Spacetimes in which the Ricci equations characterize the Riemann tensor
type: Item
instance of: scholarly article
title: Spacetimes in which the Ricci equations characterize the Riemann tensor
zbMATH Open document ID: 0777.53023
DOI: 10.1007/BF00674400
author: S. Brian Edgar
published in: International Journal of Theoretical Physics
publication date: 1993-05-16T00:00:00Z
review text: The author considers a space-time \((M,g)\) admitting a tensor \(K^ a_{bcd}\) satisfying \[ K^ a_{bcd}=-K^ a_{bdc},\;K^ a_{bcd}+K^ a_{cdb}+K^ a_{dbc}=0,\;g_{ae}K^ e_{bcd}+g_{be}K^ e_{acd}=0, \] \[ K^ a_{bcd;ef}-K^ a_{bcd;fe}=-K^ m_{bcd}K^ a_{mef}+K^ a_{mcd}K^ m_{bef}+K^ a_{bmd}K^ m_{cef}+K^ a_{bcm}K^ m_{def} \] and asks under what conditions is \(K\) the Riemann tensor \(R^ a_{bcd}\) of the metric \(g\). The author has already provided a partial answer to this general problem [J. Geom. Phys. 7, 191-200 (1990; Zbl 0727.53031)] and a counterexample showing that \(K\) is not necessarily the Riemann tensor associated with \(g\) has been given by \textit{A. D. Rendall} [Mathematical Relativity, Proc. Conf., Canberra/Aust. 1988, Proc. Cent. Math. Anal. Aust. Natl. Univ. 19, 125-136 (1989; Zbl 0686.53013)]. Here the author gives a more general account of the situation by defining a tensor \(P^ a_{bcd}=R^ a_{bcd}-K^ a_{bcd}\) and proceeds with a study of \(K\) and \(P\) which involves the Petrov algebraic types of these tensors. A topological discussion of the generic nature of the results is also given using Whitney-type topological arguments.
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 83C99
zbMATH DE Number: 167645
zbMATH Keywords: Riemann tensor
zbMATH Keywords: Petrov algebraic types
reviewed by: G. S. Hall
MaRDI profile type: Publication
cites work: Principal null directions of Petrov type I Weyl spinors: geometry and symmetry
cites work: Sufficiency of the Ricci equations for characterizing the Riemann tensor
cites work: The classification of the Ricci tensor in general relativity theory
cites work: 'Degenerate' nondegenerate spacetime metrics
cites work: The Riemann tensor, the metric tensor, and curvature collineations in general relativity
cites work: An Approach to Gravitational Radiation by a Method of Spin Coefficients
cites work: Q5562586
cites work: Curvature of generic space-times in general relativity
cites work: The continuous determination of spacetime geometry by the Riemann curvature tensor
cites work: Insufficiency of the Ricci and Bianchi identities for characterising curvature
Wikidata QID: Q126162846
description: scientific article; zbMATH DE number 167645