Quadratic Metric-Affine Gravity: Solving for the Affine-Connection

Abstract

We consider the most general 11 parameter parity even quadratic Metric-Affine Theory whose action consists of the usual Einstein-Hilbert plus the 11 quadratic terms in torsion, non-metricity as well as their mixing. By following a certain procedure and using a simple trick we are able to find the unique solution of the affine connection in terms of an arbitrary hypermomentum. Given a fairly general non-degeneracy condition our result provides the exact form of the affine connection for all types of matter. Subsequently we compute the forms of torsion and non-metricity in terms of their sources (hypermomentum tensor) and also express the metric field equations in effectively Einstein's GR with modified source terms that depend on the hypermomentum and its derivatives. We show that in the absence of matter the Theory always reduces to GR. Finally we generalize our result and find the form of the connection for a wider class of quadratic Theories. Contents B. Post-Riemannian Expansions1. Distortion vectors in terms of torsion and non-metricity vectors 2. Post-Riemannian expansions

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

“…It is obvious then, by direct substitution of the latter form of the distortion into (62) that apart from the usual energy-momentum tensor quadratic and derivative terms of the hypermomentum serve also as sources producing gravitational fields. Our result above serves as a generalization of the earlier findings of [42] where now the additional six parity odd quadratic scalars have been added into the action along with the 11 parity even ones. Our study now covers the whole spectrum of quadratic MAG theories, that is we have found the exact form of the connection and consequently that of torsion and non-metricity for the most general 17 parameter quadratic MAG action.…”

“…On the other hand, there is no parameter choice (except the trivial of vanishing coefficients) for which the parity odd terms cancel. Now, using a post-Riemannian expansion for every term in (42), after some lengthy calculations (see appendix for details) we finally arrive at…”

“…After varying independently with respect to the metric and the affine connection we derive the field equations of the quadratic theory. Then, focusing on the connection field equations, by performing a simple trick and using a recent result (see [42]) on how to solve complicated linear tensor equations, we obtain the exact a unique solution of the affine connection. In our step by step proof, we show that the latter is expressed in terms of the usual Levi-Civita part plus modifications coming from various combinations/dualizations and contractions of the hypermomentum tensor.…”

“…3 Renormalization in the presence of torsion was studied in [41]. 4 Here we wish to generalize the findings of [42] where only the parity even quadratic terms were included the vacuum version of the quadratic theory, despite its complexity, is always equivalent to GR given that there are no degeneracies. The same result continuous to hold true also for matter that does not couple to the connection (i.e vanishing hypermomentum).…”

Metric-Affine Vector-Tensor Correspondence and Implications in $F(R,T,Q,\mathcal{T},\mathcal{D})$ gravity

“…(A.3)Finally, let us write the following post-Riemannian expansions (cf. also[54]):R µν = Rµν + − (−12 − 7n + 2n 2 + n 3 ) 4[(n − 1)(n + 2)] 2 Q λ Q λ + (−10 − 7n + n 3 ) 2[(n − 1)(n + 2)] 2 Q λ q λ + 2(n + 1) [(n − 1)(n + 2)] 2 q λ q λ 3n + 2 Q λ S λ + 2(n 2 − n − 4) n 3 − 3n + 2 q λ S λ − 4(n − 2) (n − 1) 2 S λ S λ − (n + 3) 2(n − 1)(n + 2) ∇λ Q λ + (n + 1) (n − 1)(n + 2) ∇λ q λ − 2 n − 1 ∇λ S λ g µν + (n 3 − 7n − 10) 4[(n − 1)(n + 2)] 2 Q µ Q ν + 4(n + 1) [(n − 1)(n + 2)] 2 Q (µ q ν) − (n 2 + n + 2) [(n − 1)(n + 2)] 2 q µ q ν + 2(n 2 − n − 4) n 3 − 3n + 2 Q (µ S ν) + 3n + 2 q (µ S ν) + 4(n − 2) (n − 1) 2 S µ S ν + (n + 1) 2(n − 1)(n + 2) ∇µ Q ν − (n 2 − 3) 2(n − 1)(n + 2) ∇ν Q µ − 2 (n − 1)(n + 2) ∇(µ q ν) − 2(n − 2) n − 1 ∇ν S µ , (A.4) R = R − (n 2 − 5) 4(n − 1)(n + 2)…”