We study a large family of metric-affine theories with a projective symmetry, including nonminimally coupled matter fields which respect this invariance. The symmetry is straightforwardly realised by imposing that the connection only enters through the symmetric part of the Ricci tensor, even in the matter sector. We leave the connection completely free (including torsion) and obtain its general solution as the Levi-Civita connection of an auxiliary metric, showing that the torsion only appears as a projective mode. This result justifies the widely used condition of setting vanishing torsion in these theories as a simple gauge choice. We apply our results to some particular cases considered in the literature like the so-called Eddington-inspired-Born-Infeld theories among others. We finally discuss the possibility of imposing a gauge fixing where the connection is metric compatible and comment on the genuine character of the non-metricity in theories where the two metrics are not conformally related.PACS numbers: I. INTRODUCTIONThe remarkable properties of Born-Infeld electromagnetism [1], originally aimed at resolving divergences of point-like charged particles, motivated the search of a similar route to resolve the singularities of General Relativity [2]. Among the different proposals, the so-called Eddington-inspired-Born-Infeld (EiBI) theory [3] has attracted a lot of attention in the last years due to its extraordinary ability to get rid of cosmological and black hole singularities and numerous works have been devoted to constrain the model using different types of observations [4]. Extensions and modifications of that model also lead to interesting results in cosmological and black hole scenarios [5][6][7][8]. The exploration of these theories showed that their natural habitat is the framework of metric-affine geometries and precisely this formulation permitted the mentioned progress (see [9] for a review). The reason for the necessity of considering these theories in the metric-affine approach is the avoidance of ghost-like instabilities that otherwise would be present in the metric formulation of theories with non-linear curvature terms in the action [10]. The metric-affine (sometimes also called Palatini) formulation is characterised by unlocking the affine structure and disentangle it from the metric structure, which amounts to assuming that the geometry is not Riemannian a priori, but of metric-affine type, where the metric and the connection are regarded as fully independent objects. The spirit of this approach is that only the resulting field equations should specify the full geometrical structure of the spacetime and, in particular, the relation between the metric and the connection with the matter fields. In this regard, it must be noted that the EiBI theory has been systematically analysed for a constrained family of connections by assuming a vanishing torsion tensor 1 . Given the * Electronic address: viafonso@df.ufcg.edu.br † Electronic address: cbejarano@iafe.uba.ar ‡ Electronic address: jose.b...

Null tetrads are shown to be a valuable tool in teleparallel theories of modified gravity. We use them to prove that Kerr geometry remains a solution for a wide family of f (T ) theories of gravity.

Exploring the characterization of singular black hole spacetimes, we study the relation between energy density, curvature invariants, and geodesic completeness using a quadratic f (R) gravity theory coupled to an anisotropic fluid. Working in a metric-affine approach, our models and solutions represent minimal extensions of General Relativity (GR) in the sense that they rapidly recover the usual Reissner-Nordström solution from near the inner horizon outwards. The anisotropic fluid helps modify only the innermost geometry. Depending on the values and signs of two parameters on the gravitational and matter sectors, a breakdown of the correlations between the finiteness/divergence of the energy density, the behavior of curvature invariants, and the (in)completeness of geodesics is obtained. We find a variety of configurations with and without wormholes, a case with a de Sitter interior, solutions that mimic non-linear models of electrodynamics coupled to GR, and configurations with up to four horizons. Our results raise questions regarding what infinities, if any, a quantum version of these theories should regularize.

We show that McVittie geometry, which describes a black hole embedded in a FLRW universe, not only solves the Einstein equations but also remains as a non-deformable solution of f (T ) gravity. This search for GR solutions that survive in f (T ) gravity is facilitated by a null tetrad approach. We also show that flat FLRW geometry is a consistent solution of f (T ) dynamical equations not only for T = −6H 2 but also for T = 0, which could be a manifestation of the additional degrees of freedom involved in f (T ) theories.