Congruences of Null Strings and Their Relations with Weyl Tensor and Traceless Ricci Tensor

2017 ◽  
Vol 10 (2) ◽  
pp. 373
Author(s):  
A. Chudecki
Keyword(s):  
Weyl Tensor ◽  
Ricci Tensor

scholarly journals WEYL COLLINEATIONS THAT ARE NOT CURVATURE COLLINEATIONS

2005 ◽  
Vol 14 (08) ◽  
pp. 1431-1437 ◽  
Author(s):  
IBRAR HUSSAIN ◽  
ASGHAR QADIR ◽  
K. SAIFULLAH
Keyword(s):  
Linear Combination ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Weyl Tensor ◽  
Ricci Tensor ◽  
Lie Symmetries ◽  
Ricci Scalar ◽  
Finite Dimensional ◽  
Fundamental Reason ◽  
Curvature Collineations

Though the Weyl tensor is a linear combination of the curvature tensor, Ricci tensor and Ricci scalar, it does not have all and only the Lie symmetries of these tensors since it is possible, in principle, that "asymmetries cancel." Here we investigate if, when and how the symmetries can be different. It is found that we can obtain a metric with a finite dimensional Lie algebra of Weyl symmetries that properly contains the Lie algebra of curvature symmetries. There is no example found for the converse requirement. It is speculated that there may be a fundamental reason for this lack of "duality."

scholarly journals General properties of f(R) gravity vacuum solutions

2020 ◽  
Vol 29 (13) ◽  
pp. 2050089
Author(s):  
Salvatore Capozziello ◽  
Carlo Alberto Mantica ◽  
Luca Guido Molinari
Keyword(s):  
Weyl Tensor ◽  
Ricci Tensor ◽  
Curvature Scalar ◽  
General Properties ◽  
Walker Metric ◽  
Vacuum Solutions

General properties of vacuum solutions of [Formula: see text] gravity are obtained by the condition that the divergence of the Weyl tensor is zero and [Formula: see text]. Specifically, a theorem states that the gradient of the curvature scalar, [Formula: see text], is an eigenvector of the Ricci tensor and, if it is timelike, the spacetime is a Generalized Friedman–Robertson–Walker metric; in dimension four, it is Friedman–Robertson–Walker.

scholarly journals Weyl compatible tensors

2014 ◽  
Vol 11 (08) ◽  
pp. 1450070 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Luca Guido Molinari
Keyword(s):  
General Relativity ◽  
Constant Curvature ◽  
Weyl Tensor ◽  
Ricci Tensor ◽  
Algebraic Property ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Symmetric Tensors ◽  
Magnetic Part ◽  
Necessary And Sufficient

We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular, it is shown that the existence of a Weyl compatible vector implies that the Weyl tensor is algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdziński and Shen [11], Hall [15]) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as in McIntosh et al. [16, 17]). A simple example of Weyl compatible tensor is the Ricci tensor of an hypersurface in a manifold with constant curvature.

scholarly journals On conformally recurrent manifolds of dimension greater than 4

2016 ◽  
Vol 13 (05) ◽  
pp. 1650053
Author(s):  
Carlo Alberto Mantica ◽  
Luca Guido Molinari
Keyword(s):  
Explicit Expression ◽  
Riemannian Manifolds ◽  
Weyl Tensor ◽  
Ricci Tensor ◽  
Symmetric Tensor ◽  
Scalar Function ◽  
Dimension Formula ◽  
Null Recurrence ◽  
Conformally Recurrent

Conformally recurrent pseudo-Riemannian manifolds of dimension [Formula: see text] are investigated. The Weyl tensor is represented as a Kulkarni–Nomizu product. If the square of the Weyl tensor is non-zero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak’s theorem, the explicit expression of the traceless part of the Ricci tensor is obtained, up to a scalar function. The Ricci tensor has at most two distinct eigenvalues, and the recurrence vector is an eigenvector. Lorentzian conformally recurrent manifolds are then considered. If the square of the Weyl tensor is non-zero, the manifold is decomposable. A null recurrence vector makes the Weyl tensor of algebraic type IId or higher in the Bel–Debever–Ortaggio classification, while a time-like recurrence vector makes the Weyl tensor purely electric.

scholarly journals HIGHER-ORDER CURVATURE TERMS IN BORN–INFELD TYPE BRANE THEORIES

2011 ◽  
Vol 20 (01) ◽  
pp. 59-75 ◽  
Author(s):  
EFRAIN ROJAS
Keyword(s):  
Ricci Tensor ◽  
General Structure ◽  
Extrinsic Curvature ◽  
Higher Order ◽  
Square Root ◽  
Auxiliary Variables ◽  
Field Equations ◽  
Alternative Description ◽  
Brane Models ◽  
Combined Matrix

The field equations associated to Born–Infeld type brane theories are studied by using auxiliary variables. This approach hinges on the fact, that the expressions defining the physical and geometrical quantities describing the worldvolume are varied independently. The general structure of the Born–Infeld type theories for branes contains the square root of a determinant of a combined matrix between the induced metric on the worldvolume swept out by the brane and a symmetric/antisymmetric tensor depending on gauge, matter or extrinsic curvature terms taking place on the worldvolume. The higher-order curvature terms appearing in the determinant form come to play in competition with other effective brane models. Additionally, we suggest a Born–Infeld–Einstein type action for branes where the higher-order curvature content is provided by the worldvolume Ricci tensor. This action provides an alternative description of the dynamics of braneworld scenarios.

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