Ricci decomposition

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

Some Properties of the Bel and Bel± Robinson Tensors

Abstract: The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum We present some new properties of these tensors which might throw some light onto this relationship First, for any spacetime we find a decomposition of the Bel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the “pure matter” super-energy tensor and the “matter-gravity coupling” super-energy tensor We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the “square” of the usual energy-momentum tensor This, together with the fact that the Bel-Robinson tensor has dimensions of energy density square, leads us to the definition of square root for the Bel-Robinson tensor: a two-covariant symmetric traceless tensor with dimensions of energy density and such that its “square” gives the Bel-Robinson tensor We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented The properties of this new tensor are examined and some interesting explicit examples are analyzed Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves We also examine the decomposition of the whole Bel tensor for Vaidya's radiating metric and Kerr-Newman's solution Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types

Black Box Low Tensor-Rank Approximation Using Fiber-Crosses

Abstract: In this article we introduce a black box type algorithm for the approximation of tensors A in high dimension d. The algorithm adaptively determines the positions of entries of the tensor that have to be computed or read, and using these (few) entries it constructs a low rank tensor approximation X that minimizes the l 2-distance between A and X at the chosen positions. The full tensor A is not required, only the evaluation of A at a few positions. The minimization problem is solved by Newton’s method, which requires the computation and evaluation of the Hessian. For efficiency reasons the positions are located on fiber-crosses of the tensor so that the Hessian can be assembled and evaluated in a data-sparse form requiring a complexity of $\mathcal{O}(Pd)$ , where P is the number of fiber-crosses and d the order of the tensor.

A Fully Nonlinear Equation on Four-Manifolds with Positive Scalar Curvature

Abstract: We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.

Tensor Triangular Geometry

Abstract: We survey tensor triangular geometry : Its examples, early theory and first applications. We also discuss perspectives and suggest some problems. Mathematics Subject Classification (2000). Primary 18E30; Secondary 14F05, 19G12, 19K35, 20C20, 53D37, 55P42.