General relativity derived from an affine variation of a quadratic Lagrangian
I. W. Roxburgh · Journal of Physics A Mathematical and General · 1975
An alternative derivation of general relativity is presented in which the field equations are derived from a gauge invariant quadratic Lagrangian by varying the affine field Gamma bc a and keeping the geometry fixed. The field equations show that the Gamma bc a can be considered as a Christoffel connection of a tensor g ab and if, after the variation, the metric tensor is chosen to be this tensor, the field equations of general relativity with a cosmical constant are recovered.