Electrodynamics in the general theory of relativity I. Null electromagnetic field

Conditions which must be satisfied by the energy-momentum tensor of the null electromagnetic field (i.e., by a field of pure radiation) in the general theory of relativity are formulated within the framework of the Newman-Penrose formalism. If a normal geodesic congruence is permitted in the space (this is equivalent to the allowed existence of wave fronts), there can be only two types of null electromagnetic fields. The asymptotic behavior of one of these types is analyzed.Translated from Izvestiya VUZ. Fizika, No. 10, pp. 83–87, October, 1969.In conclusion the author thanks V. I. Rodichev for an intersting discussion of this study.     

Non-static electromagnetic field in general relativity VI

The equations of Rainich's already unified field theory are investigated for the typeG ₃ II acting on the spatially homogeneous hypersurfacesx ⁰=constant. Two solutions of a non-static electromagnetic field for the above case are presented here by using the exterior differential calculus. The space-time admits a three-parameter continuous group of motions, the minimum invariant varieties being the geodesically parallel parametric surfacesx ¹=constant,x ²=constant andx ³=constant orthogonal tox ⁰=constant at points of the geodesics.     

Matrix formulation of special relativity in classical mechanics and electromagnetic theory

The two-component spinor theory of van der Waerden is put into a convenient matrix notation. The mathematical relations among various types of matrices and the rule for forming covariant expressions are developed. Relativistic equations of classical mechanics and electricity and magnetism are expressed in this notation. In this formulation the distinction between time and space coordinates in the four-dimensional space-time continuum falls out naturally from the assumption that a four-vector is represented by a Hermitian matrix. The indefinite metric of tensor analysis is a derived result rather than an arbitrary ad hoc assumption. The relation to four-component spinor theory is also discussed.     

Conservation laws in the tetrad formulation of the general theory of relativity

The covariant consequences of a weak conservation law in the tetrad formulation of general relativity that do not contain noncovariant complexes of energy-momentum or external and internal spin momenta are considered. The relationship between a group of arbitrary tetrad Lorentz transformations and a generally covariant definition of the spin angular momentum of nongravitational matter is outlined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 101–105, February, 1979.     

Mean field theory and geodesics in general relativity

It is proven that all geodesics in a mean gravitational field can be interpreted locally as the averages of geodesics in the unaveraged field. The respective time-like, space-like or null character of averaged and unaveraged geodesics is discussed carefully. Finally, some important astrophysical and cosmological applications and consequences are investigated.     

Coordinate-dependent 3 + 1 formulation of the general relativity field equations

This paper presents a coordinate-dependent 3+ 1 decomposition of the general relativity field equations in terms of a scalar potentialc ²[(–g ₄₄)^1/2–1], a vector potentialA _icg _4i/(–g₄₄)^1/2, and the three-space metric _ijg _ij–g_4i g _4j/g ₄₄. The equations are exact and the form of the decomposed equations is valid in any coordinate system.     

General relativity as a conformally invariant scalar gauge field theory

The global symmetry implied by the fact that one can multiply all masses with a common constant is made into a local, gauge symmetry. The matter action then becomes Conformally invariant and it seems natural to choose for the corresponding scalar gauge field the action for a conformally invariant (massless) scalar field. The resulting conformally invariant theory turns out to be equivalent to general relativity. Since this means that the usual Einstein-Hilbert action is not, in fact, a true gauge action for the space-time geometry, the full theory ought to be supplied with such a term. Gauge-theoretic arguments and conformal invariance requirements dictate its form.     

Fundamentals of the nonholonomically covariant formulation of the general theory of relativity

The physical foundations of the nonholonomic formulation of general relativity are determined, and the role of the Fock-Ivanenko coefficients in setting up and developing the tetrad formalism in general relativity is discussed. The physical and geometrical meaning of the nonholonomic transformations used in general relativity is determined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 55–60, December, 1974.     

Lattice formulation of general relativity

It is shown that almost the entire excitation energy acquired by the fission fragments during the descent from the saddle to the scission point comes from Landau-Zener transitions. The states tractable in the first order adiabatic approximation carry an excitation energy of a few hundred keV.     

Covariant formulation of the weak conservation law in the general theory of relativity without the energy-momentum of the gravitational field

A systematic covariant formulation is given of the weak conservation law in the general theory of relativity for arbitrary coordinate transformations without introducing the unsatisfactory definition of the energy-momentum of the gravitational field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 13–16 July, 1979.     

On the invariant mass conjecture in general relativity

An asymptotic symmetries theorem is proved under certain hypotheses on the behaviour of the metric at spatial infinity. This implies that the Einstein-von Freud-ADM mass can be invariantly assigned to an asymptotically flat four dimensional end of an asymptotically empty solution of Einstein equations if the metric is a no-radiation metric or if the end is defined in terms of a collection of boost-type domains.On leave of absence from the Institute of Mathematics, Polish Academy of Sciences, Warsaw, PolandSupported in part by the Polish Ministry of Science Research Project CPBP 01.2 and by the NSF Grant PHY8503072 to Yale University     

Null electromagnetic fields in general relativity

The necessary and sufficient condition that a general Riemannian space-time $$ds^2 = g_{ij} dx^i dx^j$$ , may represent a null electromagnetic field is S = C, where $$S = R_{hijk} F^{hi} F^{jk} ,C = C_{hijk} F^{hi} F^{jk} .$$ .     

Relativistic canonical formalism and the invariant single-particle distribution function in the general theory of relativity

The relativistic canonical formalism is used to construct an eight-dimensional phase space and an invariant distribution function, and integral and differential operations in the phase space and statistical averages, associated with the field of geodesic observers, are introduced. Liouville's theorem is proved.     

Interacting electromagnetic waves in general relativity

The problem is considered of finding exact solutions of the Einstein-Maxwell equations which describe the physical situation of two colliding and subsequently interacting electromagnetic waves. The general theory of relativity predicts a nonlinear interaction between electromagnetic waves. The situation is described using an approximate geometrical method, and a new exact solution describing two interacting electromagnetic waves is given. This describes waves emitted from two sources mutually focusing each other on the opposite source.     

相对论的几何表述②

2.4 "尺缩"效应 质点在三维语言中是空间的一点,在四维语言中是时空中的一条类时线(世界线).同理,一把细尺在三维语言中是一段直线,在四维语言中则是一个二维面(称为世界面,是介于尺头、尺尾世界线之间的二维面)于是,在三维语言中非常明确的"尺长"概念在四维语言中变得含糊:哪条线的线长才是尺长?对熟悉四维语言的相对论工作者来说,细尺本来就不是一维的,它是个二维对象.这是一个绝对的对象.与参考系、坐标系以及观者无关.为什么细尺在普通语言(三维语言)中是一维对象?因为人人都站在自己所在参考系的立场上看问题,这种看法一开头就是相对的.K系的每一同时面Σt既然代表t时刻的全空间,它与细尺世界面的交线(直线段oa,见图15)自然代表"K系测得的、t时刻的细尺".     

Cosmological solutions foe a spinor field in the general theory of relativity

We consider models of the universe containing linear and nonlinear spinor matter. It is assumed that the linear spinor matter is described by the generally covariant Dirac equation, and the nonlinear by the generally covariant Ivanenko-Heisenberg equation.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 68–71, December, 1973.     

Canonical formulation of spin in general relativity

The present article aims at an extension of the canonical formalism of Arnowitt, Deser, and Misner from self‐gravitating point‐masses to objects with spin. This would allow interesting applications, e.g., within the post‐Newtonian (PN) approximation. The extension succeeded via an action approach to linear order in the single spins of the objects without restriction to any further approximation. An order‐by‐order construction within the PN approximation is possible and performed to the formal 3.5PN order as a verification. In principle both approaches are applicable to higher orders in spin. The PN next‐to‐leading order spin(1)‐spin(1) level was tackled, modeling the spin‐induced quadrupole deformation by a single parameter. All spin‐dependent Hamiltonians for rapidly rotating bodies up to and including 3PN are calculated.     

The symmetry aspects of quantum field theory in general relativity 1

Some proposed models for a quantum field theory in general relativity are briefly analyzed. Their main difficulties are a consequence of the initial choice of the group of symmetries of the (quantum) field equations. The necessity of selecting space-time isometries in a general covariant theory and the unphysical character of the Poincaré translations in a tangent plane theory are discussed. Starting from some basic requirements, a model is proposed in which the groups of symmetries are derived from the proper homogeneous groups of isometries of the minimal isometric local embedding spaces of space-times.This essay received an honorable mention (1977) from the Gravity Research Foundation-Ed.