The initial boundary value problem and quasi-local Hamiltonians in General Relativity
Zhongshan An, Michael T. Anderson
TL;DR
This work interrogates how to formulate quasi-local Hamiltonians in General Relativity in the presence of a finite time-like boundary by scrutinizing initial-boundary data choices. It shows Dirichlet and Neumann boundary data are ill-posed for the vacuum IBVP, while conformal-mean curvature data and AA boundary data offer well-posed, covariant formulations that admit consistent variational problems and Hamiltonians. By leveraging the covariant phase space framework, the authors derive boundary-dependent Hamiltonians for different data and discuss normalization schemes that yield positive, physically meaningful quasi-local energies, connecting to Brown-York, Wang-Yau, and Bartnik notions. The results highlight the central role of boundary data selection and gauge choices in obtaining robust, diffeomorphism-invariant quasi-local quantities in GR. Overall, the paper provides a cohesive, covariant pathway to defining and normalizing quasi-local energies in finite GR regions, with clear implications for energy bounds and geometric uniqueness.
Abstract
We discuss relations between the initial boundary value problem (IBVP) and quasi-local Hamiltonians in GR. The latter have traditionally been based on Dirichlet boundary conditions, which however are shown here to be ill-posed for the IBVP. We present and analyse several other choices of boundary conditions which are better behaved with respect to the IBVP and carry out a corresponding Hamiltonian analysis, using the framework of the covariant phase space method.