The affine geometry of the Lanczos H-tensor formalism
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.
scientific article; zbMATH DE number 165099
scientific article