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Geometrically the phase space of a mechanical system involves the co-tangentbundle of the configuration space. The phase space of a relativistic fieldtheory is infinite dimensional and can be endowed with a symplectic structuredefined in a perfectly co-variant manner that is very useful for discussingsymmetries and conserved quantities of the system. In general relativity thesymplectic structure takes Darboux form and it is shown in this work that thepresence of a cosmological constant does not change this conclusion.  For space-times that admit time-like Killing vectors the formalism can beused to define mass in general relativity and it is known that, forasymptotically flat black holes, this mass is identical the the usualArnowitt-Desner-Misner mass while for asymptotically anti-de Sitter Kerrmetrics it is the same as the Henneaux-Teitelboim mass. We show that the sameformalism can also be used to derive the Brown-York mass and the Bondi mass forstationary space times, in particular the Brown-York mass has a naturalinterpretation in terms of differential forms on the space of solutions of thetheory.

On the definition of mass in general relativity: Noether charges and conserved quantities in diffeomorphism invariant theories