Geometric inequivalence of metric and Palatini formulations of General Relativity

Understand

Projective invariance is a symmetry of the Palatini version of General Relativity which is not present in the metric formulation.

• The fact that the Riemann tensor changes nontrivially under projective transformations implies that, unlike in the usual metric approach, in the Palatini formulation this tensor is subject to a gauge freedom, which allows some ambiguities even in its scalar contractions.
• In this sense, we show that for the Schwarzschild solution there exists a projective gauge in which the (affine) Kretschmann scalar, $K\equiv {R^\alpha}_{\beta\mu\nu}{R_\alpha}^{\beta\mu\nu}$, can be set to vanish everywhere.
• This puts forward that the divergence of curvature scalars may, in some cases, be avoided by a gauge transformation of the connection.