Boundary and corner terms in the action for general relativity

Abstract

We revisit the action principle for general relativity, motivated by the path integral approach to quantum gravity. We consider a spacetime region whose boundary has piecewise C 2 components, each of which can be spacelike, timelike or null and consider metric variations in which only the pullback of the metric to the boundary is held fixed. Allowing all such metric variations we present a unified treatment of the spacelike, timelike and null boundary components using Cartan's tetrad formalism. Apart from its computational simplicity, this formalism gives us a simple way of identifying corner terms. We also discuss "creases" which occur when the boundary is the event horizon of a black hole. Our treatment is geometric and intrinsic and we present our results both in the computationally simpler tetrad formalism as well as the more familiar metric formalism. We recover known results from a simpler and more general point of view and find some new ones. arXiv:1612.00149v2 [gr-qc] 1 Feb 2017

Sachs’ free data in real connection variables

“…For a more general expression of Θ without a full foliation and a discussion of corner terms without any coordinate gauge fixing, and its relevance to capture the full information about the charges, see [47]. See also [43,[48][49][50] for additional discussions on corner terms.…”

Stress tensor on null boundaries

“…For the details of the calculations and specially the method by which one can take into account the contribution of joints, see Refs. [8,10,12] or [1]. We see that the variation of HE action includes both the metric and its normal derivative (the extrinsic curvature) on the boundary, thus, the variational principle is not well-posed for this action if we demand the Dirichlet boundary conditions 3 .…”

“…The proper boundary action complementing the HE action, when the boundary is timelike or spacelike, is the well-known Gibbons-Hawking-York (GHY) term [5,6]. Applying variational principle in GR for null boundaries has been a subject of investigation in recent years [7][8][9][10][11].…”

Extended thermodynamics and complexity in gravitational Chern-Simons theory

“…We know that the variation of the total action (I EH + I GCS ) with respect to the metric is well-defined if only the GHY boundary term is included [39], up to a counterterm ambiguity [37,43,56] that we discuss in Appendix B. Using this formulation then one can say that there are no other corner terms to consider, as we know that the corner terms originate from the boundary term in the action [33,57]. Using (2.11) we therefore reach the intriguing conclusion that the late-time bound is…”