Abstract
We compute the Hamiltonian surface charges of gravity for a family of conservative boundary conditions, that include Dirichlet, Neumann, and York’s mixed boundary conditions defined by holding fixed the conformal induced metric and the trace of the extrinsic curvature. We show that for all boundary conditions considered, canonical methods give the same answer as covariant phase space methods improved by a boundary Lagrangian, a prescription recently developed in the literature and thus supported by our results. The procedure also suggests a new integrable charge for the Einstein-Hilbert Lagrangian, different from the Komar charge for non-Killing and non-tangential diffeomorphisms. We study how the energy depends on the choice of boundary conditions, showing that both the quasi-local and the asymptotic expressions are affected. Finally, we generalize the analysis to non-orthogonal corners, confirm the matching between covariant and canonical results without any change in the prescription, and discuss the subtleties associated with this case.
Highlights
Effect of this process is to produce smaller values of the energy
We show that for all boundary conditions considered, canonical methods give the same answer as covariant phase space methods improved by a boundary Lagrangian, a prescription recently developed in the literature and supported by our results
Since the rationale for constructing the covariant surface charges is to use an action with a well-posed variational principle, one may wonder if the corner Lagrangian contributes to the formula for the charges as well
Summary
We briefly review the different boundary conditions we will consider in this paper. We start from the Einstein-Hilbert (EH) Lagrangian, LEH = R ,. The corner contribution can always be thought of as part of the space-like hypersurface, and its variation corresponds to a change in the state, and not of the boundary conditions [15] It is a change of state associated to a different choice of lapse and shift, and which can be considered irrelevant to characterize different physical solutions.. A similar logic can be applied to the case of York’s mixed boundary conditions Since they leave the determinant of the induced metric free, it seems reasonable to us to take δβ = 0 in this case, even though it is not required by the well-posedness of the initial value problem [27, 34]. We will refer to them as b-generalized trace-K actions
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