![]() Observe that we can invert the relations (1) to write Then S is the Pauli-Lubanski pseudo-vector, from which a magnetic dipole moment can be constructed, whilst the components of Z, which will be referred to as the Pirani vector, can be used to define an electric or mass dipole moment. In the following we take u to be the proper four-velocity of the particle. Funded by SCOAP3.īy construction both four-vectors S and Z are space-like: S » u» _ 0, Z »u» _ 0. This is an open access article under the CC BY license (). The spin degrees of freedom are described by an antisymmetric tensor 'Elxv, which can be decomposed into two space-like four-vectors by introducing a time-like unit vector u: u^ ulx = -1, and definingĠ370-2693/© 2015 The Authors. kinetic, momenta rather than canonical momenta see and references cited there for a general discussion, and for the application to spinning particles. A convenient starting point for models with gauge-field interactions is the use of covariant, i.e. Changes in the parametrization can be compensated by redefining the brackets and the hamiltonian. The parametrization of phase-space is not unique, as is familiar from the Hamilton-Jacobi theory of dynamical systems. ![]() Hamiltonian dynamical systems are specified by three sets of ingredients: the phase space, identifying the dynamical degrees of freedom, the Poisson-Dirac brackets defining a symplectic structure, and the hamiltonian generating the evolution of the system with given initial conditions by specifying a curve in the phase space passing through the initial point. In this way specific aspects of the structure can still be accounted for. One of the advantages of this description is that it can be applied to compact bodies with different types of spin dynamics, such as different gravimagnetic ratios. van Holten).Įffective hamiltonian formalism similar to the one introduced in Ref. In this letter we take the second point of view for the description of spinning test masses in curved space-time, using anĮ-mail addresses: (G. A large variety of models for spinning particles is found in the literature. This is also known as the spinning-particle approximation, and is used for the semi-classical description of elementary particles as well. The other approach is to construct effective equations of motion for point-like objects, which is an idealization of a compact body, at the price of neglecting details of the internal structure by assigning the point-like object an overall position, momentum and spin. Equations of motion for these quantities are then derived by applying the conservation law for the energy-momentum tensor of matter. The energy-momentum vector and the angular-momentum tensor can be constructed by computing integrals of components of the energy-momentum tensor and their first moments over the volume of the body, using suitable boundary conditions. One approach starts from the covariant divergence-free energy-momentum tensor of matter, which makes it possible to keep track of aspects of the structure of the body. As argued in there are two complementary approaches to the subject. The dynamics of angular momentum and spin of gravitating compact bodies has been a subject of great interest and intense investigation since the early days of relativity theory for recent overviews see. This is an open access article under the CC BY license We show that this extension respects a large class of known constants of motion for the minimal case. We also discuss a non-minimal extension of the hamiltonian giving rise to a gravitational equivalent of the Stern-Gerlach force. The analysis is illustrated by the example of motion in Schwarzschild space-time. We give full results for the simplest case with minimal hamiltonian, constructing constants of motion including spin. Different choices of hamiltonians allow for the description of different gravitating systems. ![]() The dynamics of spinning particles in curved space-time is discussed, emphasizing the hamiltonian formulation. van Holtena bĪ Nikhef, Science Park 105, Amsterdam, Netherlands b Lorentz Institute, Leiden University, Niels Bohrweg 2, Leiden, Netherlands Contents lists available at ScienceDirectĬovariant hamiltonian spin dynamics in curved space-time
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