A gauge-invariant symplectic potential for tetrad general relativity

TL;DR

This paper presents a gauge-invariant symplectic potential for tetrad-based general relativity that aligns with known results and clarifies the role of surface terms and Lorentz charges in black hole mechanics.

Contribution

It introduces a fully gauge-invariant symplectic potential for tetrad gravity, clarifying its relation to surface charges and black hole thermodynamics.

Findings

Reproduces the Einstein-Hilbert symplectic potential when torsion vanishes.

Shows the potential reduces to the Komar form under Lie derivatives.

Establishes the first law of black hole mechanics from Noether identities.

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.