Living Reviews in Relativity: "Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries"

© SpringerSchäfer, G. & Jaranowski, P., "Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries", Living Rev Relativ (2018) 21:7

Open Access | Review Article

First Online: 31 August 2018


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. Configurations are treated where the absolute values of the spin vectors can be considered constant. 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. The conservative spin-dependent dynamics finds explicit presentation in Hamiltonian form through next-to-next-to-leading-order spin–orbit and spin1–spin2 couplings and to leading-order in the cubic and quartic in spin interactions. 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.

The Authors:

Gerhard Schäfer is Emeritus Professor of Relativistic Astrophysics at Friedrich Schiller University, Jena, Germany. His scientific interests include gravitational physics, quantum field theory, and fundamental physics.

Piotr Jaranowski is Professor at the Department of Astronomy and Astrophysics of the University of Bialystok, Poland. His research is focused on the detection of gravitational waves and the relativistic two-body problem.