularization and the violation of the Leibniz rule by the distributional derivative. The final equations of motion at the 3PN order are invariant under global Lorentz transformations, and admit a conserved energy (neglecting the radiation reaction force at the 2.5PN order). However, they are not fully determined, as they depend on one arbitrary constant, which probably reflects a physical incompleteness of the point-mass regularization. The results of this paper should be useful when comparing theory to the observations of gravitational waves from binary systems in future detectors VIRGO and LIGO.