193 research outputs found
The Tulczyjew triple for classical fields
The geometrical structure known as the Tulczyjew triple has proved to be very
useful in describing mechanical systems, even those with singular Lagrangians
or subject to constraints. Starting from basic concepts of variational
calculus, we construct the Tulczyjew triple for first-order Field Theory. The
important feature of our approach is that we do not postulate {\it ad hoc} the
ingredients of the theory, but obtain them as unavoidable consequences of the
variational calculus. This picture of Field Theory is covariant and complete,
containing not only the Lagrangian formalism and Euler-Lagrange equations but
also the phase space, the phase dynamics and the Hamiltonian formalism. Since
the configuration space turns out to be an affine bundle, we have to use affine
geometry, in particular the notion of the affine duality. In our formulation,
the two maps and which constitute the Tulczyjew triple are
morphisms of double structures of affine-vector bundles. We discuss also the
Legendre transformation, i.e. the transition between the Lagrangian and the
Hamiltonian formulation of the first-order field theor
Hamiltonian of a spinning test-particle in curved spacetime
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.Comment: Corrected typo in the ADM Hamiltonian at 3.5 PN order (eq. 6.20
Higher-order spin effects in the dynamics of compact binaries I. Equations of motion
We derive the equations of motion of spinning compact binaries including the
spin-orbit (SO) coupling terms one post-Newtonian (PN) order beyond the
leading-order effect. For black holes maximally spinning this corresponds to
2.5PN order. Our result for the equations of motion essentially confirms the
previous result by Tagoshi, Ohashi and Owen. We also compute the spin-orbit
effects up to 2.5PN order in the conserved (Noetherian) integrals of motion,
namely the energy, the total angular momentum, the linear momentum and the
center-of-mass integral. We obtain the spin precession equations at 1PN order
beyond the leading term, as well. Those results will be used in a future paper
to derive the time evolution of the binary orbital phase, providing more
accurate templates for LIGO-Virgo-LISA type interferometric detectors.Comment: transcription error in Eqs. (2.17) correcte
The de Sitter Relativistic Top Theory
We discuss the relativistic top theory from the point of view of the de
Sitter (or anti de Sitter) group. Our treatment rests on Hanson-Regge's
spherical relativistic top lagrangian formulation. We propose an alternative
method for studying spinning objects via Kaluza-Klein theory. In particular, we
derive the relativistic top equations of motion starting with the geodesic
equation for a point particle in 4+N dimensions. We compare our approach with
the Fukuyama's formulation of spinning objects, which is also based on
Kaluza-Klein theory. We also report a generalization of our approach to a 4+N+D
dimensional theory.Comment: 25 pages, Latex,commnets and references adde
Motion of a Vector Particle in a Curved Spacetime. II First Order Correction to a Geodesic in a Schwarzschild Background
The influence of spin on a photon's motion in a Schwarzschild and FRW
spacetimes is studied. The first order correction to the geodesic motion is
found. It is shown that unlike the world-lines of spinless particles, the
photons world-lines do not lie in a plane.Comment: 14 pages, LaTeX2e, second paper in the series (the first one:
gr-qc/0110067), replaced with typos and style corrected version, accepted in
MPL
The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory
We present a general definition of the Poisson bracket between differential
forms on the extended multiphase space appearing in the geometric formulation
of first order classical field theories and, more generally, on exact
multisymplectic manifolds. It is well defined for a certain class of
differential forms that we propose to call Poisson forms and turns the space of
Poisson forms into a Lie superalgebra.Comment: 40 pages LaTe
Motion of test bodies in theories with nonminimal coupling
We derive the equations of motion of test bodies for a theory with nonminimal
coupling by means of a multipole method. The propagation equations for
pole-dipole particles are worked out for a gravity theory with a very general
coupling between the curvature scalar and the matter fields. Our results allow
for a systematic comparison with the equations of motion of general relativity
and other gravity theories.Comment: 5 pages, RevTex forma
Multipole particle in relativity
We discuss the motion of extended objects in a spacetime by considering a
gravitational field created by these objects. We define multipole moments of
the objects as a classification by Lie group SO(3). Then, we construct an
energy-momentum tensor for the objects and derive equations of motion from it.
As a result, we reproduce the Papapetrou equations for a spinning particle.
Furthermore, we will show that we can obtain more simple equations than the
Papapetrou equations by changing the center-of-mass.Comment: 22 pages, 2 figures. Accepted for publication in Phys. Rev.
Dirac equations in curved space-time versus Papapetrou spinning particles
We find out classical particles, starting from Dirac quantum fields on a
curved space-time, by an eikonal approximation and a localization hypothesis
for amplitudes. We recover the results by Mathisson-Papapetrou, hence
establishing a fundamental correspondence between the coupling of classical and
quantum spinning particles with the gravitational field.Comment: 6 pages, 1 figure, accepted for publication in Europhysics Letter
Dynamics of test bodies with spin in de Sitter spacetime
We study the motion of spinning test bodies in the de Sitter spacetime of
constant positive curvature. With the help of the 10 Killing vectors, we derive
the 4-momentum and the tensor of spin explicitly in terms of the spacetime
coordinates. However, in order to find the actual trajectories, one needs to
impose the so-called supplementary condition. We discuss the dynamics of
spinning test bodies for the cases of the Frenkel and Tulczyjew conditions.Comment: 11 pages, RevTex forma
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