The basic equations of the dynamics of the continuous distribution of dislocations I. General theory

The basic equations of the dynamics of the continuous distribution of dislocations analogical to Maxwell's equations are derived in a series of papers I, II, III. The analogy of the elastic and electromagnetic fields is analyzed. In part I the basic equations of the continuous distribution of dislocations are derived for the dynamic case (especially for small deformations and for the Poisson ratio equal to zero in a continuum, infinite with respect to space and time, from the variational problem by means of the formal apparatus of the special theory of relativity.

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I.

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The basic equations of the dynamics of the continuous distribution of dislocations III. special problems

The basic equations of the dynamics of the continuous distribution of dislocations analogical to Maxwell equations are derived in a series of papers [I, II, III]. The analogy of the elastic and electromagnetic fields is analyzed. In part [III] some special problems are discussed, such as the density of the forces acting on the dislocations, the energy dissipation during the movement of dislocations, which is expressed by an equation analogical to Ohm's law. The equations derived in the previous parts in four-dimensional symbolics are considered in the three-dimensional differential and integral form. It is found that in special cases the relations become the known ones of elastodynamics, hydrodynamics and the static theory of the continuous distribution of dislocations. It is found that Kröner's method of integrating the equations of the dislocation field by means of so-called incompatibility tensors is analogical to the integration of the Maxwell equations by means of Hertz vectors. The analogy between the elastic dislocation field and the electromagnetic field is discussed in detail.

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III.

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A new approach to the theory of relativity. II. The general theory of relativity

The considerations of Part I are extended and the experimental data and hypotheses that led to the establishment of the general theory of relativity are analyzed. It is found that one of the fundamental assumptions is that light is propagated homogeneously; i.e., by using arbitrary systems of coordinates, propagation of light can be represented by a homogeneous quadratic form. This is shown to be an assumption that can be verified by experiment, at least in principle. As a result of adding a number of further assumptions to this, the usual formalism of the general theory of relativity can be established. In the above point of view, the general theory of relativity—like any other theory—cannot be built upad hoc, but is built on distinct physical hypotheses, each of which can be subjected to test by experiment.     

The general class of the vacuum spherically symmetric equations of the general relativity theory

The system of the spherical-symmetric vacuum equations of the General Relativity Theory is considered. The general solution to a problem representing two classes of line elements with arbitrary functions g ₀₀ and g ₂₂ is obtained. The properties of the found solutions are analyzed.     

The dynamics of charged dust in the general theory of relativity

The spherically symmetric motion of charged dust in its own gravitational field (the analog of Tolman's problem) is investigated. A conclusion is reached about the cessation of the contraction of the charged dust cloud.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 63–67, December, 1973.     

Republication of: Contributions to the theory of pure gravitational radiation. Exact solutions of the field equations of the general theory of relativity II

This is an English translation of a paper by Pascual Jordan, Juergen Ehlers and Rainer Sachs, first published in 1961 in the proceedings of the Academy of Sciences and Literature in Mainz (Germany). The original paper was part 2 of a five-part series of articles containing the first summary of knowledge about exact solutions of Einstein’s equations found until then. (Parts 1 and 4 of the series have already been reprinted, parts 3 and 5 will be printed as Golden Oldies in near future.) This second paper discusses the geometry of geodesic null congruences, the algebraic classification of the Weyl tensor by spinor methods, and applies these to a study of the propagation of gravitational and electromagnetic radiation. It has been selected by the Editors of General Relativity and Gravitation for republication in the Golden Oldies series of the journal. The republication is accompanied by an editorial note written by Malcolm A. H. MacCallum and Wolfgang Kundt.     

Macroscopic Maxwell's equations of the general theory of relativity

Covariant Maxwell's equations of the general theory of relativity for a system of electromagnetically and gravitationally interacting particles of the form

A generalized field theory. II. Linearized field equations

A thorough examination of the generalized field theory, formulated by the same authors in a previous paper, is being carried out in the absence of feedback effects. The results obtained are found to be in complete agreement with those of linear field theories of gravity and electromagnetism. Strict functions, which serve as indicators of the strength of the two fields, are being identified. This study reveals also two interesting results: the first is the classification of tetrad vector fields used, the second is the definite appearance of a mutual interaction between gravitational and electromagnetic fields.     

Method of green's functions in the theory of quantum kinetic equations. II

The kinetic and antikinetic equations are obtained for the single-particle Wigner function in the context of the method of Green's time-temperature functions for an inhomogeneous system of weakly interacting particles situated in a time-dependent electric field. The kinetic equation is derived here from the equation of motion for Green's function, satisfying the causality condition.     

Nonlinear effects in the relative dynamics of trial particles in the general theory of relativity II. Effect of small perturbing forces

The effect of small perturbing forces in the relative dynamics of trial particles is studied. An asymptotic approximation is obtained for the solution of the problem with accuracy to within ² in the nonresonant case and in the case of the fundamental and off-multiple resonances. Phenomena associated with nonlinear resonance (a cutoff of the oscillation amplitude and parametric resonance) arise in the system.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 21–30, December, 1978.     

The gauge theory of dislocations: Static solutions of screw and edge dislocations

We investigate the T(3)-gauge theory of static dislocations in continuous solids. We use the most general linear constitutive relations in terms of the elastic distortion tensor and dislocation density tensor for the force and pseudomoment stresses of an isotropic solid. The constitutive relations contain six material parameters. In this theory, both the force and pseudomoment stresses are asymmetric. The theory possesses four characteristic lengths ?₁, ?₂, ?₃ and ?₄, which are given explicitly. We first derive the three-dimensional Green tensor of the master equation for the force stresses in the translational gauge theory of dislocations. We then investigate the situation of generalized plane strain (anti-plane strain and plane strain). Using the stress function method, we find modified stress functions for screw and edge dislocations. The solution of the screw dislocation is given in terms of one independent length ?₁ = ?₄. For the problem of an edge dislocation, only two characteristic lengths ?₂ and ?₃ arise with one of them being the same ?₂ = ?₁ as for the screw dislocation. Thus, this theory possesses only two independent lengths for generalized plane strain. If the two lengths ?₂ and ?₃ of an edge dislocation are equal, we obtain an edge dislocation, which is the gauge theoretical version of a modified Volterra edge dislocation. In the case of symmetric stresses, we recover well-known results obtained earlier.     

A background-dependent approach to the theory of gravitation II. Relativistic gravitational equations

On the basis of the results of Paper I and guided by a Machian view of nature, we find new gravitational equations which are background dependent. Such equations describe a purely geometrical theory of gravitation, and their dependence on the background structure is through the total energy-momentum tensor on the past sheet of the light cone of each space-time pointx [θ^μν x, say], i.e., through the integral on the past sheet of the light cone ofx of the parallel transport of the energy-momentum tensor from the space-time point in which it is defined tox along the geodesic connecting the two space-time points. Following Gürsey, we assume that the source of the De Sitter metric is not the cosmological term, but, rather, the energy-momentum tensor of a “uniform distribution of mass scintillations” [T ^μν x, say].T ^μν x, indeed, turns out to be equal to the metric tensor times a constant factor. As a consequence, in any local inhomogeneity A of a space-time whose background structure is determined by the Perfect Cosmological Principle,θ ^μν turns out to be approximately equal to the metric tensor times a constant factor, providedT=g _αβ T ^αβ is sufficiently small and the structure of the past sheet of the light cones of the space-time points belonging to Λ is not too much perturbed by the local gravitational field. As a consequence, in Λ the new equations approximately reduce to Einstein's equations. If one considers a “superuniverse model” in which our universe is considered as a local inhomogeneity in a De Sitter background, then from the above result there follows a fortiori the agreement of the new gravitational equations with the classical tests of gravitation. Furthermore, the dependence on the background structure is such that the new equations (i) incorporate the idea that the frame has to be fixeddirectly in connection with cosmological observations, and (ii) are singular in the absence of matter in the whole space-time. Moreover, (iii) the coupling constant turns out to be dimensionless in natural units (c=1=?), and (iv) a local inertial frame in a De Sitter background is determined by the condition that with respect to it the background structure is homogeneous in space and in time and is Lorentz invariant.     

Non-equilibrium statistical mechanics in the general theory of relativity III. Collisional stellar dynamics

This paper assembles and extends earlier results to formulate a coherent theory of relativistic stellar dynamics appropriate for comparatively small systems of stars in which relativistic effects can be important. The structure of the Newtonian theory is outlined, culminating in the “collisional Boltzmann” or Fokker-Planck equation appropriate for an unconfined system of point masses. The theory of relativistic Fokker-Planck equations is then developed for general Lorentz-covariant interactions such as electromagnetism or scalar fields. The basic physical ingredients of Newtonian stellar dynamics are identified, and it is indicated how they can be reformulated relativistically. These considerations are then used to construct a relativistic Fokker-Planck equation appropriate for the evolution of a collection of point mass stars. The analysis is then generalized to allow, both Newtonianly and relativistically, for the effects of direct physical collisions between stars of finite size. By way of conclusion and illustration, the theory is applied to the study of a prototypical dense galactic nucleus which could evolve to contain a massive black hole. The paper ends by enumerating a number of tractable unsolved problems deserving of further consideration.     

Fundamental concepts of the theory of dislocations

The fundamental concepts of the theory of dislocations are given in an exact way and some theorems are proved.

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In conclusion the author would like to thank Professor M. Valouch and Professor L. Zachoval for the interest they took in this work and for remarks and advice which helped to improve it.     

Dynamics of solidification in binary melts during rapid cooling II. Analysis of basic equations

In the first part of our paper we have derived a set of stochastic differential equations which describe the solidification of binary melts. The equations have been derived within the framework of the model in which the mass and heat transport and the kinetics of the phase transition is considered. In the second part of our paper we present the analysis of the set of general equations. On the basis of this analysis it will be obvious which approximations can be used for the solution of the basic equations in a particular regime of solidification. The adiabatic approximation is one of them. Another situation occurs when the thermodynamic conditions of the phase transformation change fast with time. The system not only moves away from thermodynamic equilibrium but also we can observe inertia of the system, which results in a delay of the evolution of the system respecting the steady-state regime (e.g. the nucleation processes) and the adiabatic approximation cannot be assumed. Concluding this paper, the method used in paper [13] to describe the nucleation in binary systems is presented as an example of the solution of the set of general equations in the case where the adiabatic approximation cannot be adopted.