Comparing metrics at large: harmonic vs quo-harmonic coordinates

To compare two space-times on large domains, and in particular the global structure of their manifolds, requires using identical frames of reference and associated coordinate conditions. In this paper we use and compare two classes of time-like congruences and corresponding adapted coordinates: the harmonic and quo-harmonic classes. Besides the intrinsic definition and some of their intrinsic properties and differences we consider with some detail their differences at the level of the linearized approximation of the field equations. The hard part of this paper is an explicit and general determination of the harmonic and quo-harmonic coordinates adapted to the stationary character of three well-know metrics, Schwarzschild's, Curzon's and Kerr's, to order five of their asymptotic expansions. It also contains some relevant remarks on such problems as defining the multipoles of vacuum solutions or matching interior and exterior solutions.Comment: 27 pages, no figure

A Unified Algebraic Approach to Classical Yang-Baxter Equation

In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical r-matrix constructed from a left-symmetric algebra which corresponds to a parak\"ahler structure in geometry. Moreover, the former in a special case gives an algebraic interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a classical r-matrix.Comment: To appear in Journal of Physics A: Mathematical and Theoretica

Gauged motion in general relativity and in Kaluza-Klein theories

In a recent paper [1] a new generalization of the Killing motion, the {\it gauged motion}, has been introduced for stationary spacetimes where it was shown that the physical symmetries of such spacetimes are well described through this new symmetry. In this article after a more detailed study in the stationary case we present the definition of gauged motion for general spacetimes. The definition is based on the gauged Lie derivative induced by a threading family of observers and the relevant reparametrization invariance. We also extend the gauged motion to the case of Kaluza-Klein theories.Comment: 42 pages, revised version, typos correction along with some minor changes, Revtex forma

Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type

We consider the special type of the field-theoretical Symplectic structures called weakly nonlocal. The structures of this type are in particular very common for the integrable systems like KdV or NLS. We introduce here the special class of the weakly nonlocal Symplectic structures which we call the weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then the connection of such structures with the Whitham averaging method and propose the procedure of "averaging" of the weakly nonlocal Symplectic structures. The averaging procedure gives the weakly nonlocal Symplectic Structure of Hydrodynamic Type for the corresponding Whitham system. The procedure gives also the "action variables" corresponding to the wave numbers of $m$-phase solutions of initial system which give the additional conservation laws for the Whitham system.Comment: 64 pages, Late

From Rota-Baxter Algebras to Pre-Lie Algebras

Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension $\leq 3$ and their corresponding pre-Lie algebras.Comment: 23 pages, appear in Journal of Physics A; Mathematical and Theoretica

Jordan polynomials can be analytically recognized

We prove that there exists a real or complex central simple associative algebra M with minimal one-sided ideals such that, for every non-Jordan associative polynomial p, a Jordan-algebra norm can be given on M in such a way that the action of p on M becomes discontinuous