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.

Spatially covariant gravity with nonmetricity

“…Generally, the affine connection Γ c ab is treated as an independent variable, which extends the geometric description of gravitation. The theory with a general affine connection is thus dubbed metric-affine gravity (MAG), which has been studied extensively [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. By coupling MAG to scalar field(s), one gets the so-called "metric-affine scalar-tensor theory" [56][57][58][59][60], which can be seen as an analogue of the scalar-tensor theory in Riemannian geometry.…”

Circularly polarized scalar induced gravitational waves from the Chern-Simons modified gravity

“…When torsion is tuned on, the simplest term that corresponds to the CS term is the so-called Nieh-Yan term [80]. The chiral GWs have also been extensively studied with NY term and its extensions [81][82][83][84][85][86][87][88][89][90], as well as in more general models with non-vanishing torsion and/or non-metricity tensors [91][92][93][94][95][96][97]. To our knowledge, the previous studies with the PV modes dealt with only the primordial/linear GWs, which exhibit interesting features such as the velocity and amplitude birefringence phenomenons of GWs [55,69,71,98], see also [99][100][101][102].…”