A gauge-invariant symplectic potential for tetrad general relativity

TL;DR

The paper addresses gauge-invariance issues in the covariant phase space for tetrad general relativity by constructing a gauge-invariant symplectic potential through the addition of a boundary exact form. It shows that the corrected potential $\Theta'_{\rm EC}$ reproduces the Komar form for Lie-derivative variations and, in the torsionless limit, coincides with the Einstein–Hilbert symplectic potential, including proper 2d corner contributions. The authors prove equivalence to the EH potential for general variations via geometric identities involving extrinsic curvature, and they verify that asymptotic Poincaré charges are preserved. They also demonstrate that the first law of black hole mechanics follows from the Noether identity with covariant Lie derivatives and remains invariant under the cohomology ambiguity when Lorentz charges are properly accounted. Overall, the gauge-invariant potential clarifies the role of internal Lorentz charges, aligns tetrad gravity with metric GR in the torsionless limit, and strengthens the covariant phase-space formulation for gravitational theories with boundaries.

Abstract

We identify a symplectic potential for general relativity in tetrad and connection variables that is fully gauge-invariant, using the freedom to add surface terms. When torsion vanishes, it does not lead to surface charges associated with the internal Lorentz transformations, and reduces exactly to the symplectic potential given by the Einstein-Hilbert action. In particular, it reproduces the Komar form when the variation is a Lie derivative, and the geometric expression in terms of extrinsic curvature and 2d corner data for a general variation. The additional surface term vanishes at spatial infinity for asymptotically flat spacetimes, thus the usual Poincare charges are obtained. We prove that the first law of black hole mechanics follows from the Noether identity associated with the covariant Lie derivative, and that it is independent of the ambiguities in the symplectic potential provided one takes into account the presence of non-trivial Lorentz charges that these ambiguities can introduce.