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 usual Einstein–Hilbert plus terms in torsion, non-metricity as well their mixing. By following a certain procedure and using simple trick we are able to find unique solution affine connection an arbitrary hypermomentum. Given fairly non-degeneracy condition our result provides exact form for all types matter. Subsequently compute forms torsion sources (hypermomentum tensor) also express metric field equations effectively Einstein’s GR with modified source that depend on hypermomentum its derivatives. show absence matter always reduces GR. Finally generalize wider class Theories.