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.

14 pages, no figures

References in corpus (5)

• Metric-affine Gravity and Inflation
• Probing non-Riemannian spacetime geometry
• Metric-Affine Gravity and Cosmology/Aspects of Torsion and non-Metricity in Gravity Theories
• Cosmology of Quadratic Metric-Affine Gravity
• (3+1)-Formulation for Gravity with Torsion and Non-Metricity II: The Hypermomentum Equation

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• Spatially covariant gravity with nonmetricity
• Covariant spin-parity decomposition of the Torsion and Path Integrals
• Violating Lorentz invariance minimally by the emergence of nonmetricity? A Perspective
• Radiative corrections to the $R$ and $R^2$ invariants from torsion fluctuations on maximally symmetric spaces
• On the geometric origin of the energy-momentum tensor improvement terms