Showing papers on "Ricci decomposition published in 1987"

Nonlinear gravitational Lagrangians

Abstract: “Alternative gravitational theories” based on Lagrangian densities that depend in a nonlinear way on the Ricci tensor of a metric are considered. It is shown that, provided certain weak regularity conditions are met, any such theory is equivalent, from the Hamiltonian point of view, to the standard Einstein theory for a new metric (which, roughly speaking, coincides with the momentum canonically conjugated to the original metric), interacting with external matterfields whose nature depends on the original Lagrangian density.

Inhomogeneous Einstein metrics on complex line bundles

Abstract: Recent developments in higher-dimensional unified field theories have led to a great deal of interest in compact spaces admitting Einstein metrics. Almost all the physics literature on such spaces has been concerned with the very atypical case in which the space is homogeneous. The authors present a very simple construction of a wide class of inhomogeneous compact Einstein spaces with positive Ricci curvature found earlier by Berard-Bergery, (1982) which arise as certain 2-sphere bundles over an arbitrary Einstein-Kahler base space of positive Ricci curvature. Solutions on complete non-compact manifolds also exist, with negative or zero Ricci curvature.

Three-dimensional space-times

Abstract: A real version of the Newman-Penrose formalism is developed for (2+1)-dimensional space-times. The complete algebraic classification of the (Ricci) curvature is given. The field equations of Deser, Jackiw, and Templeton, expressing balance between the Einstein and Bach tensors, are reformulated in triad terms. Two exact solutions are obtained, one characterized by a null geodesic eigencongruence of the Ricci tensor, and a second for which all the polynomial curvature invariants are constant.

Representations for Isotropic and Anisotropic Non-Polynomial Tensor Functions

Abstract: Material symmetries of a continuum impose definite restrictions on the form of constitutive relations. The restrictions are specified in the representations of isotropic and anisotropic tensor functions and indi­cate the type and the number of independent variables involved in a cons­titutive relation. Thus, in a properly written constitutive equation, the material symmetries are automatically verified.

Some perfect fluid solutions of Einstein's field equations without symmetries

Abstract: Metrics of the form ds2=N2(x1, xn)(dx1)2+gab(xn) xadxb are considered which are subject to the conditions that the time-like 3-space (with metric gab) is conformally flat, that its Ricci tensor has at most two different eigenvalues and that the 4-velocity is an eigenvector of this Ricci tensor. The perfect fluid (or dust) solutions are necessarily of Petrov type D or O, and in the general case they do not admit a Killing vector. All rotating solutions are given explicitly. The non-rotating solutions are either conformally flat (and thus known) or (if of type D) contained in the class of solutions investigated, for example, by Szekeres (1975), Tomimura (1977), and Szafron and Wainwright (1977).

Two important invariant identities

Abstract: Two important invariant identities about the products of the Riemann tensor R/sub cap alpha//sub beta//sub gamma//sub lambda/ and the Ricci tensor R/sub cap alpha//sub beta/ and the scalar curvature R are derived by the technique of Weyl decomposition of the Riemann tensor and by the spinor formalism These identities are very useful in four dimensions for simplifying the final expression of the a/sub 3/ coefficient of the scalar fields and for simplifying the evaluation of the vacuum-polarization energy-momentum tensor This result is of relevance to the work of Jack and Parker on the summed form of the heat kernel

Adaptive tensor product grids for singular problems

Abstract: We consider piecewise polynomial approximation of order M in N-dimensions with a tensor product partition of the space. We assume that the partition is to be chosen to minimize ------'the-maximum----em>r---in-the-approximation.------1be---optimal---Iate---gf--cgrwetgence-foyieGeWisef--------polynomial approximation to a smooth function for unconstrained partitions is known to be order ~/N where K is the number of elements in the partition. This rate of convergence is achieved K by a uniform grid which may be taken to be a tensor product. In 1979 de Boor and Rice gave an adaptive algorithm which achieves this same order of convergence for a wide variety of singular functions. We now study whether this optimal order of convergence can be achieved by partitions constrained to be tensor products. We show that the optimal order of convergence is achieved by tensor product grids (partitions) for functions with point or boundary layer singularities. For some other singularities, the tensor product constraint reduces the order of convergence substantially_ This wol1r.was supported in part by the National Science POODdation graDtMS-83-01589

Space-time curvature and the sources of gravity

Abstract: The behaviour of the derivatives of the metric tensor under coordinate transformations is studied and some conclusions regarding the coordinate-invariant quantities that can be formed from these are drawn. It is pointed out that the curvature tensor is the simplest invariant related to the second derivatives of the metric. A proper choice of coordinates at a given point is shown to greatly simplify many formulae and thus some light is thrown on what the sources of the gravitational field are when one considers the metric to be the potential of the field.

The gravitational field in the wave zone II. Consequences of asymptotic structure

Abstract: We consider an asymptotically flat and empty space-time generated by a bounded source of perfect fluid. The vanishing of the conformal Weyl tensor onI+ and of the Ricci tensor nearI+ are used to simplify the expression obtained in the previous paper for the coefficient ofr− of the metric tensor after an expansion in powers ofc−1. The result is a very simple expression for the dominant term ofgαβ,0 in the radiation zone in terms of the quadrupole moment of the source. Using this expression and an invariant definition of the total energy, we calculate in the framework of full general relativity the radiated energy per unit time and prove that the first term is identical with the quadrupole radiation as given by the linearized version of general relativity.

Tensor analysis and curvature in quantum space-time

Abstract: Introducing quantum space-time into physics by means of the transformation language of noncommuting coordinates gives a simple scheme of generalizing the tensor analysis. The general covariance principle for the quantum space-time case is discussed, within which one can obtain the covariant structure of basic tensor quantities and the motion equation for a particle in a gravitational field. Definitions of covariant derivatives and curvature are also generalized in the given case. It turns out that the covariant structure of the Riemann-Christoffel curvature tensor is not preserved in quantum space-time. However, if the curvature tensor $$\hat R$$ μvλϰ (z) is redetermined up to the value of theL 2 term, then its covariant structure is achieved, and it, in turn, allows us to reconstruct the Einstein equation in quantum space-time.