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Noether Charges, Brown-York Quasilocal Energy and Related Topics

The Lagrangian proposed by York et al. and the covariant first order Lagrangian for General Relativity are introduced to deal with the (vacuum) gravitational field on a reference background. The two Lagrangians are compared and we show that the first one can be obtained from the latter under suitable hypotheses. The induced variational principles are also compared and discussed. A conditioned correspondence among Noether conserved quantities, quasilocal energy and the standard Hamiltonian obtained by 3+1 decomposition is also established. As a result, it turns out that the covariant first order Lagrangian is better suited whenever a reference background field has to be taken into account, as it is commonly accepted when dealing with conserved quantities in non-asymptotically flat spacetimes. As a further advantage of the use of a covariant first order Lagrangian, we show that all the quantities computed are manifestly covariant, as it is appropriate in General Relativity.