Republication of: On the general relativity theory

This English translation of the paper by H. Weyl, “Zur allgemeinen Relativitätstheorie”, Physikalische Zeitschrift 24, 230–232 (1923), in which he formulated the geometrical foundations of a model of an expanding Universe, has been selected by the Editors of General Relativity and Gravitation for publication in the Golden Oldies series of the journal. The paper is accompanied by an editorial note written by Juergen Ehlers and by Weyl’s brief biography compiled by Andrzej Krasiński from internet sources, with corrections provided by Weyl’s son and grandson.     

Observables in the general theory of relativity

The theory of finite deviations of geodesics as directly observable variables in general relativity theory is elaborated in the proper coordinates of the reference frame of a solitary observer.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 80–83, July, 1976.     

Interacting fields in general relativity theory

The direct interaction of a massless neutral scalar field with an electromagnetic field is investigated with regard for the proper gravitational field. The interaction Lagrangian is chosen in the form L_int=FF, =e–1, where the parameter characterizes the interaction force. Exact static spherically and cylindrically symmetric solutions are obtained. A solution with a finite total field energy is extracted. A comparison is made with the corresponding system in flat space-time. It is concluded that the gravitational field performs a regulatory function.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 25–30, September, 1977.The authors are indebted to Yu. S. Vladimirov for valuable comments.     

Quantum theory and Einstein's general relativity

We discuss the meaning and prove the accordance of general relativity, wave mechanics, and the quantization of Einstein's gravitation equations themselves. Firstly, we have the problem of the influence of gravitational fields on the de Broglie waves, which influence is in accordance with Eeinstein's weak principle of equivalence and the limitation of measurements given by Heisenberg's uncertainty relations. Secondly, the quantization of the gravitational fields is a quantization of geometry. However, classical and quantum gravitation have the same physical meaning according to limitations of measurements given by Einstein's strong principle of equivalence and the Heisenberg uncertainties for the mechanics of test bodies.     

Between general relativity and quantum theory

Some possibilities of reconciling general relativity with quantum theory are discussed. The procedure of quantization is certainly not unique, but depends upon the choice of the coordinate conditions. Most versions of quantization predict the existence of gravitons, but it is also possible to formulate a quantum theory with a classical gravity whereby the expectation values ofT _µv constitute the sources of the classical metric field.     

Gravitational waves in the general theory of relativity

The problem of gravitational radiation is discussed with the help of the Rodichev energy-momentum tensor. An invariant criterion is formulated for finding radiation in a space without sources. In general, the question of the presence of radiation in a system is resolved by a local study of the behavior of the energy-momentum tensor in the comoving tetrad. The results are applied to the study of certain exact solutions of the Einstein equations.     

Gravitational waves in the general theory of relativity

The methods and capabilities of the invariant theory of gravitational radiation derived previously are demonstrated for an exact solution of the Kaigorodov type III. Analysis of this solution shows that it describes gravitational radiation with sources.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 121–125, November, 1969.In conclusion, we thank V. I. Rodichev for constant interest in this study.     

Energy problem in the general theory of relativity

The structure of the gravi-inertial field in a bounded (sufficiently small) region is similar to the structure of a quasistationary electromagnetic field, which allows formulating a local conservation law in the same formas in electrodynamics. Some ideas with respect to a global conservation law are presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 20–25, October, 1978.     

Energy complex in the general theory of relativity

In the tetradic formulation of the theory of gravitation an energy complex is proposed which makes it possible to calculate unambiguously the energy enclosed in any volume, in any co-ordinate mesh, with an arbitrary orientation of local, Cartesian, three-dimensional frames. In deriving the complex the analogy between the tetradic equations of the gravitational field and the covariant form of the equations of electrodynamics is exploited. As an example, a number of exact solutions of the Einstein equations are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 10, pp. 57–62, October, 1971.     

Plane waves in the general theory of relativity

The Newman-Penrose method is used to study the class of gravitational fields in a vacuum which permit a normal congruence of isotropic geodesies. The energy-momentum tensor is used in tetrad form to prove that if the nondegeneratemetric of these fields depends only on a single isotropic coordinate, the solutions will describe plane gravitational waves.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, Vol. 12, No. 4, pp. 20–23, April, 1969.     

Kinetic equation in the general theory of relativity

An attempt is made to discover the physical content of the general-relativistic theory of gases. Under an invariant interpretation of the collision term, this theory does not satisfy the correspondence principle for classical theory. The collision term in the classical Boltzmann form is meaningful only in an isolated reference frame realized by a locally inertial frame with the origin at the collision point. With this formulation of the kinetic equations, the class of equilibrium states in GR expands considerably and also covers nonstationary distributions. A specific example of such a distribution that is locally Maxwellian is given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 72–76, February, 1979.In conclusion, the author considers it his pleasant duty to thank Prof. N. A.Chernikov for useful discussion of certain problems touched on this paper, and A. V. Zakharov, who made this result of his article koown to the author.     

Electromagnetic radiation in the general theory of relativity

Spaces allowing a normal, shear-free, expanding congruence of isotropic geodesics are considered. A metric form allowing the simultaneous solution of the Einstein and Maxwell solutions for an isotropic electromagnetic field is derived in explicit form.Some of the nomenclature and definitions encountered in this paper were specified in Part I [1].Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 32–35, May, 1973.     

Magnetic monopole in the general theory of relativity

Equations of motion are obtained for Yang-Mills and Higgs fields using the Georgi-Glashow SO(3)-model in a curved spherically symmetric space-time. Magnetic monopole and dion solutions and the space-time metric are found in the intrinsic gravitational field. A particlelike solution of the magnetic monopole and dion type is constructed in the external field of a static universe in the Bogomol'nyi-Prasad-Sommerfield limit.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 89–93, November, 1982.     

Non-equilibrium statistical mechanics in the general theory of relativity: II. Linear fields in a kinetic approximation

The first paper in this series introduced a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object of this second paper is to apply that formalism to the evolution of a collection of particles that interact via linear fields in a fixed curved background spacetime. Given the viewpoint adopted here, the fundamental objects of the theory are a many-particle distribution function, which lives in a many-particle phase space, and a many-particle conservation equation which this distribution satisfies. By viewing a composite N-particle system as interacting one- and (N ? 1)-particle subsystems, one can derive exact coupled equations for appropriately defined reduced one- and (N ? 1)-particle distribution functions. Alternatively, by treating all the particles on an identical footing, one can extract an exact closed equation involving only the one-particle distribution. The implementation of plausible assumptions, which constitute straightforward generalizations of standard non-relativistic “kinetic approximations”, then permits the formulation of an approximate kinetic equation for the one-particle distribution function. In the obvious non-relativistic limit, one recovers the well-known Vlasov-Landau equation. The explicit form for the relativistic expression is obtained for three concrete examples, namely, interactions via an electromagnetic field, a massive scalar field, and a symmetric second rank tensor field. For a large class of interactions, of which these three examples are representative, the kinetic equation will admit a relativistic Maxwellian distribution as an exact stationary solution; and, for these interactions, an H-theorem may be proved.     

Bianci-IX cosmological models with rotation in general relativity theory

Bianci-IX metrics have been used to obtain new exact solutions to the GRT equations, which may describe the early stages in the evolution of an expanding and rotating universe.Kaluga Branch, Bauman Moscow State Technical University. M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnyk Zavedenii, Fizika, No. 9, pp. 122–127, September, 1994.     

Bianchi-II cosmological models with rotation in general relativity theory

We obtain a broad class of new exact solutions in Bianchi-II cosmologies with rotation and expansion. We show that within the framework of general relativity theory, we can construct realistic cosmological models in good agreement with current observational data.N. É. Bauman State Technical University, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 54–63, October, 1994.