The role of nonmetricity in metric-affine theories of gravity

Abstract

The intriguing choice to treat alternative theories of gravity by means of the Palatini approach, namely elevating the affine connection to the role of independent variable, contains the seed of some interesting (usually under-explored) generalizations of General Relativity, the metric-affine theories of gravity. The peculiar aspect of these theories is to provide a natural way for matter fields to be coupled to the independent connection through the covariant derivative built from the connection itself. Adopting a procedure borrowed from the effective field theory prescriptions, we study the dynamics of metric-affine theories of increasing order, that in the complete version include invariants built from curvature, nonmetricity and torsion. We show that even including terms obtained from nonmetricity and torsion to the second order density Lagrangian, the connection lacks dynamics and acts as an auxiliary field that can be algebraically eliminated, resulting in some extra interactions between metric and matter fields.Dedicated to the memory of Francesco Caracciolo

The Weyl–Cartan Gauss–Bonnet gravity

“…For this case we will just add the Gauss-Bonnet term which we know does not produce any ghost for the metric perturbations. Adding non-metricity as well as torsion has been studied in [21]. The author has shown that at the second order gravity level, connection has no dynamics which is same as the results in [14].…”

“…The other approach, which is more similar to [25], is to write all the possible terms constructed by non-metricity and torsion in addition to the metric itself. This second approach was done in [21,22].…”

Family of scalar-nonmetricity theories of gravity

“…The work was supported by the Estonian Research Council through the Institutional Research Funding project IUT02-27 and the Personal Research Funding project PUT790 (start-up project), as well as by the European Regional Development Fund through the Center of Excellence TK133 "The Dark Side of the Universe". Let us calculate the contractions of G λ µν ω σρ , defined by (16). A straightforward calculation yields…”

Metric-affine gravity and inflation

“…In this paper, we will use a formalism called metricaffine geometry, in which the metric and the connection are independent variables [35][36][37][38][39][40][41]. We consider theories given only by the curvatures, but not by the function of the connections such as torsions.…”