Showing papers on "Ricci decomposition published in 1972"

Scalar-Tensor Theory of Gravitation

Abstract: A scalar-tensor theory of gravitation is constructed using the Weyl formulation of Riemannian geometry. The scalar field is given an important geometrical role to play and is related to the integrable change in length of a vector as it is transported from point to point in space-time. The geometry uses modified covariant derivatives and a metric tensor which is not covariantly constant. The field equations can be written down very simply in terms of a modified curvature tensor. The theory agrees with the usual Lagrangian formalism in its experimental predictions and offers a reformulation or reinterpretation of the transformation of units considered by Dicke.

The Algebra of the Riemann Curvature Tensor in General Relativity: Preliminaries

Abstract: In a four-dimensional curved space-time it is well-known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley-Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.

On a horizontal conformal Killing tensor of degreep in a sasakian space

Abstract: We deal with a horizontal conformal Killing tensor of degree p in a Sasakian space. After some preparations we prove that a horizontal conformal Killing tensor of odd degree is necessarily Killing. Moreover, we consider horizontal conformal Killing tensor of even degree. The form of the associated tensor is determined completely and a decomposition theorem is proved. Then we give the examples of a conformal Killing tensor of even degree and a special Killing tensor of odd degree with constant l.

The Algebra of the Riemann Curvature Tensor in General Relativity: The Relation of the Invariants of the Einstein Curvature Tensor to the Invariants Describing Matter

Abstract: In this paper, the second in a series of papers on the algebra of the Riemann curvature tensor, we relate the algebraic invariants of the Einstein curvature tensor to the algebraic invariants of the trace-free part of the Ricci tensor, and, consequently, to the trace-free part of the stress-energy tensor for the physical system. We also show explicitly how all of the components of the trace-free part of the Ricci tensor transform under a Lorentz transformation.

The Riemann tensor and stress singularities

Abstract: The algebraic invariants of the Riemann tensor for a static axially symmetric gravitational field are examined. Contrary to common expectation they are shown to be well defined in a region of the field known to be physically singular.