The energy tensor of matter in the projectve theory of relativity

In the projective theory of relativity the 5-dimensional field equation \(_{\mu \nu } \) and the resulting equation of motion Tμυ_;μ = 0 are investigated. There Tμυ stands for the 5-dimensional tensor of macroscopic matter. The 4-dimensional field equations and equation of motion obtained by projection are a generalization of Einstein's theory of general relativity and Maxwell's electrodynamics, involving a scalar field φ.They contain a single constant φ₀.The weak field approximation is investigated for the case of an ideal fluid and leads to Newton's mechanics, including Newton's gravitational law, and to Maxwell's electrodynamics. For the constant φ₀ one obtains the approximate value φ₀c⁴/γ_N with Newton's gravitational constant γ_N.For homogeneous and isotropic cosmological models consisting of matter only the general solution for the radius K of curvature is given. This solution is independent of the equation of state of matter For a pure dust universe the general solution for the scalar field φ is given. For a closed universe a power law φ ?K^?1 is valid which leads to Mach's principle. The calculation of the age of a closed universe yields over 7×10⁹y,if one uses mean values of the present cosmological data.     

The cosmic field tensor in bimetric general relativity

We construct all cosmic field tensors which are symmetric rank-two tensor concomitants of a metric and a background metric and which have zero divergence when the background metric satisfies the generalized De Donder condition. The resulting background cosmic field represents an Einstein space-time.     

Self-gravitating static massive vector field in general relativity

A static spherically symmetric solution is found to the combined system of Einstein's equations and the equation of a massive vector field, the cosmological term being taken into account. It is shown that such a self-gravitating configuration forms a closed inhomogeneous cosmological model.Translated from Izvestiya Vysshlkh Uchebnykh Zavedenii, Fizika, No. 4, pp. 56–59, April, 1980.     

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.     

On a new variational principle in general relativity and the energy of the gravitational field

A new variational principle based on the affine connection in space-time is proposed. This leads to a new formulation of general relativity. The gravitational field is a field of inertial frames in space-time. The metricg appears as a momentum canonically conjugate to the gravitational field. In the case of simple matter fields, e.g., scalar fields, electromagnetic fields, Proca fields, or hydrodynamical matter, the new formulation is equivalent to the traditional one. A new formulation of conservation laws is proposed.     

Charged spinning fluids in general relativity

The treatment of neutral spinning fluids in general relativity due to Ray and Smalley is generalized, by considering the fluid as charged and having a magnetic dipole moment proportional to the spin.     

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.     

Conformal tensor discontinuities in general relativity

The postulate is made that across a given hypersurfaceN the metric and its first derivatives are continuous. This postulate is used to derive conditions which must be satisfied by discontinuities in the Riemann tensor acrossN. These conditions imply that the conformal tensor jump is uniquely determined by the stress-energy tensor discontinuity ifN is non-null (and to within an additive term of type Null ifN is lightlike). Alternatively, and [R] determine ifN is non-null. These relationships between the conformal tensor and stress-energy tensor jumps are given explicitly in terms of a three-dimensional complex representation of the antisymmetric tensors. Application of these results to perfect-fluid discontinuities is made: is of type D across a fluid-vacuum boundary and across an internal, non-null shock front. is of type I (non-degenerate) in general across fluid interfaces across which no matter flows, except for special cases.The research reported herein was supported in part by the Atomic Energy Commission under contract number AT (11-1)-34, Project Agreement No. 125, while the author was at the University of California at Berkeley, and in part by National Science Foundation Grant GU-1598-University Science Development Program.     

Collineations of the curvature tensor in general relativity

Curvature collineations for the curvature tensor, constructed from a fundamental Bianchi Type-V metric, are studied. We are concerned with a symmetry property of space-time which is called curvature collineation, and we briefly discuss the physical and kinematical properties of the models.     

Summary of classical general relativity workshop

This is a summary of the presentations at the parallel session in the classical general relativity workshop of the ICGC-2004.     

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.     

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.     

Analysing the elasticity difference tensor of general relativity

The elasticity difference tensor, used in [1] to describe elasticity properties of a continuous medium filling a space-time, is here analysed. Principal directions associated with this tensor are compared with eigendirections of the material metric. Examples concerning spherically symmetric and axially symmetric space-times are then presented.     

An algebraic approach to quantum field theory and commutation relations for energy momentum in the framework of general relativity

We present a consistent set of commutation relations (C.R.) for a quantum system immersed in a classical gravitational field. The gravity field is described by metric tensorg _ik (x) andg ₀₀(x) with coordinate gaugeg _i0=0. The Hamiltonian of the system is found to be a linear function of [–g ₀₀(x)]^1/2. Its properties we define by C.R. avoiding explicit expression in terms of fields, as well as its splitting into free and interaction parts. In this way a consistent set of C.R., which are equally simple for a flat and curvilinear space, can be established. To stress the main idea of our approach, we consider the simple but still nontrivial example of a scalar electrodynamics immersed in a gravity field. The electromagnetic current operator we define by its C.R. and not explicitly. An interesting feature of this approach is that the Poisson equation follows from the consistency of the C.R. The C.R. for the energy and momentum operators of the system in a gravity field are established which generalize the usual Poincare group generators C.R. For example, we find (i/hc ²)[H _(x) ,H _(x) ]=P _– , whereH _(x) is the Hamiltonian of the system, which is a linear functional of (x)[–g ₀₀(x)]^1/2 andP _s(x) represents the momentum-density operator [averaged with the classical functions(x)].     

On five-dimensional unified field theory and projective relativity

The choice between the Kaluza-Klein and the Jordan-Thiry forms of five-dimensional unified field theory has been considered by McInnes. Here a reply is made to criticisms by McInnes of a previous paper. In addition, the implications of projective relativity are considered.     

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.     

Some remarks on the energy-momentum tensor in general relativity

In general relativity, the energy-momentum tensor of a classical tensor field can be constructed by varying the action of the field with respect to the background metric. This paper suggests an alternative interpretation of the construction which also makes sense for spinor fields, and which gives some insight into the locality of energy-momentum operators in generally covariant quantum field theory.     

Positive energy in general relativity

An argument is given in favor of the conjecture that an isolated system in general relativity must have nonnegative total energy.