Foundations of the unified theory of gravitational,electromagnetic, and scalar fields

A particular case of bimetrical unified field theory is considered, which is based on Hilbert's proposal of obtaining a complete system of independent equations for unified theory. The action depends on two symmetrical tensors g_μν and g _μν ^° , the second leading to a zero curvature tensor, which results in the theory being invariant under the Poincaré group, and in ten conservation laws. The field equations obtained when varying the action with respect to g_μν have the form of Einstein equations whose righthand side is not defined independently, but is rather a function of g_μν and g _μν ^° . The vector and scalar gauge transformations corresponding to variations δS of special form are defined. With the aid of these transformations, the electromagnetic and scalar fields are introduced within the framework of the unified theory. The basic equations of the theory under consideration contain a new dimensional physical constant, which connects gravitation and electromagnetism. A numerical estimate of this constant is given.     

Necessary and sufficient conditions for trivial solutions in supergravity

We determine the conditions necessary for a solution of the supergravity field equations with infinitesimal spin-3/2 field to be a pure gauge transformation of an Einstein vacuum field. The analysis depends on the Petrov classification of the curvature tensor and uses two-component spinor calculus. For general type I, the type II, and typeD, the necessary conditions found are also shown to be sufficient, and the explicit form of the gauge transformation can be given.Work supported in part by the Einstein Memorial Foundation.     

Poisson theory and integration method of Birkhoffian systems in the event space

<正>This paper focuses on studying the Poisson theory and the integration method of a Birkhoffian system in the event space.The Birkhoff's equations in the event space are given.The Poisson theory of the Birkhoffian system in the event space is established.The definition of the Jacobi last multiplier of the system is given,and the relation between the Jacobi last multiplier and the first integrals of the system is discussed.The researches show that for a Birkhoffian system in the event space,whose configuration is determined by(2n + 1) Birkhoff's variables,the solution of the system can be found by the Jacobi last multiplier if 2n first integrals are known.An example is given to illustrate the application of the results.     

Metric and Ricci tensors for a certain class of space-times ofD type

The class of space-times has been determined at the connection level, assuming the existence of some symmetrical relations between the Ricci rotation coefficients. It has been assumed, for instance, that at least two shear-free congruences of null geodesics exist. We have shown that onlyD type or conformally flat space-times can belong to this class. The theorem has been proved that a system of coordinates exists in which the metric tensor can depend on two coordinates, only. The metric tensor has been determined with an accuracy to two functions, each of which is a function of only one coordinate. Linear, second-order differential expressions have been found for these two functions. They determine the Ricci tensor. Several solutions of the Einstein-Maxwell equations with a cosmological constant are given.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

Bipartite Matching and Van der Waerden Conjecture

In this paper I show that the free energy F and the cost C associated to a bipartite matching problem can be explicitly estimated in term of the solution of a suitable system of equations (cavity equations in the following). The proof of these results relies on a well known result in combinatorics: the Van der Waerden conjecture (Egorychev–Falikman Theorem). Cavity equations, derived by a mean field argument by Mèzard and Parisi, can be considered as a smoothed form of the dual formulation for the bipartite matching problem. Moreover cavity equation are the Euler–Lagrange equations of a convex functional G parameterized by the temperature T. In term of their unique solution it is possible to define a free-energy-like function of the temperature g(T). g is a strictly decreasing concave function of T and C=g(0). The convexity of G allows to define an explicit algorithm to find the solution of the cavity equations at a given temperature T. Moreover, once the solution of the cavity equations at a given temperature T is known, the properties of g allow to find exact estimates from below and from above of the cost C.     

Differential Equations Compatible with KZ Equations

We define a system of dynamical differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra g. These are equations on a function of n complex variables z _i taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the dual variable in the Cartan subalgebra of g. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions.     

Theory of Gravitational-Inertial Field of Universe. I. Gravitational-Inertial Field Equations

In the series of present articles the original proposition is a generalization of the real world tensor by the introduction of a inertial field tensor. From this generalization it follows, particularly, that ?_ig_lm ? g_lm;i ≠ 0. This allows to use as a Lagrangian density of the field the expression A_g = k₁ g_lm;ig^lm _;kg^ik. On the basis of variational equations a system of more general covariant equations of the gravitational-inertial field is obtained. In the Einstein approximation these equations reduce to the field equations of Einstein. The solution of fundamental problems in the general theory of relativity by means of the new equations gives the same results as the solution by means of Einstein's equations. However, application of these equations to the cosmologic problem gives a result different from that obtained by Friedmann's theory. In particular, the solution gives the Hubble law as the law of motion of a free body in the inertial field - in contrast to Galileo-Newton's law.     

Ferromagnetic resonance in the polycrystalline hexagonal ferrites Co2−x ZnxW

Experimental studies of the ferromagnetic resonance spectra of polycrystalline hexagonal ferrites of the system Co_2−x Zn_xW in the frequency range 16–32 GHz are described. It is shown that interpretation of the experimental data requires the assumption of an anisotropic effective magnetomechanical ratio (or g-factor). The results of these experiments are compared with calculations based on the equations of motion, which ensure the conservation of the mechanical moment length. Concentration dependences are determined for the components of the magnetomechanical ratio tensor and the anisotropy field of this system of hexagonal ferrites at room temperature. Possible reasons why anisotropy fields measured in the vicinity of a spin reorientation transition will differ from results given by other methods are discussed. Fiz. Tverd. Tela (St. Petersburg) 41, 1050–1053 (June 1999)     

Fermion Fields,Boson Fields,and Gravitational Field

If a tetrad theory is derivable from a variational principle with a Lagrangian ?? of the form ?? = ??_F+??_M 6 tetrad components will be defined by the vacuum equations if the energy momentum tensor is symmetric. Therefore, we look for a realisation of a programme proposed in a little different way by TREDER according to which the 16 tetrad field equations should degenerate to 10 equations for the Riemannian metric if boson fields are the only source of the gravitational field.     

The GHP conformai Killing equations and their integrability conditions,with application to twisting type N vacuum spacetimes

The conformai Killing equations in resolved form and their first and second integrability conditions are obtained in the compact spin coefficient formalism for arbitrary spacetimes. To facilitate calculations an operatorL is introduced which agrees with the Lie derivative only when operating on quantities with GHP weights (0,0). The resulting equations are used to find the conditions for the existence of a two dimensional non-Abelian group of homothetic motions in a twisting typeN vacuum spacetime. The equivalence of two such sets of metrics is established, metrics that were recently the subject of independent investigations by Herlt on the one hand and by Ludwig and Yu on the other.     

Static characteristics of Josephson interferometers

This investigation deals with the range in operating currents for which a Josephson interferometer, sometimes also referred to as Superconducting QUantum Interference Device (SQUID), may remain in the zero-voltage Josephson condition. An interferometer consists of one or more inductive loops each of which contains two Josephson junctions or other weak links. Two types of current are considered. Gate currentI _gpasses the junctions in parallel. Control currentI _cgenerates magnetic flux via inductive coupling in the loops. Zero-voltage operation is possible within certain areas of theI _g,I _cplane. These areas are manifestations of flux-quantum states and their boundary lines are referred to as static characteristics. In view of the nonlinearity of the constituting equations, not all their formal solutions are physically realizable. A stability analysis yields criteria which permit the identification of realizable operating conditions. The static characteristics comprise operating conditions where the limit of stability is reached. To obtain the static characteristics, linearized equations may be utilized if theLI _o product, a measure for the size of an interferometer, is large compared to the flux quantumΦ ₀, whereL is the inductance per loop, andI _o the maximum Josephson current per junction. As a general method of solving system of transcendental equations, continuation is discussed. The utilization of continuation for obtaining interferometer characteristics is explained. It is shown that some changes in the gate-current feed arrangement are equivalent to shearing the characteristics in theI _g,I _cplane. Analytical results are given on extrema, inflexion points, and singularities in the shape of cusps which conceptually relate to the existence and connectivity of flux-quantum states. Experimental static characteristics are presented on two-and four-junction interferometers. They are in agreement with characteristics computed on the basis of simple lumped circuit models. Relevant circuit parameters are obtained from the experimental characteristics.     

E.S.R. Study of copper and silver N,N-dialkyldiselenocarbamates

Single crystal E.S.R. studies of Cu(II) and Ag(II) dialkyldiselenocarbamates, diluted in the corresponding Ni(II) complexes, are reported. A method has been developed which removes the necessity to measure the spectra on the same single crystal and in three mutually orthogonal planes. Equations have been derived for the resonance fields and the transition probabilities, taking into account the full spin hamiltonian and without assuming coinciding tensor axes. Applying these equations the g tensor and the metal hyperfine and quadrupole coupling tensors are obtained in high precision. In contrast to the situation in the corresponding dithiocarbamates, it was found for all diselenocarbamates studied that the axes of the g and metal hyperfine tensor do not coincide and that the relative orientation depends on the central metal ion and on the molecular symmetry. The highest g value was always found in the molecular plane, whereas in the dithiocarbamates the highest value is found perpendicular to the molecular plane. The spectra reveal the presence of two sets of two equivalent Se atoms, so that the molecule has inversion symmetry. Especially from the measured ⁷⁷Se hyperfine splittings it could be concluded that the built in guest molecule accepts the structure of the host crystal. Also measurements are reported on powders, glasses and liquid solutions. Evidence has been obtained that in the latter two media the g and metal hyperfine principal axes coincide.     

Algebraic connectivity analysis in molecular electronic structure theory I: coulomb potential,tensor connectivity, ε-approximation

In this (first) paper we attempt to generalize the notion of tensor connectivity, subsequently studying how this property is affected in different tensorial operations. We show that the often implied corollary of the linked diagram theorem, namely individual size-extensivity of arbitrary connected closed diagrams, can be violated in Coulomb systems. In particular, the assumption of the existence of localized Hartree–Fock orbitals is generally incompatible with the individual size-extensivity of connected closed diagrams when the interaction tensor is generated by the true two-body part of the electronic Hamiltonian. Thus, in general, size-extensivity of a many-body method may originate in specific cancellations of super-extensive quantities, breaking the convenient one-to-one correspondence between the connectivity of arbitrary many-body equations and the size-extensivity of the expectation values evaluated by those equations (for example, when certain diagrams are discarded from the method). Nevertheless, assuming that many-body equations are evaluated for a stable many-particle system, it is possible to introduce a workaround, called the ε-approximation, which restores the individual size-extensivity of an arbitrary connected closed diagram, without qualitatively affecting the asymptotic behavior of the computed expectation values. No assumptions concerning the periodicity of the system and its strict electrical neutrality are made.     

ON THE ADM EQUATIONS FOR GENERAL RELATIVITY

The Arnowitt–Deser–Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of quasi-linear partial differential equations, supplemented by the constraint equations. Such equations are here mapped into another first-order system. In particular, an evolution equation for the trace of the extrinsic-curvature tensor K is obtained whose solution is related to a discrete spectral resolution of a three-dimensional elliptic operator of Laplace type. Interestingly, all nonlinearities of the original equations give rise to the potential term in . An example of this construction is given in the case of a closed Friedmann–Lemaitre–Robertson–Walker universe. Eventually, the ADM equations are re-expressed as a coupled first-order system for the induced metric and the trace-free part of K. Such a system is written in a form which clarifies how a set of first-order differential operators and their inverses, jointly with spectral resolutions of operators of Laplace type, contribute to solving, at least in principle, the original ADM system.     

Über die Randwertaufgabe im Zusammenhang mit kraftfreien Magnetfeldern im rotationssymmetrischen Fall

The question of the existence and uniqueness of the solutions of the system in case of rotational symmetry can be reduced to the question of the resolvibility of a nonlinear elliptic differential equation for α which contains an arbitrary function g(α). Under rather general suppositions the boundary values of α and the function g(α) are given by the boundary values of H⊥ and H φ (the component of H which is perpendicular to the boundary). From the theory of non-linear elliptic equations one obtains: In sufficiently small domains the boundary value problem has a unique solution. For large domains these exist estimations about the diameter and area which guarantee the existence and uniqueness of the solutions.     

Momentum Space Diffusion from Spatially Amplified Waves in a Plasma in a Magnetic Field

Quasilinear equations for relativistic plasmas in external magnetic fields are derived for the case of spatially growing wave turbulence. This generalizes the well-known quasilinear approach to an amplifying relativistic plasma. The equations can be cast in condensed momentum-space diffusive form. The diffusion tensor is given. As a result an inhomogeneity is produced in the space charge distribution giving rise to the development of a second-order field-aligned dc electric field. A general formula for the electrostatic potential is presented, which is specialised to the case of ion-cyclotron turbulence. Here the field points out of the interaction region.     

A study of the magnetic,molecular and electronic structure of some alkali biphenyl ion pairs in the solid state

A magnetic resonance and susceptibility study of the molecular and magnetic structure of some alkali biphenyl (Bp) polyglyme single crystals and powders has been performed. For RbBp · 2Ttg (Ttg=tetraglyme), space group C2/c, and NaBp · 2Tg (Tg=triglyme), space group P2₁, the average g tensors have been measured. From the average g tensor and the known crystal structure of RbBp · 2Ttg the molecular g tensor of the Bp anion could be derived. The molecular g tensor was calculated with Stone's theory. Excellent agreement was found with the experimental values. With the aid of the molecular g tensor of the Bp anion and the average g tensor of NaBp · 2Tg, for which the crystal structure is unknown, the orientation of the symmetry axes of Bp with respect to the crystal axes could be calculated. Susceptibility measurements reveal ferromagnetic coupling in both KBp · 2Ttg and NaBp · 2Tg, whereas antiferromagnetic coupling exists in RbBp · 2Ttg. The results of alkali N.M.R. experiments are related to the molecular structure. A lower limit is found for the electrical conductivity of NaBp · 2Tg.     

Dynamical mechanism for varying light velocity as a solution to cosmological problems

A dynamical model for varying light velocity in cosmology is developed, based on the idea that there are two metrics in spacetime. One metric g_μν describes the standard gravitational vacuum, and the other describes the geometry through which matter fields propagate. Matter propagating causally with respect to can provide acausal contributions to the matter stress-energy tensor in the field equations for g_μν, which, as we explicitly demonstrate with perfect fluid and scalar field matter models, provides a mechanism for the solution of the horizon, flatness and magnetic monopole problems in an FRW universe. The field equations also provide a ‘graceful exit' to the inflationary epoch since below an energy scale (related to the mass of ψ_μ) we recover exactly the standard FRW field equations.     

Specific features of the EPR line shape of dimers due to spin-lattice relaxation

The magnetization relaxation in a coupled system of two spins has been shown to induce magnetization transfer between the observed transitions. In the degeneracy region of the transitions, the magnetization transfer becomes most effective and leads to the collapse of the magnetic resonance lines. In the electron paramagnetic resonance (EPR) spectra calculated for a polycrystal, this effect manifests itself in the form of additional peaks in the absorption line. The effect of anisotropy of the g factors of the dimer spins and misalignment of the axes of the g tensor and the interaction tensor on the appearance of additional lines in the EPR spectra has been analyzed.     

Mathematical Structures of Space-Time

At first we introduce the space-time manifold and we compare some aspects of Riemannian and Lorentzian geometry such as the distance function and the relations between topology and curvature. We then define spinor structures in general relativity, and the conditions for their existence are discussed. The causality conditions are studied through an analysis of strong causality, stable causality and global hyperbolicity. In looking at the asymptotic structure of space-time, we focus on the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar's spinorial variables. Finally, the question of a rigorous theory of singularities in space-times with torsion is addressed, describing in detail recent work by the author. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors do, because their definition of geodesics only involves the Christofel symbols, though studying theories with torsion. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychauduri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Our result can also be interpreted as a no-singularity theorem if the torsion tensor does not obey some additional conditions. Namely, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work is to be compared with important previous papers. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy-momentum tensor, and we emphasize the role played by the full extrinsic curvature tensor and by the variation formulae.