Dimensional regularization of the gravitational interaction of point masses in the ADM formalism

TL;DR

This work demonstrates that dimensional regularization resolves 3PN ambiguities in the ADM two-body Hamiltonian by taking the $d\to3$ limit of the $d$-dimensional calculation, with pole terms cancelling and a finite Hamiltonian obtained. It shows agreement between multiple calculation methods for the $\Delta H_{\text{3PN}}$ and fixes the ambiguity parameters to $\omega_{\text{kinetic}} = 41/24$ and $\omega_{\text{static}} = 0$. The result solidifies the consistency of dimensional continuation with Hadamard regularization outcomes and provides a robust foundation for high-precision modeling of compact binaries in general relativity. The methodology and findings have direct implications for accurate gravitational-wave predictions and tests of GR in the strong-field regime.

Abstract

The ADM formalism for two-point-mass systems in $d$ space dimensions is sketched. It is pointed out that the regularization ambiguities of the 3rd post-Newtonian ADM Hamiltonian considered directly in $d=3$ space dimensions can be cured by dimensional continuation (to complex $d$'s), which leads to a finite and unique Hamiltonian as $d\to3$. Some so far unpublished details of the dimensional-continuation computation of the 3rd post-Newtonian two-point-mass ADM Hamiltonian are presented.