Showing papers on "Ricci decomposition published in 1976"

Left-Degenerate Vacuum Metrics

Abstract: For all complex space-times in which the self-dual part of the Weyl tensor is algebraically degenerate, Einstein's vacuum equations are reduced to a single differential equation of the second order and second degree.

On asymptotically flat space-times

Abstract: The allowed asymptotic behavior of the Ricci tensor is determined for asymptotically flat space-times. With the aid of Penrose's conformai technique the asymptotic behavior of the components of the metric tensor, Weyl tensor, and spin coefficients in a suitable frame is calculated for such a space-time. For Einstein-Maxwell space-times these results reduce to those of Exton, Newman, Penrose, Unti, and Kozarzewski.

The classification of the Ricci tensor in general relativity theory

Abstract: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory.

Differential geometry on Grassmann algebras

Abstract: H. C. Lee [1] developed the analogue of Riemannian geometry on a real symplectic manifold — the fundamental skew two-form taking the place of the symmetric tensor. The usual Riemannian concepts do not adapt themselves very well, thus ‘curvature’ is represented by a tensor of the third rank and ‘Killing's equations’ now involve this ‘curvature tensor’. The immediate reason for this is that otherwise familiar terms appear with the wrong sign. We have found that these unaesthetic features disappear, and formal elegance is marvellously restored, when the manifold is replaced by a Grassmann algebra. The connection with supersymmetry is explained but applications are not reported here.

Conformally flat manifolds and a pinching problem on the Ricci tensor

Abstract: There is a formal similarity between the theory of hypersurfaces and conformally flat d-dimensional spaces of constant scalar curvature provided d > 3. For, then, the symmetric linear transformation field Q defined by the Ricci tensor satisfies Codazzi's equation (Vx Q)Y = (Vy Q)X. This observation leads to a pinching theorem on the length of the Ricci tensor. 1. Statement of results. Recently, one of the authors [1] obtained THEOREM G. Let M be a d-dimensional compact conformallyflat manifold with definite Ricci curvature. If the scalar curvature r is constant and if the square of the length of the Ricci tensor is not greater than r2/(d 1), d > 3, then M is a space of constant curvature. Note that the square length of the Ricci tensor is greater than or equal to r2ld, so the Ricci tensor has been "pinched". In the present paper the following two theorems are proved, the first of which generalizes Theorem G. THEOREM 1. Let M be a d-dimensional compact conformally flat manifold with constant scalar curvature r. If the length of the Ricci tensor is less than r/ d -1, d > 3, then M is a space of constant curvature. THEOREM 2. In a d-dimensional compact conformally flat manifold M, if the length of the Ricci tensor is constant and less than r/ d -1, then M is a space of constant curvature. 2. Conformally flat manifolds. Let M be a Riemannian manifold of dimension d > 3. We cover M by a system of local coordinate neighborhoods (U, xh), and denote by gji, V>, Rk_h, and R1i the Riemannian metric, the operator of covariant differentiation in terms of the Riemannian connection, the curvature tensor and the Ricci tensor, respectively. We say that M is conformally flat if its Riemannian metric is conformally related to a locally Euclidean metric. In a conformally flat manifold, Received by the editors June 9, 1975 and, in revised form, October 15, 1975. AMS (MOS) subject classifications (1970). Primary 53A30, 53B20, 53C20.

A theorem on stress–energy tensors

Abstract: The equality of the symmetrized Noether stress–energy tensor (Belinfante’s tensor) and the canonical stress–energy tensor (functional derivative of the Lagrangian density with respect to the metric) is established by methods based on the formalism of tetrads and Ricci rotation coefficients. The result holds for any Lagrangian which contains no derivatives of the fields higher than first order.

Divergence-Free Third Order Concomitants of the Metric Tensor in Three Dimensions

Abstract: Publisher Summary This chapter discusses the concept of divergence-free third-order concomitants of the metric tensor in three dimensions. It defines a tensor density in a three-dimensional Riemannian space with metric tensor and local coordinates. In this condition, tensor density has the following properties: (1) it is a concomitant of the metric tensor together with its first three derivatives, (2) it is symmetric, (3) it is symmetric, and (4) it is trace-free. Recently, the tensor density in the given condition has arisen in two different contexts, in theoretical physics and in pure mathematics. It has occurred in the work of York who has discussed the physical significance of a tensor with particular reference to the Einstein initial-value problem for gravity. It has also arisen in the calculus of variations in the following manner.

Tensor spherical harmonics and tensor multipoles. II. Minkowski space

Abstract: The bases of tensor spherical harmonics and of tensor multipoles discussed in the preceding paper are generalized in the Hilbert space of Minkowski tensor fields. The transformation properties of the tensor multipoles under Lorentz transformation lead to the notion of irreducible tensor multipoles. We show that the usual 4‐vector multipoles are themselves irreducible, and we build the irreducible tensor multipoles of the second order. We also give their relations with the symmetric tensor multipoles defined by Zerilli for application to the gravitational radiation.

New identities on the Riemann tensor

Abstract: A set of new identities which involve second covariant derivatives and quadratic forms of the Riemann tensor are proved. These new identities can be thought of as integrability conditions derived from the equations that define the Rieman tensor in terms of the affine connections.