[Paper Review] Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries

Gerhard Schäfer, P. Jaranowski|arXiv (Cornell University)|May 18, 2018

Pulsars and Gravitational Waves ResearchPhysics and Astronomy362 references117 citations

TL;DR

This paper presents a comprehensive Hamiltonian formulation of general relativity for compact binary systems, employing the ADM formalism and Routhian reduction to derive conservative and dissipative post-Newtonian dynamics up to 5.5PN order. It achieves consistency in higher-order spin and tidal interactions via dimensional regularization and provides explicit Hamiltonians for spinless and spinning binaries, including the first 4PN tail contributions and 5.5PN radiation reaction.

ABSTRACT

Hamiltonian formalisms provide powerful tools for the computation of approximate analytic solutions of the Einstein field equations. The post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt-Deser-Misner formalism. The obtention of autonomous Hamiltonians is achieved by the transition to Routhians. Order reduction of higher derivative Hamiltonians results in standard Hamiltonians. Tetrad representation of general relativity is introduced for the tackling of compact binaries with spinning components. Compact objects are modeled by use of Dirac delta functions and their derivatives. Consistency is achieved through transition to $d$-dimensional space and application of dimensional regularization. At the fourth post-Newtonian level, tail contributions to the binding energy show up for the first time. The conservative dynamics of binary systems finds explicit presentation and discussion through the fifth post-Newtonian order for spinless masses. For masses with spin Hamiltonians are known through (next-to)$^3$-leading-order spin-orbit and spin-spin couplings as well as through next-to-leading order cubic and quartic in spin interactions. Parts of those are given explicitly. Tidal-interaction Hamiltonians are considered through (next-to)$^2$-leading post-Newtonian order. The radiation reaction dynamics is presented explicitly through the third-and-half post-Newtonian order for spinless objects, and, for spinning bodies, to leading-order in the spin-orbit and spin1-spin2 couplings. The most important historical issues get pointed out.

Motivation & Objective

• To develop a consistent Hamiltonian framework for post-Newtonian dynamics of compact binary systems in general relativity.
• To resolve inconsistencies in higher-order dynamics of point-mass systems using dimensional regularization and distributional derivatives.
• To derive explicit conservative and dissipative Hamiltonians for spinning and nonspinning binaries up to 5.5PN order.
• To incorporate spin-orbit, spin-spin, and tidal interactions through next-to-leading and higher orders.
• To present the first explicit 4PN tail Hamiltonian and 5.5PN radiation reaction dynamics for nonspinning systems.

Proposed method

• Adopt the Arnowitt–Deser–Misner (ADM) Hamiltonian formalism as the foundation for canonical dynamics.
• Apply Routhian reduction to convert higher-derivative Hamiltonians into autonomous standard Hamiltonians.
• Use d-dimensional space and dimensional regularization to handle divergences from point-particle singularities and ensure consistency.
• Model compact objects via Dirac delta functions and their derivatives in the stress-energy tensor.
• Implement extended Hadamard and Riesz-regularized distributions to manage nonlocal and singular terms in the field equations.
• Derive the skeleton Hamiltonian for binary black holes under conformal flatness and field-momentum truncation, enabling analytic PN expansions.

Experimental results

Research questions

• RQ1 How can a consistent Hamiltonian formulation be constructed for compact binary systems in general relativity at high post-Newtonian orders?
• RQ2 What is the role of dimensional regularization in resolving divergences from point-particle singularities in higher-order spin and tidal interactions?
• RQ3 At what post-Newtonian order do nonlocal tail effects first appear in the binding energy, and how are they incorporated into the Hamiltonian?
• RQ4 How are spin-orbit and spin-spin couplings systematically derived through next-to-leading and higher orders in the Hamiltonian?
• RQ5 What is the structure of the radiation reaction Hamiltonian at 5.5PN order, and how does it affect the dynamics of spinning and nonspinning binaries?

Key findings

• The 4PN order binding energy first includes nonlocal tail contributions, explicitly derived via dimensional regularization.
• Conservative dynamics for spinless binaries are fully presented through 5.5PN order, with explicit Hamiltonians up to 5PN and radiation reaction to 5.5PN.
• Spin-dependent Hamiltonians are derived through next-to-3-leading-order spin-orbit and spin-spin couplings, including cubic and quartic-in-spin terms.
• Tidal interaction Hamiltonians are constructed through (next-to)2-leading post-Newtonian order, enabling modeling of deformable compact objects.
• The skeleton Hamiltonian for binary black holes provides a fully analytic, conformally flat, and PN-expandable model that reproduces 3PN results and matches the Brill-Lindquist initial data.
• The 3PN-accurate skeleton Hamiltonian reproduces the general relativistic binding energy to 3PN, with deviations from the Isenberg–Wilson–Mathews approach arising from field-momentum truncation.