A previously found momentum-dependent regularization ambiguity in the third post-Newtonian two pointmass Arnowitt-Deser-Misner Hamiltonian is shown to be uniquely determined by requiring global Poincaré invariance. The phase-space generators realizing the Poincaré algebra are explicitly constructed.PACS number͑s͒: 04.25.Nx, 04.20.Fy, 04.30.Db, 97.60.Jd The equations of motion of a gravitationally interacting two point-mass system have been derived some years ago up to the 5/2 post-Newtonian ͑2.5PN͒ approximation, 1 in harmonic coordinates ͓1-3͔. Recently, it has been possible to derive the third post-Newtonian ͑3PN͒ Hamiltonian of a two point-mass system ͓4͔ within the canonical formalism of Arnowitt, Deser and Misner ͑ADM͒ ͓5͔. It was found that, at the 3PN level, the use of Dirac-delta-function sources to model the two-body system causes the appearance of badly divergent integrals which ͑contrary to what happened at the 2.5PN ͓3,6͔ and 3.5PN ͓7͔ levels͒ cannot be unambiguously regularized ͓4,8,9͔. The ambiguities in the regularization of the 3PN divergent integrals are parametrized by two quantities: static and kinetic .Prompted by a recent remark ͓10͔, the purpose of this work is to show that requiring the ͑global͒ Poincaré invariance of the 3PN ADM Hamiltonian dynamics uniquely determines one ͑and only one͒ of these regularization ambiguities: namely, the ''kinetic ambiguity'' parameter kinetic . ͓The ''static ambiguity'' static remains unconstrained because it parametrizes a O(c Ϫ6 ) Galileo-invariant additional contribution to the 3PN Hamiltonian.͔ Parallel work in the harmonic-coordinates approach to 3PN dynamics has recently obtained similar results ͓18͔.Note that general relativity admits ͑when considering isolated systems͒ the full Poincaré group as a global symmetry.

We show how to use dimensional regularization to determine, within the Arnowitt-Deser-Misner canonical formalism, the reduced Hamiltonian describing the dynamics of two gravitationally interacting point masses. Implementing, at the third post-Newtonian (3PN) accuracy, our procedure we find that dimensional continuation yields a finite, unambiguous (no pole part) 3PN Hamiltonian which uniquely determines the heretofore ambiguous "static" parameter: namely, ωs = 0. Our work also provides a remarkable check of the perturbative consistency (compatibility with gauge symmetry) of dimensional continuation through a direct calculation of the "kinetic" parameter ω k , giving the unique answer compatible with global Poincaré invariance (ω k = 41 24 ) by summing ∼ 50 different dimensionally continued contributions.

The paper presents the conservative dynamics of two-body point-mass systems up to the third post-Newtonian order (1/c 6 ). The two-body dynamics is given in terms of a higher order ADM Hamilton function which results from a third post-Newtonian Routh functional for the total field-plus-matter system. The applied regularization procedures, together with making use of distributional differentiation of homogeneous functions, give unique results for the terms in the Hamilton function apart from the coefficient of the term (νp i ∂ i ) 2 r −1 . The result suggests an invalidation of the binary point-mass model at the third post-Newtonian order.PACS number(s): 04.

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We complete the analytical determination, at the 4th post-Newtonian (4PN) approximation, of the conservative dynamics of gravitationally interacting two-point-mass systems. This completion is obtained by resolving the infra-red ambiguity which had blocked a previous 4PN calculation [P. Jaranowski and G. Schäfer, Phys. Rev. D 87, 081503(R) (2013)] by taking into account the 4PN breakdown of the usual near-zone expansion due to infinite-range tail-transported temporal correlations found long ago [L. Blanchet and T. Damour, Phys. Rev. D 37, 1410 (1988)]. This leads to a Poincaré-invariant 4PN-accurate effective action for two masses, which mixes instantaneous interaction terms (described by a usual Hamiltonian) with a (time-symmetric) nonlocal-in-time interaction.