paper

A unified view of curvature and torsion in metric-affine gauge theory of gravity through affine-vector bundles

arXiv:2008.07750 · doi:10.1088/1361-6382/ac08a5

Abstract

One of the most appealing results of metric-affine gauge theory of gravity is a close parallel between the Riemann curvature two-form and the Cartan torsion two-form: While the former is the field strength of the Lorentz-group connection one-form, the latter can be understood as the field strength of the coframe one-form. This parallel, unfortunately, is not fully established until one adopts Trautman's idea of introducing an affine-vector-valued zero-from, the meaning of which has not been satisfactorily clarified. This paper aims to derive this parallel from first principles without any ad hoc prescriptions. We propose a new mathematical framework of an associated affine-vector bundle as a more suitable arena for the affine group than a conventional vector bundle, and rigorously derive the covariant derivative of a local section on the affine-vector bundle in the formal Ehresmann-connection approach. The parallel between the Riemann curvature and the Cartan torsion arises naturally on the affine-vector bundle, and their geometric and physical meanings become transparent. The clear picture also leads to a conjecture about a kinematical effect of the Cartan torsion that in principle can be measured à la the Aharonov-Bohm effect.

arXiv comment35 pages, 2 figures; contains quotes from Refs. [3] (arXiv:gr-qc/9602013), [8] (arXiv:gr-qc/9402012) and [26] (MTW); v2: various minor improvements made, version to appear in CQG

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