Conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity

The fourth post-Newtonian (4PN) two-body dynamics has been recently tackled by several different approaches: effective field theory, Arnowitt-Deser-Misner Hamiltonian, action-angle-Delaunay averaging, effective-one-body, gravitational self-force, first law of dynamics, and Fokker action. We review the achievements of these approaches and discuss the complementarity of their results. Our main conclusions are: (i) the results of the first complete derivation of the 4PN dynamics [T. Damour, P. Jaranowski, and G. Sch\"afer, Phys. Rev. D 89, 064058 (2014)] have been, piecewise, fully confirmed by several subsequent works; (ii) the results of the Delaunay-averaging technique [T. Damour, P. Jaranowski, and G. Sch\"afer, Phys. Rev. D 91, 084024 (2015)] have been confirmed by several independent works; and (iii) several claims in a recent harmonic-coordinates Fokker-action computation [L. Bernard et al., arXiv:1512.02876v2] are incorrect, but can be corrected by the addition of a couple of ambiguity parameters linked to subtleties in the regularization of infrared and ultraviolet divergences.