# Geometric inequivalence of metric and Palatini formulations of General Relativity

## TL;DR

This paper explores the fundamental differences between metric and Palatini formulations of General Relativity, highlighting how projective invariance in the Palatini approach introduces gauge freedoms affecting curvature scalars.

## Contribution

It demonstrates that in the Palatini formulation, the Kretschmann scalar can be gauge-transformed to vanish, revealing a geometric inequivalence with the metric formulation.

## Key findings

- The Kretschmann scalar can be set to zero via a gauge in the Palatini approach.
- Projective invariance introduces gauge freedom in the Riemann tensor.
- Curvature scalar divergences may be gauge-dependent in Palatini GR.

## Abstract

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.