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 Li 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 Poincaré 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.
Highlights
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
We show that our symplectic potential permits to derive the first law of black hole mechanics from the Noether charge associated with a Lie derivative, just like in the metric case [25]
In this paper we have proposed a gauge-invariant symplectic potential for tetrad general relativity, implementing what discussed for generic gauge theories in [14]
Summary
We consider the following first order action for Einstein-Cartan gravity (for a review, see [15]). The power of this formalism for diff-invariant theories is that it allows one to define quasi-local Hamiltonian charges for diffeomorphisms as the canonical generators in the covariant phase space.4 They are given by the pre-symplectic form with one variation being a Lie derivative δξ = £ξ,. As further support for the use of (2.34), we remark that it matches the boundary term derived in the Hamiltonian analysis of [13], starting from the requirement of having a canonical transformation of connection variables to the ADM phase space in the presence of corners We derived it from the requirement of full gauge-invariance of the presymplectic structure in the covariant phase space.. ΘE′ C(Lξ) + 2PIJKLeI ∧ T J ωKL ξ. 14When our paper appeared on the archives, Matthias Blau showed us some unpublished notes where he had constructed the same gauge-invariant potential and proved the property (2.38) [30]
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