\(Gl(n,\mathbb{R})\)-invariant tetrad theory and its relation to the purely affine formulation of general relativity
The authors review the purely affine formulation of general relativity in which the Lagrangian \(\mathcal L\) is a function of the components of a symmetric linear connection and its derivatives but only via the symmetrized Ricci tensor \(K_{ij}\). The metric is a derived quantity introduced via a Legendre transformation. The form of \(\mathcal L\) that leads to the vacuum Einstein equations with cosmological constant is then the square root of the determinant of \(K_{ij}\). This theory is compared to a tetrad (more generally \(n\)-frame) theory of gravitation in which the dynamical variables are linear frame fields, the connection, their rotation coefficients and the Lagrangian is the square root of the determinant of a symmetric tensor field made up of the torsion of this connection. The curvature tensor is assumed to vanish. The authors reformulate this theory by decomposing the connection into a symmetric part, a traceless antisymmetric field, and a vector field. Imposing a flat curvature tensor via Lagrange multipliers, they find again the Einstein vacuum equations with cosmological constant and additional constraints on the connection that have not yet been analyzed.
scientific article; zbMATH DE number 1087282