Dimensional regularization of the gravitational interaction of point masses

TL;DR

This paper applies dimensional regularization within the ADM formalism to derive a finite, unambiguous 3PN Hamiltonian for two point masses in gravity, resolving previous ambiguities and confirming gauge consistency.

Contribution

It introduces a novel use of dimensional regularization at 3PN order to obtain a unique Hamiltonian and resolve longstanding ambiguities in gravitational two-body dynamics.

Findings

Dimensional continuation yields a finite 3PN Hamiltonian without ambiguities.

The static parameter $\omega_s$ is determined to be zero.

The kinetic parameter $\omega_k$ is found to be 41/24, consistent with Poincaré invariance.

Abstract

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, $\omega_s=0$. 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 $\omega_k$, giving the unique answer compatible with global Poincar\'e invariance ($\omega_k={41/24}$) by summing $\sim50$ different dimensionally continued contributions.