Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.

The Cotton tensor in Riemannian spacetimes

Abstract: Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n(n2 − 4)/3 for the first time. Subsequently, we show its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw and Templeton. For each class examples are given. Finally, we investigate the relation between the Cotton tensor and the energy–momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.

Generalized quasi-Einstein manifolds with harmonic Weyl tensor

Abstract: In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n − 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.

Averaging out the Einstein equations

Abstract: A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.

When is gttgrr = -1?

Abstract: The Schwarzschild metric, its Reissner–Nordstrom–de Sitter generalizations to higher dimensions and some further generalizations all share the feature that gttgrr = −1 in Schwarzschild-like coordinates. In this pedagogical note we trace this feature to the condition that the Ricci tensor (and stress–energy tensor in a solution to Einstein's equation) has vanishing radial null–null component, i.e., is proportional to the metric in the t–r subspace. We also show that this condition holds if and only if the area–radius coordinate is an affine parameter on the radial null geodesics.

Prescribing symmetric functions of the eigenvalues of the Ricci tensor

Abstract: We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.