ADM canonical formalism for gravitating spinning objects

2008/05/16 by Jan Steinhoff, Gerhard Schäfer, Steven Hergt · 5 citations
Mathematics · Physics and Astronomy · #Classical mechanics #Cosmology and Gravitation Theories #Covariant transformation #Formalism (music) #General relativity #Gravitation #Hamiltonian (control theory) #Isotropy #Mathematical physics #Mathematics #Minkowski space #Noncommutative and Quantum Gravity Theories #Physics #Pulsars and Gravitational Waves Research #Quantum mechanics #Spacetime #Theoretical physics #gr-qc

paper · pdf · doi:10.1103/physrevd.77.104018

published as Phys.Rev.D77:104018,2008 · REVTeX4, 18 pages. v1: published version. v2: corrected misprints in (8.4) and (9.3), updated references

openalex publication_date 2008/05/16 · arxiv created 2009/10/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In general relativity, systems of spinning classical particles are implemented into the canonical formalism of Arnowitt, Deser, and Misner [R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962), p. 227; arXiv:gr-qc/0405109]. The implementation is made with the aid of a symmetric stress-energy tensor and not a 4-dimensional covariant action functional. The formalism is valid to terms linear in the single spin variables and up to and including the next-to-leading order approximation in the gravitational spin-interaction part. The field-source terms for the spinning particles occurring in the Hamiltonian are obtained from their expressions in Minkowski space with canonical variables through 3-dimensional covariant generalizations as well as from a suitable shift of projections of the curved spacetime stress-energy tensor originally given within covariant spin supplementary conditions. The applied coordinate conditions are the generalized isotropic ones introduced by Arnowitt, Deser, and Misner. As applications, the Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling, recently obtained by Damour, Jaranowski, and Sch"afer [Phys. Rev. D 77, 064032 (2008)], is rederived and the derivation of the next-to-leading order gravitational spin(1)-spin(2) Hamiltonian, shown for the first time in [J. Steinhoff, S. Hergt, and G. Sch"afer, Phys. Rev. D 77, 081501(R) (2008)], is presented.

Citations

• Calculation of the First Nonlinear Contribution to the General-Relativistic Spin-Spin Interaction for Binary Systems
• Localized States for Elementary Systems
• Tangent Euler Top in General Relativity
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• Dimensional regularization of the gravitational interaction of point masses
• Quantum Theory of Gravity. I. The Canonical Theory
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• L'intégrale de Riemann-Liouville et le problème de Cauchy
• The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles
• Binary black-hole problem at the third post-Newtonian approximation in the orbital motion: Static part
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• Spin effects on gravitational waves from inspiraling compact binaries at second post-Newtonian order