The full quadratic metric-affine gravity (including parity odd terms): exact solutions for the affine-connection

Abstract

We consider the most general Quadratic Metric-Affine Gravity setup in the presence of generic matter sources with non-vanishing hypermomentum. The gravitational action consists of all 17 quadratic invariants (both parity even and odd) in torsion and non-metricity as well as their mixings, along with the terms that are linear in the curvature namely the Ricci scalar and the totally antisymmetric Riemann piece. Adding also a matter sector to the latter we first obtain the field equations for the generalized quadratic Theory. Then, using a recent Theorem, we successfully find the exact form of the affine connection under some quite general non-degeneracy conditions. Having obtained the exact and unique solution of the affine connection we subsequently derive the closed forms of spacetime torsion and non-metricity and also recast the metric field equations into a GR form with modified source terms that are quadratic in the hypermomentum and linear in its derivatives. We also study the vacuum quadratic Theory and prove that in this instance, or more generally for vanishing hypermomentum, the connection becomes the Levi-Civita one. Therefore, we also find exactly to what does the quadratic vacuum Theory correspond to. Finally, we generalize our result even further and also discuss the physical consequences and applications of our study.

Metric-Affine Cosmologies: Kinematics of Perfect (Ideal) Cosmological Hyperfluids and First Integrals

“…More information about the form of ( 121) and ( 82) can be extracted if further dynamical characteristics are known. For instance, for the wide class of quadratic (in torsion and non-metricity) MAG Theories all distortion variables are linearly related to the hypermomentum sources (see [5]), here ψ, and consequently we find the net density and pressure…”

“…The MAG framework has attracted a lot of attention lately[5][6][7][8][9][10][11][12][13][14][15][15][16][17][18][19][20][21][22][23][24][25][26] 2 This definition is given in Weinberg's book[27].…”

Coupling Metric-Affine Gravity to a Higgs-Like Scalar Field

“…Based on [85], solutions for torsion and non-metricity in terms of the energy-momentum-and hypermomentum-tensors were derived in this model in [86]. An explicit computation of the equivalent metric theory was only performed in [50], where theories of a scalar field coupled to gravity in the metric-affine formulation were studied, but solely a specific subset of possible contributions due to torsion and non-metricity was included.…”

“…Fist, we generalize the scalar part of [69] by including non-metricity in addition to torsion. Secondly, we develop further [86] by making the matter sector explicit -using a scalar field as example -and then deriving the equivalent metric theory. Finally, our work generalizes the paper [50], where only a subset of terms was included in the study of a scalar field non-minimally coupled to metric-affine gravity.…”

“…The matter sector was not made explicit in [49], and so the functions Ω 2 (h), K(h) and A i (h) were not present. The action proposed in [49] was further studied in [86] and solutions for torsion and non-metricity were derived in terms of energy-momentum-and hypermomentum-tensors. Unlike in the present work, the paper [86] employed a method for finding solutions that does not require the separation of pure tensor parts [191,192].…”