Showing papers on "Ricci decomposition published in 1970"

Irreducible Cartesian Tensors. II. General Formulation

Abstract: A general formulation is given of a method of reduction of Cartesian tensors, by Cartesian tensor operations, to tensors irreducible under the three‐dimensional rotation group. The criterion of irreducibility is that a tensor be representable as a traceless symmetric tensor, its reduced or natural form, invariantly embedded in the space of appropriate order. The general formulation exploits the properties of invariant linear mappings between tensor spaces. Considered abstractly, such mappings bring out the structure of the theory and illuminate the relation to spherical tensor theory. On the other hand, any linear invariant mapping between tensor spaces is equivalent to a combination of operations with the elementary invariant tensors U and e. The general abstract formation therefore has a direct operational representation in terms of the ordinary tensor operations of contraction and permutation of indices. An analogous formulation is given for spinors, and the relations between spinors, Cartesian tensors, and spherical tensors is discussed in the language of the present formalism. Lastly, several examples are given as to how the general formalism may be applied to groups other than the rotation group.

Einstein Tensor and 3‐Parameter Groups of Isometries with 2‐Dimensional Orbits

Abstract: The algebraic classification of the Weyl and Ricci tensors and the relation between them in a Riemann space with an isometry group possessing a nontrivial isotropy group are reviewed. All metrics with Minkowski signature, invariant under a 3‐parameter isometry group with 2‐dimensional orbits having nondegenerate metrics, are constructed from the group properties and are shown to have Ricci tensors with a double eigenvalue, and the orbits are shown to be surfaces of constant curvature. The null orbits are shown to have a triply degenerate eigenvalue of the Ricci tensor. The various additionally degenerate metrics are classified in further detail, extending the work of Plebanski and Stachel.

Characteristic Classes and Weyl Tensor: Applications to General Relativity

Abstract: In a recent paper, Chern and Simons proved that the Pontrjagin forms of a Riemannian manifold remain invariant under a conformal deformation. We show that these forms can be expressed uniquely in terms of the conformal curvature tensor: this provides a new proof of their result. Similar techniques can be applied to Euler-Poincare characteristic class, as suggested to me by A. Taub. We obtain the following: If the Weyl tensor of a compact space time is of type III of Bel-Petrov, then it cannot carry a perfect fluid + electromagnetic field.

Some properties of gravitational fields of magnetic type.

Abstract: In this paper we have investigated some properties of the gravitational fields of magnetic type, using timelike congruences and the Bianchi identities. Null eigendirections of the conformal tensor are determined without any assumption regarding the nature of the sources of the gravitational field. Some relations between the kinematical quantities of the timelike congruences and the curvature of space-time have been derived. Our results reduce to those of empty space when the Ricci tensor is taken to be zero. These emptyspace equations lead us to a scalar potential which satisfies the field equation of Newton's theory of gravitation.

Die Einstein‐Rosen‐Materie

Abstract: The Einstein-Rosen matter is given by a space-time V4 in which the domains z 0 are isometric But, on the hypersurface z = 0 the Riemannian tensor Riklm and the Ricci tensor Rkl become delta-like infinite On this surface the first derivations of the metric tensor gik have essential discontinuities Therefore, it is impossible to cover the small domain –e ≤ z ≤ +e with one coordinate system of the class C2

A generalization of metric tensor and its application

Abstract: The definition of metric tensor and the well known Ricci theorem are modified. The Einstein tensor and Maxwell equations are generalized. The field equations of gravitation and electricity are unified. The introduction of a metric in the space with symmetric connection is defined.

Equations of the de Broglie Wave Field and Their Relationship to Riemann's Curvature Tensor

Abstract: The equations of the de Broglie wave field (field equations) [J Kulhanek, Nuovo Cimento Supp 4, 172 (1966)] under special conditions require a very particular geometry together with a specific interpretation of the curvature scalar The purpose of the present paper is to show that the same condition turns the conservation law (which is a consequence of the field equations) into an identity and that the Rainich [Nature 115, 498 (1925)] decomposition of Riemann's curvature tensor gives only one component

Decomposition of the Tensor Product of two Imaginary Mass Representations of the Poincaré Group

Abstract: The reduction of the tensor product is carried out by a method that is completely analogous to that used for real mass representations in the helicity coupling. This is possible through the use of a continuous basis for the UIRs of the little group SU(1,1). The Clebsch-Gordan coefficients are obtained in the same simple form as in the real mass case.