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

TL;DR

The study demonstrates that dimensional regularization can consistently regularize the 3PN dynamics of two point masses in harmonic coordinates by absorbing pole terms into shifts of the world-lines, yielding finite, Lorentz-invariant equations of motion. By comparing with Hadamard regularization and carefully handling distributional derivatives, the authors fix the previously undetermined 3PN parameter lambda to -1987/3080, in agreement with ADM results and surface-integral methods. The work provides a nontrivial consistency check for dim. reg. in classical gravity, and its results reinforce the reliability of 3PN templates for gravitational-wave astronomy. Moreover, it establishes a controlled framework to extend dimensional regularization to higher PN orders and to regularize gravitational radiation quantities (e.g., at 3.5PN).

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 1/(d-3)] 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_munu(x). 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 heretofore unknown 3PN parameter lambda to be: lambda = - 1987/3080. This value is fully consistent with the recent determination of the equivalent 3PN static ambiguity parameter, 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.

Figures (8)

• Figure 1: Diagrams representing "compact" contributions to acceleration $\mathbf{a}_1$. The dotted line represents $\Box^{-1}$, the cross represents the field point $\mathbf{x}$ (here taken on the first worldline), and the bullet represents either a source point or (in the Figures below) an intermediate nonlinear vertex.
• Figure 2: Quadratically non-compact contributions to acceleration $\mathbf{a}_1$.
• Figure 3: Cubically non-compact contributions to acceleration $\mathbf{a}_1$.
• Figure 4: Other non-linear contributions to acceleration $\mathbf{a}_1$.
• Figure 5: Quartically non-compact contributions which do not occur in our calculation of acceleration $\mathbf{a}_1$ at the 3PN order.