Third-order tensor potentials for the Riemann and Weyl tensors. II: Singular solutions
type: Item
instance of: scholarly article
title: Third-order tensor potentials for the Riemann and Weyl tensors. II: Singular solutions
zbMATH Open document ID: 0543.53026
DOI: 10.1007/BF00762334
author: Franco Bampi
author: Giacomo Caviglia
published in: General Relativity and Gravitation
publication date: 1984-01-01T00:00:00Z
review text: \textit{E. Udeschini Brinis} [Rend., Sci. Mat. Fis. Chim. Geol. 111, 466-475 (1977; Zbl 0414.53010)] conjectured that the Riemann tensor might be derivable as the covariant derivative of a third rank tensor suitably skewed and symmetrized. In 1983 the authors of this paper showed [Gen. Relativ. Gravitation 15, 375-386 (1983; Zbl 0508.53021)] that the integrability conditions prevent any ordinary (in the sense of Cartan) solutions. There can be singular solutions and in four dimensions these are not constrained by any integrability condition. (However the paper by \textit{E. Massa} and \textit{E. Pagani} announced below shows that one of the original conditions reduces, for solutions of the vacuum field equations with or without a cosmical term, to a severe condition on the space, which is not fulfilled by the Schwarzschild, Kerr or de Sitter solutions, nor in general for any metrics of Petrov type different from III, N or 0. It is to be hoped that this will be the end of the story.)
Mathematics Subject Classification ID: 53B50
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 83C05
zbMATH DE Number: 3864023
zbMATH Keywords: Riemann tensor
zbMATH Keywords: covariant derivative of a third rank tensor
zbMATH Keywords: integrability conditions
zbMATH Keywords: vacuum field equations
reviewed by: C. W. Kilmister
MaRDI profile type: Publication
cites work: Third-order tensor potentials for the Riemann and Weyl tensors
cites work: The Splitting of the Riemann Tensor
cites work: E. Cartan's geometrie theory of partial differential equations
cites work: On E. Cartan's Prolongation Theorem of Exterior Differential Systems
cites work: A natural scalar field in the Einstein gravitational theory
cites work: On the relativistic extension of the equations of the classical gravitational potential
full work available at URL: https://doi.org/10.1007/bf00762334
OpenAlex ID: W2145996755
description: scientific article; zbMATH DE number 3864023