Metric-affine gravity and the Nester-Witten 2-form

TL;DR

This paper redefines the metric-affine Hilbert Lagrangian using spin-connection and tetrad, deriving a global superpotential and recovering important forms like the Nester-Witten 2-form through gauge-natural bundle methods.

Contribution

It introduces a new gauge-natural bundle approach to derive gravitational superpotentials, unifying and extending previous results including the Nester-Witten 2-form.

Findings

Derived a global gravitational superpotential.

Recovered Kijowski's result for the metric Hilbert Lagrangian.

Obtained the Nester-Witten 2-form using a different gauge lift.

Abstract

In this paper we redefine the well-known metric-affine Hilbert Lagrangian in terms of a spin-connection and a spin-tetrad. On applying the Poincar\'e-Cartan method and using the geometry of gauge-natural bundles, a global gravitational superpotential is derived. On specializing to the case of the Kosmann lift, we recover the result originally found by Kijowski (1978) for the metric (natural) Hilbert Lagrangian. On choosing a different, suitable lift, we can also recover the Nester-Witten 2-form, which plays an important role in the energy positivity proof and in many quasi-local definitions of mass.