Quasilocal energy and conserved charges derived from the gravitational action

Abstract

The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on 3 B, the history of the system's boundary. Energy density, momentum density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate 3 B. The integral of the energy density over such a two-surface B is the quasilocal energy associated with a spacelike threesurface Σ whose intersection with 3 B is the boundary B. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary B as a surface embedded in Σ. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper time translations on 3 B in the direction orthogonal to B. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on 3 B. For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt-Deser-Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.

Free boundary conditions and the AdS3/CFT2 correspondence

“…We show that AdS 3 with these boundary conditions is dual to 2D quantum gravity in either the conformal or lightcone gauges. The dual theory is formally described by the partition function 6) where φ a represents the matter and ghost fields that result from gauge fixing and S m+g is the sum of their actions. The central charge of the matter and ghost systems is denoted by c M = 0 which implies that the matter plus ghost theory, assumed to be conformallyinvariant at the classical level, has a non-vanishing Weyl anomaly.…”

“…Here S G is the total gravitational action, 1 γ ij is the metric at the boundary, and T ij is the Brown-York stress-energy tensor [6,7]. Making the boundary metric dynamical means…”

Black hole mass and Hamilton-Jacobi counterterms

“…One approach to dealing with this problem, as suggested by Brown and York [5], is to subtract the divergent action of a reference spacetime from the action for the spacetime of interest. In many cases this technique is sufficient, but it suffers from two main drawbacks.…”

Flat-space energy-momentum tensor from BMS/GCA correspondence

“…The warm up calculation is finding one-point function of energy momentum tensor which can be used in the bulk side for the calculation of conserved charges. The simple method for this calculation is given by [22] and [23] which uses Brown and York's proposal [25] for the definition of quasilocal stress tensor. According to [25], the Brown and York's quasi-local energy-momentum tensor is given by…”

“…We deduce that the ultra-relativistic field theory must live on a flat spacetime ∂M with line element ds 2 = −du 2 + G 2 dφ 2 . Now we can use components of our energy-momentum tensor (4.13) for finding conserved charges of symmetry generators (2.14) by making use of Brown and York definition [25] 18) where Σ is the spacelike surface of ∂M (u = constant surface), σ ab is metric of Σ i.e. σ ab dx a dx b = G 2 dφ 2 and v µ is the unit timelike vector normal to Σ.…”