  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.
