On Gauge Gravity's 'Hypermomentum Challenge' to the Geometric Trinity

Abstract

Claims concerning a dynamical equivalence between general relativity and two reformulations of it, in which gravity is encoded in torsion and non-metricity respectively, have attracted some attention. If taken to be true, this 'Geometric Trinity' suggests that it is impossible to determine, even in principle, whether one inhabits a spacetime solution possessing curvature, torsion or non-metricity (insofar as the probes of geometry are paths of test particles). This claim is in direct tension with the long tradition of studying gravity as a gauge theory, in which distinct geometric attributes couple to distinct properties of matter (such as energy-momentum to curvature and spin angular momentum to torsion). In this short paper, I compare these two conflicting frameworks for geometrizing spacetime and describe the precise source, nature and implications of their disagreement. The equivalence claim holds only under a substantive restriction on the matter sector, namely that matter does not couple to an independent affine connection. Because the relevant couplings remain beyond current empirical reach, the choice between the frameworks turns on a trade-off among ontological, epistemic, and methodological forms of parsimony.