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A gauge-invariant symplectic potential for tetrad general relativity. (English) Zbl 1395.83046

Summary: 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.

MSC:

83C57 Black holes
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C40 Gravitational energy and conservation laws; groups of motions
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