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. 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.
Highlights
The problem of motion of many-body systems is an important issue in GR
Where ∂i f on the lhs denotes the derivative of f considered as a distribution, while ∂i f on the rhs denotes the derivative of f considered as a function, Σ is any smooth close surface surrounding the origin and dσi is the surface element on Σ
The observables of two-body systems that can be measured from infinity by, say, gravitational-wave observations, are describable in terms of dynamical invariants, i.e., functions which do not depend on the choice of phase-space coordinates
Summary
The problem of motion of many-body systems is an important issue in GR (see, e.g., Damour 1983a, 1987b). Where the “Newtonian” mass density ∗ = √−g u0/c [g = det(gμν), u0 is the time component of the four-velocity field uμ, uμuμ = −c2] fulfills the metric-free continuity equation. The correct expression for the rest mass contrarily reads, at the 1PN level, m = ̇ d3x ∗ 1 + 1 Π − U ,. For pressureless (dust-like) matter, the correct 1PN expression is given by m = d3x ∗ = ̇ d3x det(gi j ) = dV ,. The error in question slept into second of two sequential papers by de Sitter (1916a, b, 1917) when calculating the 1PN equations of motion for a many-body system. LeviCivita (1937b) used the correct rest mass formula for dusty bodies. Einstein criticized the calculations by Levi-Civita because he was missing pressure for stabilizing the bodies. Levi-Civita argued with the “effacing principle”, inaugurated by
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
