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

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ErratumPhysRevD.85.pdf

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##### Citation

Barausse, E., Racine, E., & Buonanno, A. (2012). Hamiltonian of a spinning test-particle
in curved spacetime [Erratum: Phys. Rev. D 80, 104025 (2009)].* Physical Review D,* *85*: 069904. doi:10.1103/PhysRevD.85.069904.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0015-8498-2

##### 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 (SSC),
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
(ADM) 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 S_Kerr S (and S_Kerr^2) through 3PN order, S_Kerr
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.