Hamiltonian of a spinning test particle in curved spacetime

Enrico Barausse, Etienne Racine, and Alessandra Buonanno
Phys. Rev. D 80, 104025 – Published 17 November 2009; Erratum Phys. Rev. D 85, 069904 (2012)

Abstract

Using a Legendre transformation, we compute the unconstrained Hamiltonian of a spinning test particle in a curved spacetime at linear order in the particle spin. The equations of motion of this unconstrained Hamiltonian coincide with the Mathisson-Papapetrou-Pirani equations. We then use the formalism of Dirac brackets to derive the constrained Hamiltonian and the corresponding phase space algebra in the Newton-Wigner spin supplementary condition, suitably generalized to curved spacetime, and find that the phase space algebra (q,p,S) is canonical at linear order in the particle spin. We provide explicit expressions for this Hamiltonian in a spherically symmetric spacetime, both in isotropic and spherical coordinates, and in the Kerr spacetime in Boyer-Lindquist coordinates. Furthermore, we find that our Hamiltonian, when expanded in post-Newtonian (PN) orders, agrees with the Arnowitt-Deser-Misner canonical Hamiltonian computed in PN theory in the test particle limit. Notably, we recover the known spin-orbit couplings through 2.5PN order and the spin-spin couplings of type SKerrS (and SKerr2) through 3PN order, SKerr being the spin of the Kerr spacetime. Our method allows one to compute the PN Hamiltonian at any order, in the test particle limit and at linear order in the particle spin. As an application we compute it at 3.5PN order.

  • Received 3 August 2009

DOI:https://doi.org/10.1103/PhysRevD.80.104025

©2009 American Physical Society

Erratum

Erratum: Hamiltonian of a spinning test particle in curved spacetime [Phys. Rev. D 80, 104025 (2009)]

Enrico Barausse, Etienne Racine, and Alessandra Buonanno
Phys. Rev. D 85, 069904 (2012)

Authors & Affiliations

Enrico Barausse, Etienne Racine, and Alessandra Buonanno

  • Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

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Issue

Vol. 80, Iss. 10 — 15 November 2009

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