Dimensional regularization of the third post-Newtonian dynamics of point particles in harmonic coordinates

Abstract

Dimensional regularization is used to derive the equations of motion of two point masses in harmonic coordinates. At the third post-Newtonian (3PN) approximation, it is found that the dimensionally regularized equations of motion contain a pole part [proportional to $(d\ensuremath{-}{3)}^{\ensuremath{-}1}]$ which diverges as the space dimension d tends to 3. It is proven that the pole part can be renormalized away by introducing suitable shifts of the two world-lines representing the point masses, and that the same shifts renormalize away the pole part of the ``bulk'' metric tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}{(x}^{\ensuremath{\lambda}}).$ The ensuing, finite renormalized equations of motion are then found to belong to the general parametric equations of motion derived by an extended Hadamard regularization method, and to uniquely determine the 3PN ambiguity parameter $\ensuremath{\lambda}$ to be $\ensuremath{\lambda}=\ensuremath{-}1987/3080.$ This value is fully consistent with the recent determination of the equivalent 3PN ``static ambiguity'' parameter, ${\ensuremath{\omega}}_{s}=0,$ by a dimensional-regularization derivation of the Hamiltonian in Arnowitt-Deser-Misner coordinates. Our work provides a new, powerful check of the consistency of the dimensional regularization method within the context of the classical gravitational interaction of point particles.