Defining Euler-Lagrange fields in terms of almost tangent structures

It is shown that a differentiable manifold with an almost tangent structure provides a suitably general setting for lagrangian dynamics, in analogy to the way that a symplectic manifold provides a suitably general setting for hamiltonian dynamics; and it is shown how a closed 1-form on a manifold with almost tangent structure determines a vector field, its Euler-Lagrange field, whose integral curves are solutions of the Euler-Lagrange equations for a suitable lagrangian function.     

Field theory onR×S 3 topology. IV: Electrodynamics of magnetic moments

The equations of electrodynamics for the interactions between magnetic moments are written on R×S³ topology rather than on Minkowskian space-time manifold of ordinary Maxwell's equations. The new field equations are an extension of the previously obtained Klein-Gordon-type, Schrödinger-type, Weyl-type, and Dirac-type equations. The concept of the magnetic moment in our case takes over that of the charge in ordinary electrodynamics as the fundamental entity. The new equations have R×S³ invariance as compared to the Lorentz invariance of Maxwell's equations. The solutions of the new field equations are given. In this theory the divergence of the electric field vanishes whereas that of the magnetic field does not.Research supported in part by the Colgate Research Council and by the Center for Theoretical Physics, University of Maryland.     

Higher-spin fields in a curved space-time

Algebraic constraints are derived for higher-spin fields in a curved space-time manifold. Comparison is made with previously obtained results. A particular solution of the zero-restmass field equations is given for the plane wave Einstein-Maxwell space-times.     

Yukawa couplings between (2, 1)-forms

The compactification of superstrings leads to an effective field theory for which the space-time manifold is the product of a four-dimensional Minkowski space with a six-dimensional Calabi-Yau space. The particles that are massless in the four-dimensional world correspond to differential forms of type (1, 1) and of type (2, 1) on the Calabi-Yau space. The Yukawa couplings between the families correspond to certain integrals involving three differential forms. For an important class of Calabi-Yau manifolds, which includes the cases for which the manifold may be realized as a complete intersection of polynomial equations in a projective space, the families correspond to (2, 1)-forms. The relation between (2, 1)-forms and the geometrical deformations of the Calabi-Yau space is explained and it is shown, for those cases for which the manifold may be realized as the complete intersection of polynomial equations in a single projective space or for many cases when the manifold may be realized as the transverse intersection of polynomial equations in a product of projective spaces, that the calculation of the Yukawa coupling reduces to a purely algebraic problem involving the defining polynomials. The generalization of this process is presented for a general Calabi-Yau manifold.     

Linear double universal Grassmann manifold method for the stationary axisymmetric vacuum gravitational field equations

Nagatomo's universal Grassmann manifold scheme is extended to a double form, which is used to find the exact solutions of the stationary axisymmetric vacuum gravitational field equations. Some new results are given.     

Cosmological Models and Centre Manifold Theory

Centre manifold theory is applied to some dynamical systems arising from spatially homogeneous cosmological models. Detailed information is obtained concerning the late-time behaviour of solutions of the Einstein equations of Bianchi type III with collisionless matter. In addition some statements in the literature on solutions of the Einstein equations coupled to a massive scalar field are proved rigorously.     

Field theory onR×S 3 topology. V:SU 2 gauge theory

A gauge theory on R×S ³ topology is developed. It is a generalization to the previously obtained field theory on R×S ³ topology and in which equations of motion were obtained for a scalar particle, a spin one-half particle, the electromagnetic field of magnetic moments, and a Shrödinger-type equation, as compared to ordinary field equations defined on a Minkowskian manifold. The new gauge field equations are presented and compared to the ordinary Yang-Mills field equations, and the mathematical and physical differences between them are discussed.On leave from Center for Theoretical Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel.     

Exact Solitary Wave Solutions of the Coupled Nonlinear Scalar Field Equations

Because of the arbitrary property of the singular manifold, the usual singularity analysis can be extended to a different form. Using a nonstandard truncation approach, five special types of exact solution of the coupled nonlinear scalar field equations are obtained.     

Solutions of Euler-Lagrange equations for self-interacting field of linear frames on product manifold of group space

We investigate a model of self-interacting field of linear frames on the product manifold M × G, where G is a semisimple Lie group acting freely and transitively on a manifold M. We find two families of solutions of the Euler-Lagrange equations for the field of frames.     

Application of Lie Algebroid Structures to Unification of Einstein and Yang-Mills Field Equations

Yang-Mills field equations describe new forces in the context of Lie groups and principle bundles. It is of interest to know if the new forces and gravitation can be described in the context of algebroids. This work was intended as an attempt to answer last question. The basic idea is to construct Einstein field equation in an algebroid bundle associated to space-time manifold. This equation contains Einstein and Yang-Mills field equations simultaneously. Also this equation yields a new equation that can have interesting experimental results.     

Anisotropic scalar field with cosmical time

A static, plane-symmetric scalar field of long range is considered in general relativity, and a one-parameter class of exact solutions with cosmical time is studied in harmonic coordinates. The geodetic equations are solved. A velocity-dependent acceleration field is found, acting attractively on the component of the velocity normal to the plane of symmetry, and repulsively on the component parallel to that plane. The manifold is complete. Test particles at rest are insensitive to it.     

On the phase manifold geometry of the two-dimensional Burgers equations

A two space dimensional Burgers hierarchy is constructed and analyzed in terms of an invariant, Nijenhuis mixed tensor field on the phase manifold. The analysis extends to Burgers equations in higher dimension as well.     

Hamiltonian systems with constraints: A geometric approach

Hamilton-Dirac equations for a constrained Hamiltonian system are deduced from a variational principle. In the local problem for such systems an algorithm is proposed to obtain the final constraint manifold and the dynamical vector field on it using vector fields on the phase space. The global problem is solved in terms of fiber bundles associated with the problem.     

Stability for the equilibrium state manifold of relativistic Birkhoffian systems

In this paper, the stability of equilibrium state manifold for relativistic Birkhoffian systems in studied. The equilibrium state equations, the disturbance equations and their first approximation are presented. The criteria of stability for the equilibrium state manifold are obtained. The relationship between the stability of the equilibrium-state manifold of relativistic Birkhoffian systems and that of classical Birkhoffian systems is discussed. An example is given to illustrate the results.     

Regge-Palatini calculus

We formulate a Palatini version of the Regge calculus by constructing a discrete torsion field on the simplicial manifold. The action has two components, the original Regge action and an additional action for the torsion field. In the absence of matter the variational equations reduce the torsion field to zero. Matter fields can act as sources of torsion.     

Averaging out the Einstein equations

A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.     

Tunneling method for Hawking radiation in the Nariai case

We revisit the tunneling picture for the Hawking effect in light of the charged Nariai manifold, because this general relativistic solution, which displays two horizons, provides the bonus to allow the knowledge of exact solutions of the field equations. We first perform a revisitation of the tunneling ansatz in the framework of particle creation in external fields à la Nikishov, which corroborates the interpretation of the semiclassical emission rate \({\varGamma }_{emission}\) as the conditional probability rate for the creation of a couple of particles from the vacuum. Then, particle creation associated with the Hawking effect on the Nariai manifold is calculated in two ways. On the one hand, we apply the Hamilton–Jacobi formalism for tunneling, in the case of a charged scalar field on the given background. On the other hand, the knowledge of the exact solutions for the Klein–Gordon equations on Nariai manifold, and their analytic properties on the extended manifold, allow us a direct computation of the flux of particles leaving the horizon, and, as a consequence, we obtain a further corroboration of the semiclassical tunneling picture from the side of S-matrix formalism.     

Geometrization of the classical field theory and its application in Yang-Mills field theory

The paper contains presentation of the finite-dimensional approach to the classical field theory based on the geometry of differential manifolds and forms. Geometrical construction of a symplectic structure and Poisson brackets on the space of initial conditions are realized. This space is not a manifold but it can be furnished with a structure of a differential space.The structural n+1 form for the Yang-Mills field theory is constructed. This gives automatically equations of motion and equations for initial conditions. The parasymplectic structure is computed. The directions of degeneration appear to be exactly the directions of infinitesimal gauge transformations. The Poisson bracket for Yang-Mills field theory is obtained.