Quasilocal Hamiltonians in general relativity

Abstract

Abstract. We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consistent definition of quasi-local energy in GR requires, at a minimum, a framework based on the (currently unknown) geometric well-posedness of the initial boundary value problem for the Einstein equations.The analysis of the gravitational field by Arnowitt-Deser-Misner [1] has led to a clear and welldefined construction of the Hamiltonian, and resulting definitions of energy, linear and angular momentum in the context of asymptotically flat spacetimes. These concepts are obviously of basic importance in understanding the physics of such (infinite) isolated gravitating systems. Nevertheless, infinite systems are idealizations of more realistic physical situations, and it is desirable to have available a similar analysis in the case of physical systems of finite extent.However the understanding of this issue for domains of finite extent is much less satisfactory. Despite numerous proposals, from a number of different viewpoints, a consensus has not yet been reached on a suitable definition of the Hamiltonian or energy of a finite system, i.e. a quasi-local Hamiltonian; cf.[2] for an excellent detailed survey of the current state of the art.In this paper, we first examine and comment on the approach to the definition of energy of a finite region of spacetime based on the Hamiltonian formulation of GR. This is essentially based on a localization of the approach taken by ADM [1] and Regge-Teitelboim [3], keeping careful track of the boundary terms that arise in a Hamiltonian or Hamilton-Jacobi analysis. This approach was initiated and pioneered by Brown-York (BY) [4]. To keep the discussion focused on the central issue, we only consider the gravitational field, (so other matter fields are set to zero); in addition, we consider only the energy and not related concepts such as linear and angular momentum, although this could be done without undue difficulty. Finally most all of the discussion below applies also to more recent modifications of the BY approach by several authors, cf.[5]-[8]; however again for clarity and simplicity we focus on the Brown-York Hamiltonian and leave it to the reader to extend the analysis to the more recent alternatives.We first recall the set-up. Let M be a spacetime region, topologically of the form I × Σ, with I = [0, 1] parametrizing time and Σ a compact 3-manifold with boundary S; typically S = ∂Σ is a 2-sphere and Σ a 3-ball. The boundary ∂M of M is a union of two spatial hypersurfaces Σ 0 ∪ Σ 1 and the timelike boundary T = I × ∂S = ∪S t . These boundaries meet at the seams or corners S 0 and S 1 . The Einstein-Hilbert action is then given by (setting 8πG = 1),where g is a smooth Lorentz metric on M . The Hamiltonian in GR plays two important but a priori distin...

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…Conversely, quasi-local Hamiltonian techniques could potentially be used to identify a large class of boundary conditions that are compatible with the evolution equation. (For a discussion of such a potential link between the two appraches, see e.g., [502, 16]). Moreover, in the quasi-local Hamiltonian approach we might hope to be able to associate nontrivial observables (and, in particular, conserved quantities) with localized systems in a natural way.…”

Brown-York charges with mixed boundary conditions

“…These two options are probably the ones most commonly considered. 6 However as pointed out in [27,28], a better choice of boundary conditions is York's mixed boundary conditions [29,30], which hold fixed the conformal metric qµν := q −1/3 q µν and K, because they lead to a better posed initial boundary value problem (See also [31][32][33][34]). With this choice, one needs to add a boundary Lagrangian given by [30]…”

“…First, they provide additional support for the prescription of [16], by showing that it reproduces the canonical results for different boundary Lagrangian, and how to amend it in the extension to nonorthogonal corners. Second, they bring more attention to the charges associated with York's mixed boundary conditions [34], which have been argued to give a better posed initial-boundary value problem [27,28]. Third, they will hopefully encourage discussions about the dependence of the energy on the boundary conditions.…”

Cylinder transition amplitudes in pure AdS3 gravity

“…19 of that same reference. 11 However, there is an important point to consider now: we want to fix the complex/conformal structure at the initial and final surfaces, which implies that in order to have a well-defined variational principle, the Gibbons-Hawking term should be multiplied by 1/2 [27,28]. Moreover, there is no a priori reason to include the area terms I B , so we discard them.…”