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Dimensional regularization of the gravitational interaction of point masses

Thibault Damour, Piotr Jaranowski, Gerhard Schäfer

TL;DR

The paper uses dimensional regularization within the ADM formalism to derive the conservative 3PN Hamiltonian for two gravitationally interacting point masses. It proves that the dimensionally continued Hamiltonian is finite as $d\to3$ with no pole terms, uniquely fixing the kinetic and static 3PN parameters by enforcing gauge and Poincaré invariance, yielding $ω_k=41/24$ and $ω_s=0$. This resolves longstanding ambiguities from previous Hadamard/Riesz approaches, strengthens the perturbative consistency of the method, and has significant implications for accurate gravitational-wave modeling and initial-data construction in numerical relativity. The work also highlights the inconsistency of conformally flat truncations at 3PN and outlines a robust path for higher-order PN calculations and flux computations.

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, $ω_s=0$. 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 $ω_k$, giving the unique answer compatible with global Poincaré invariance ($ω_k={41/24}$) by summing $\sim50$ different dimensionally continued contributions.

Dimensional regularization of the gravitational interaction of point masses

TL;DR

The paper uses dimensional regularization within the ADM formalism to derive the conservative 3PN Hamiltonian for two gravitationally interacting point masses. It proves that the dimensionally continued Hamiltonian is finite as with no pole terms, uniquely fixing the kinetic and static 3PN parameters by enforcing gauge and Poincaré invariance, yielding and . This resolves longstanding ambiguities from previous Hadamard/Riesz approaches, strengthens the perturbative consistency of the method, and has significant implications for accurate gravitational-wave modeling and initial-data construction in numerical relativity. The work also highlights the inconsistency of conformally flat truncations at 3PN and outlines a robust path for higher-order PN calculations and flux computations.

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, . 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 , giving the unique answer compatible with global Poincaré invariance () by summing different dimensionally continued contributions.

Paper Structure

This paper contains 4 sections, 16 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Dimensional continuation of the 3PN ADM Hamiltonian
  3. Finiteness of $H_{\rm ADM}(d)$ as $d\to3$
  4. Dimensional-continuation determination of the Hamiltonian

Figures (1)

  • Figure 1: "Three-loop diagram" representing the contribution (of order $G^4\,m_1^3\,m_2^2$) proportional to $\int{d^dx\,\phi_1\,\partial_k h_{(40)ij}^{\rm TT}\,\partial_k h_{(40)ij}^{\rm TT}}$ in the Hamiltonian, where $h_{(40)ij}^{\rm TT}\equiv {-2\,c(d)\,\Delta^{-1}(\partial_i\phi_1\,\partial_j\phi_2)^{\rm TT}}$. The solid lines represent the "propagator" $\Delta^{-1}$ of $\phi$, the helicoidal lines represent the "propagator" $(\Delta^{-1})^{\rm TT}$ of $h_{(4)ij}^{\rm TT}$, and the circles labelled 1 and 2 denote the sources $m_1\,\delta_1$ and $m_2\,\delta_2$.