universe Article The Geometrical Trinity of Gravity Jose Beltrán Jiménez 1,*, Lavinia Heisenberg 2,* and Tomi S. Koivisto 3,4,* 1 Departamento de Física Fundamental and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain 2 Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland 3 Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden 4 Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia * Correspondence: [email protected] (J.B.J.); [email protected] (L.H.); [email protected] (T.S.K.) Received: 5 June 2019; Accepted: 5 July 2019; Published: 15 July 2019 Abstract: The geometrical nature of gravity emerges from the universality dictated by the equivalence principle. In the usual formulation of General Relativity, the geometrisation of the gravitational interaction is performed in terms of the spacetime curvature, which is now the standard interpretation of gravity. However, this is not the only possibility. In these notes, we discuss two alternative, though equivalent, formulations of General Relativity in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity, thus putting forward the existence of three seemingly unrelated representations of the same underlying theory. Based on these three alternative formulations of General Relativity, we then discuss some extensions. Keywords: general relativity; modified gravity; torsion; non-metricity 1. Introduction Gravity and geometry have accompanied each other from the very conception of General Relativity (GR) brilliantly formulated by Einstein in terms of the spacetime curvature. This inception of identifying gravity with the curvature has since grown so efficiently that it is now a common practice to recognise the gravitational phenomena as a manifestation of having a curved spacetime. As Einstein ingeniously envisioned, the existence of a geometrical formulation of gravity is granted by the equivalence principle that renders the gravitational interaction oblivious to the specific type of matter and hints towards an intriguing relation of gravity with inertia. Thus, the motion of particles can be naturally associated with the geometrical properties of spacetime. If we embrace the geometrical character of gravity advocated by the equivalence principle, it is pertinent to explore in which equivalent manners gravity can be geometrised. It is then convenient to recall at this point that a spacetime can be endowed with a metric and an affine structure [1–4] determined by a metric a tensor gmn and a connection G mn, respectively. These two structures, although completely independent, enable the definition of geometrical objects that allow for conveniently classify geometries. The failure of the connection to be metric is encoded in the non-metricity Q g , (1) amn ≡ ra mn while its antisymmetric part defines the torsion Ta 2Ga . (2) mn ≡ [mn] Universe 2019, 5, 173; doi:10.3390/universe5070173 www.mdpi.com/journal/universe Universe 2019, 5, 173 2 of 17 Among all possible connections that can be defined on a spacetime, the Levi–Civita connection is the unique connection that is symmetric and metric-compatible. These two conditions fix the Levi–Civita connection to be given by the Christoffel symbols of the metric n o 1 a = galg + g g . (3) mn 2 ln,m ml,n − mn,l The corresponding covariant derivative will be denoted by so that we will have agmn = 0.A a D D general connection G mn then admits the following convenient decomposition: a n a o a a G mn = mn + K mn + L mn (4) with 1 a 1 a Ka = Ta + T La = Qa Q (5) mn 2 mn (m n) mn 2 mn − (m n) the contortion and the disformation pieces, respectively. Notice that, while the Levi–Civita part is non-tensorial, the contortion and the disformation have tensorial transformation properties under changes of coordinates. The curvature is determined by the usual Riemann tensor Ra (G) = ¶ Ga ¶ Ga + Ga Gl Ga Gl . (6) bmn m nb − n mb ml nb − nl mb In Figure1 we give a geometrical intuition for the different objects associated to the affine structure. A relation that will be useful later on is how the Riemann tensor transforms under a shift of the ˆ a a a a connection G mn = G mn + W mn, with W mn an arbitrary tensor. Under such a shift, the Riemann tensor becomes ˆ a a l a a a l R bmn = R bmn + T mnW lb + 2 [mW n]b + 2W [m l W n]b , (7) r j j where Rˆ a and Ra are the Riemann tensors of Gˆ and G, respectively, and is the covariant bmn bmn r derivative associated with G. ↵ R βµ⌫ <latexit 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{ The rotation of a vector transported along a closed curve is given by the curvature: General Relativity. { <latexit sha1_base64="/Qc/70FUMVg/W6zRp2zmxqTMW5U=">AAAB6XicbZC7SgNBFIbPxlsSb1EbwWYwCFZh18aUQRvLKOYCSQizk9lkyOzsMnNWiEvAB7CxUMTWN/BR7AQfxsml0MQfBj7+cw5zzu/HUhh03S8ns7K6tr6RzeU3t7Z3dgt7+3UTJZrxGotkpJs+NVwKxWsoUPJmrDkNfckb/vByUm/ccW1EpG5xFPNOSPtKBIJRtNZNO+0Wim7JnYosgzeHYuXw/jv38HFR7RY+272IJSFXyCQ1puW5MXZSqlEwycf5dmJ4TNmQ9nnLoqIhN510uumYnFinR4JI26eQTN3fEykNjRmFvu0MKQ7MYm1i/ldrJRiUO6lQcYJcsdlHQSIJRmRyNukJzRnKkQXKtLC7EjagmjK04eRtCN7iyctQPyt5lq9tGmWYKQtHcAyn4ME5VOAKqlADBgE8wjO8OEPnyXl13matGWc+cwB/5Lz/ANm2kII=</latexit> <latexit 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