Conformally invariant generalization of Einstein equations and the causality principle
type: Item
instance of: scholarly article
title: Conformally invariant generalization of Einstein equations and the causality principle
zbMATH Open document ID: 0996.83044
DOI: 10.1023/A:1015335326235
author: M. V. Gorbatenko
author: A. V. Pushkin
published in: General Relativity and Gravitation
publication date: 2002-07-09T00:00:00Z
review text: The authors propose a conformal invariant generalization of the Einstein equations with has the form \[ R_{ij}-\tfrac 12 g_{ij}= -2A_iA_j -g_{ij}A^2-2g_{ij}A^k_k + A_{i;j} + A_{j;i} \] where the semicolon means covariant differentiation with respect to the Levi-Civita connection and \(A_i\) is a 1-form which transforms as a potential \(A_i \rightarrow A_i + \partial f/ \partial x^i\) under a conformal change \(g_{ij} \rightarrow \exp (-2f)g_{ij}\) of the metric \(g_{ij}\). To obtain this equation, the authors start from a Lorentz metric \(g_{ij}\) and a linear connection \(A^i_{jk}\) and apply a variant of the Palatini variational method to the standard Hilbert action (defined by the scalar curvature of \(g_{ij}\)) together with some non-holonomic constraints. The Euler-Lagrange equations consist of of the aforementioned generalized Einstein equation together with an equation which expresses the connection \(A^i_{jk} \) as a sum of the Levi-Civita connection of \(g_{ij}\) and a tensor which depends linearly on \(A_i\) and \(g_{ij}\). The consistency of the proposed variational procedure with the causality principle and the correctness of the Cauchy problem in a synchronous coordinate system are discussed.
Mathematics Subject Classification ID: 83D05
Mathematics Subject Classification ID: 83C05
zbMATH DE Number: 1765068
zbMATH Keywords: generalized Einstein equation
zbMATH Keywords: conformal invariance
zbMATH Keywords: causality principle
zbMATH Keywords: Palatini variational method
zbMATH Keywords: Cauch problem
reviewed by: Dmitri Alekseevsky
MaRDI profile type: Publication
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full work available at URL: a:1015335326235
OpenAlex ID: W1501751802
description: scientific article; zbMATH DE number 1765068