Nonmetricity formulation of general relativity and its scalar-tensor extension

Abstract

Einstein's celebrated theory of gravitation can be presented in three forms: general relativity, teleparallel gravity, and the rarely considered before symmetric teleparallel gravity. Extending the latter, we introduce a new class of theories where a scalar field is coupled nonminimally to nonmetricity Q, which here encodes the gravitational effects like curvature R in general relativity or torsion T in teleparallel gravity. We point out the similarities and differences with analogous scalarcurvature and scalar-torsion theories by discussing the field equations, role of connection, conformal transformations, relation to f (Q) theory, and cosmology. The equations for spatially flat universe coincide with those of teleparallel dark energy, thus allowing to explain accelerating expansion. II. CONNECTIONS, GEOMETRIES, AND GRAVITATIONAL THEORIESA. Decomposition of affine connection

Parametrized post-Newtonian limit of general teleparallel gravity theories

“…In particular, one may consider more general theories, for example derived from a general constitutive relation [39], possibly including also parity-odd terms, or coupling to scalar fields [40][41][42][43], up to Horndeski-like teleparallel theories [44,45]. Further, taking inspiration from the socalled trinity of gravity [1], one may consider extensions to the symmetric teleparallel equivalent of gravity [46], and apply the parameterized post-Newtonian formalism to generalized theories based on the symmetric teleparallel geometry [47][48][49][50][51]. Another possible extension would be studying the motion of compact objects at higher orders in the post-Newtonian expansion, in order to derive the emitted gravitational waves [52].…”

New class of ghost- and tachyon-free metric affine gravities

“…In the literature, more attention has been given to theories with torsion, but recently there has been a great deal of interest for MAGs with non-metricity, see e.g. [3][4][5][6][7][8][9][10][11][12].…”

Higgs inflation and teleparallel gravity

“…There the metric and the connection are independent variables. The Palatini formulation is equivalent to the metric formulation (and to the other two formulations discussed above) for the Einstein-Hilbert action with minimally coupled matter, but becomes physically distinct when the gravitational action is more complicated [14,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] or matter couples to the connection [32][33][34][35][36][37][38][39][40][41], as is the case in Higgs inflation [35,36,[38][39][40][42][43][44]. In the metric theory, quantum corrections induce higher order curvature terms, and new geometric terms formed from torsion and non-metricity are similarly expected to arise in the teleparallel and the symmetric teleparallel formulation.…”