New Directions in NonRelativistic and Relativistic Rotational and Multipole Kinematics for N-Body and Continuous Systems (original) (raw)
Related papers
2002
In the Wigner-covariant rest-frame instant form of dynamics it is possible to develop a relativistic kinematics for the N-body problem which solves all the problems raised till now on this topic. The Wigner hyperplanes, orthogonal to the total timelike 4-momentum of any N-body configuration, define the intrinsic rest frame and realize the separation of the center-of-mass motion. The point chosen as origin of each Wigner hyperplane can be made to coincide with the covariant non-canonical Fokker-Pryce center of inertia. This is distinct from the canonical pseudo-vector describing the decoupled motion of the center of mass (having the same Euclidean covariance as the quantum Newton-Wigner 3-position operator) and the non-canonical pseudo-vector for the Møller center of energy. These are the only external notions of relativistic center of mass, definable only in terms of the external Poincaré group realization. Inside the Wigner hyperplane, an internal unfaithful realization of the Poincaré group is defined while the analogous three concepts of center of mass weakly coincide due to the first class constraints defining the rest frame (vanishing of the internal 3-momentum). This unique internal center of mass is consequently a gauge variable which, through a gauge fixing, can be localized atthe origin of the Wigner hyperplane. An adapted canonical basis of relative variables is found by means of the classical counterpart of the Gartenhaus-Schwartz transformation. The invariant mass of the N-body configuration is the Hamiltonian for the relative motions. In this framework we can introduce the same dynamical body frames, orientation-shape variables, spin frame and canonical spin bases for the rotational kinematics developed for the non-relativistic N-body problem.
Multipolar Expansions for the Relativistic N-Body Problem in the Rest-Frame Instant Form
2001
Dixon's multipoles for a system of N relativistic positive-energy scalar particles are evaluated in the rest-frame instant form of dynamics. The Wigner hyperplanes (intrinsic rest frame of the isolated system) turn out to be the natural framework for describing multipole kinematics. In particular, concepts like the {\it barycentric tensor of inertia} can be defined in special relativity only by means of the quadrupole moments of the isolated system.
2000
After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the non-relativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of {\it body frame} for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian {\it left action}. The analysis of the SO(3) coadjoint orbits contained in the N-body phase space implies the existence of a {\it spin frame} for the N-body system. Then, the existence of appropriate non-symmetry Hamiltonian {\it right actions} for non-rigid systems leads to the construction of a N-dependent discrete number of {\it dynamical body frames} for the N-body system, hence to the associated notions of {\it dynamical} and {\it measurable} orientation and shape variables, angular velocity, rotational and vibrational configurations. For N=3 the dynamical body frame turns out to be unique and our approach reproduces the {\it xxzz gauge} of the gauge theory associated with the {\it orientation-shape} SO(3) principal bundle approach of Littlejohn and Reinsch. For Ngeq4N \geq 4Ngeq4 our description is different, since the dynamical body frames turn out to be {\it momentum dependent}. The resulting Darboux bases for Ngeq4N\geq 4Ngeq4 are connected to the coupling of the {\it spins} of particle clusters rather than the coupling of the {\it centers of mass} (based on Jacobi relative normal coordinates). One of the advantages of the spin coupling is that, unlike the center-of-mass coupling, it admits a relativistic generalization.
Generalized Relativistic Kinematics
2011
We propose a method for deforming an extended Galilei algebra that leads to a nonstandard realization of the Poincaré group with the Fock-Lorentz linear fractional transformations. The invariant parameter in these transformations has the dimension of length. Combining this deformation with the standard one (with an invariant velocity c) leads to the algebra of the symmetry group of the anti-de Sitter space in Beltrami coordinates. In this case, the action for free point particles contains the dimensional constants R and c. The limit transitions lead to the ordinary (R→∞) or alternative (c→∞) but nevertheless relativistic kinematics.
General-relativistic celestial mechanics. I. Method and definition of reference systems
Physical Review D, 1991
We present a new formalism for treating the general-relativistic celestial mechanics of systems of N arbitrarily composed and shaped, weakly self-gravitating, rotating, deformable bodies. This formalism is aimed at yielding a complete description, at the first post-Newtonian approximation level, of (i) the global dynamics of such N-body systems (" external problem"), (ii) the local gravitational structure of each body (" internal problem"), and, (iii) the way the external and the internal problems fit together ("theory of reference systems"). This formalism uses in a complementary manner N+1 coordinate charts (or "reference systems"): one "global" chart for describing the overall dynamics of the N bodies, and N "local" charts adapted to the separate description of the structure and environment of each body. The main tool which allows us to develop, in an elegant manner, a constructive theory of these N+1 reference systems is a systematic use of a particular "exponential" parametrization of the metric tensor which has the effect of linearizing both the field equations, and the transformation laws under a change of reference system. This linearity allows a treatment of the first post-Newtonian relativistic celestial mechanics which is, from a structural point of view, nearly as simple and transparent as its Newtonian analogue. Our scheme differs from previous attempts in several other respects: the structure of the stress-energy tensor is left completely open; the spatial coordinate grid (in each system) is fixed by algebraic conditions while a convenient "gauge" flexibility is left open in the time coordinate [at the order 6t =O(c)]; the gravitational field locally generated by each body is skeletonized by particular relativistic multipole moments recently introduced by Blanchet and Damour, while the external gravitational field experienced by each body is expanded in terms of a particular new set of relativistic tidal moments. In this first paper we lay the foundations of our formalism, with special emphasis on the definition and properties of the N local reference systems, and on the general structure and transformation properties of the gravitational field. As an illustration of our approach we treat in detail the simple case where each body can, in some approximation, be considered as generating a spherically symmetric gravitational field. This "monopole truncation" leads us to a new (and, in our opinion, improved) derivation of the Lorentz-Droste-Einstein-Infeld-Hoffmann equations of motion. The detailecl treatment of the relativistic motion of bodies endowed with arbitrary multipole structure will be the subject of subsequent publications.
The relativity of velocity is not completed without reduced momentum relative to many-body mass-center. I present the intrinsic center-of-inertia of many-body interacting (bound) system for the case of finite radiation-speed. The concept of a center-of-inertia is not possible within theory of relativity postulating that each pair of reference system must be related by Lorentz isometry-group transformation. I am showing that center-of-mass is well defined concept within groupfree homological algebra approach postulating energy-momentum conservation.
Coordinate Systems in the General Relativistic Framework
Relativity in Celestial Mechanics and Astrometry, 1986
The treatment of the coordinate systems is briefly reviewed in the Newtonian mechanics, in the special theory of relativity, and in the general relativistic theory, respectively. Some reference frames and coordinate systems proposed within the general relativistic framework are introduced. With use of the ideas on which these coordinate systems are based, the proper reference frame comoving with a system of masspoints is defined as a general relativistic extension of the relative coordinate system in the Newtonian mechanics. The coordinate transforma tion connecting this and the background coordinate systems is presented explicitly in the post-Newtonian formalism. The conversion formulas of some physical quantities caused by this coordirate transformation are discussed. The concept of the rotating coordinate system is reexamined within the relativistic framework. A modification of the introduced proper reference frame is proposed as the basic coordinate system in the astrometry. The relation between the solar system barycentric coordinate system and the terrestrial coordinate system is given explicitly.
Physical Review D, 2000
We extract all the invariants (i.e. all the functions which do not depend on the choice of phasespace coordinates) of the dynamics of two point-masses, at the third post-Newtonian (3PN) approximation of general relativity. We start by showing how a contact transformation can be used to reduce the 3PN higher-order Hamiltonian derived by Jaranowski and Schäfer [2] to an ordinary Hamiltonian. The dynamical invariants for general orbits (considered in the center-of-mass frame) are then extracted by computing the radial action variable prdr as a function of energy and angular momentum. The important case of circular orbits is given special consideration. We discuss in detail the plausible ranges of values of the two quantities ωstatic, ω kinetic which parametrize the existence of ambiguities in the regularization of some of the divergent integrals making up the Hamiltonian. The physical applications of the invariant functions derived here (e.g. to the determination of the location of the last stable circular orbit) are left to subsequent work.