Conformally invariant generalization of Einstein equations and the causality principle
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.
scientific article; zbMATH DE number 1765068