The affine geometry of the Lanczos H-tensor formalism
type: Item
instance of: scholarly article
title: The affine geometry of the Lanczos H-tensor formalism
zbMATH Open document ID: 0776.53013
DOI: 10.1007/BF00756929
author: Kevin S. Hammon
author: L. K. Norris
published in: General Relativity and Gravitation
publication date: 1993-05-23T00:00:00Z
review text: The difference of two affine connections \((\Gamma_ 1)^{\;\mu}_{\nu\;\lambda}-(\Gamma_ 2)^{\;\mu}_{\nu\;\lambda}\) is a tensor \(T^{\;\mu}_{\nu\;\lambda}\). This remark leads to the notion of an affine space in which the \(\Gamma^{\;\mu}_{\nu\;\lambda}\)s are the points and their differences the vectors. Then by use of an affine frame bundle \(A^ 1_ 2M\) and its structure group \(A^ 1_ 2(4)=GL(4) \circledS T^ 1_ 2 \mathbb{R}^ 4\) over the space-time manifold \(M\) generalized affine connections are introduced. In the arena of this affine space the \(H\)-tensor theory of \textit{C. Lanczos} [Rev. Mod. Phys. 34, 379-389 (1962; Zbl 0106.417)] is investigated and interpreted. It is shown that the Lanczos \(H\)-tensor arises from a gauge fixing condition imposed on the generalized affine connection. In case of a Riemannian structure the Lanczos quantities \(Q_{\mu \nu}\) and \(q\) are constructed from the Riemannian affine connection. The Lanczos Lagrangian is reconstructed from affine invariant quantities. The field equations derived from the Lagrangian yield the Bianchi and Bach-Lanczos identities, and the Lanczos potential equation.
Mathematics Subject Classification ID: 53B15
zbMATH DE Number: 165099
zbMATH Keywords: Lanczos gauge
zbMATH Keywords: affine connections
zbMATH Keywords: Bach-Lanczos identities
reviewed by: Lajos Tamássy
MaRDI profile type: Publication
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cites work: Integral Formalism for Gauge Fields
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cites work: Affine connection structure of the charged symplectic 2-form
cites work: Q3263435
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description: scientific article; zbMATH DE number 165099