Geometric inequivalence of metric and Palatini formulations of General Relativity

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≡RαβμνRαβμν, 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.C. B. is funded by the National Scientific and Technical Re-search Council (CONICET). AD and AJC are supported by a PhD contract of the program FPU 2015 (Spanish Ministry of Econ-omy and Competitiveness) with references FPU15/05406 and FPU15/02864, respectively. GJO is funded by the Ramon y Cajal contract RYC-2013-13019 (Spain). DRG is funded by the Atracción de Talento Investigador programme of the Comunidad de Madrid No. 2018-T1/TIC-10431, and acknowledges support from the Fundação para a Ciência e a Tecnologia (FCT, Portugal) research grants Nos. PTDC/FIS-OUT/29048/2017 and PTDC/FIS-PAR/31938/2017. Thiswork is supported by the Spanish projects FIS2017-84440-C2-1-P, FIS2014-57387-C3-1-P (MINECO/FEDER, EU) and i-LINK1215 (CSIC), the project H2020-MSCA-RISE-2017 Grant FunFiCO-777740, the project SEJI/2017/042 (Generalitat Valenciana), the Consolider Program CPANPHY-1205388, and the Severo Ochoa grant SEV-2014-0398 (Spain)