Showing papers on "Ricci decomposition published in 1982"

The Riemann tensor, the metric tensor, and curvature collineations in general relativity

Abstract: The equation xμνRμ λαβ+xμλRμ ναβ = 0, where xμν and Rμ ναβ are the components of an arbitrary symmetric tensor and of the Riemann tensor formed from the metric tensor gμν, is trivially satisfied by xμν = φgμν. Nontrivial solutions are important in various areas of general relativity such as in the study of curvature collineations, and also in the study of algebraic methods given by Hlavatý and Ihrig for the determination of gμν, from a given set of Rμ ναβ. We have found all Rμ ναβ for which there exist nontrivial solutions of the above equation, and we have given the form of the xμν in each case. Various examples of space–times for explicit nontrivial solutions are discussed.

Relativistic segnificance of curvature tensors

Abstract: In thi paper new curvature tensors have been defined on the lines of Weyl's projective curvature tensor and it has been shown that the “distribution” (order in which the vectors in question are arranged before being acted upon by the tensor in question) of vector field over the metric potentials and matter tensors plays an important role in shaping the various physical and geometrical properties of a tensor viz the formulation of gravitational waves, reduction of electromagnetic field to a purely electric field, vanishing of the contracted tensor in an Einstein Space and the cyclic property.

Mappings of empty space-times leaving the curvature tensor invariant

Abstract: It is shown that if in some local coordinate system the componentsRijkl of the curvature tensor of an empty space-time are known, then, provided the space-time is not of Petrov typeN with hypersurface orthogonal geodesic rays, the components of the metric tensor are uniquely determined up to a trivial constant scaling factor. The Petrov type-N empty space-times with hypersurface orthogonal geodesic rays are investigated. The most general mappings leaving the curvature tensorRijkl invariant are found for each class of these space-times.

Ōtsukische Räume mit Einem Zweifach Rekurrenten Metrischen Grundtensor

Abstract: In this note necessary and sufficient conditions are determined for Weyl—Ōtsukispaces to have a birecurrent metric, i.e.,∇ m ∇ k g ij =γ km g ij . It is proved that in this space the metric tensor is an eigen-tensor. The special caseP = ϱ(x)δ is examined and we prove that in this case the recurrent metric tensor is likewise birecurrent.

Conformally flat spaces and a pinching problem on the Ricci tensor

Abstract: Recent results of S. I. Goldberg on conformally flat manifolds and hypersurfaces of Eucidean space are extended.

Kaehlerian manifolds with constant scalar curvature whose Bochner curvature tensor vanishes

Abstract: Publisher Summary This chapter proves the theorem corresponding to that of Ryan, replacing the vanishing of the Weyl conformal curvature tensor in a Riemannian manifold by that of the Bochner curvature tensor in a Kaehlerian manifold. The chapter proves some lemmas that are used in the proof of the theorem. In a Kaehlerian manifold M of dimension n, the scalar curvature is constant, the Bochner curvature tensor vanishes and the Ricci tensor is positive semi-definite. From the method of the proof, it is easily see that the conclusion of the theorem is also valid if the assumptions of compactness and constant scalar curvature are replaced by local homogeneity of M.

Differential rotation of an electrically conducting fluid in general relativity

Abstract: Starting from the Ricci identity for the 4-velocity vectoru a ,a mathematical identity is derived, in terms of the kinematic quantities and the Riemann curvature tensor, for ω,a H a , the derivative along a magnetic field line of the magnitude of the vorticity of an electrically conducting fluid. Maxwell's field equations are not used in the derivation and the result is true for a fluid with finite, infinite, or even nonuniform, electric conductivity. Previous results [3–5] derived for an infinitely conducting fluid are obtained as special cases. By expressing the Riemann curvature tensor in terms of the Ricci and Weyl tensors, Einstein's field equations are introduced and the role played by the free gravitational field is examined. It is found that ω,aHa does not depend on the electric part of the Weyl tensor and that, for an infinitely conducting fluid satisfying certain “steady state” conditions, ω,aHa is independent of that part of the curvature determined locally through Einstein's field equations.

Tensor fields in crystals and colour groups

Abstract: The theory of permutational colour groups is used to construct tables of the k = 0 irreducible representations whose basis functions are linear combinations of the components of tensor fields defined on the atoms of a crystal. As examples of their use, these tables are shown to be applicable in determining the k = 0 vibrational modes of a crystal and in applying the Tensor Field Criterion in the Landau theory of continuous phase transitions.