Action and Energy of the Gravitational Field

TL;DR

This work develops the canonical quasilocal formalism (cqf) for general relativity by applying a field-theoretic Hamilton–Jacobi viewpoint to a bounded spacetime region. By using the Trace-K action and a double-foliation boundary setup, the authors define quasilocal energy–momentum surface densities for two observer frames (barred and unbarred) and derive exact boost relations and invariants connecting these densities. The formalism yields explicit expressions for quasilocal quantities in a variety of settings, including static spacetimes, boosted Schwarzschild, isotropic cosmologies, and cylindrical waves, and clarifies the relationship between quasilocal quantities and spatial infinity concepts such as ADM and Ashtekar–Hansen energies. A key feature is the subtraction (reference) term, whose choice is tied to isometric embeddings (Weyl problem) and directly affects the energy zero-point while leaving momentum densities invariant under Euclidean references. The results illuminate how boundary data govern gravitational energy-momentum in bounded regions and provide a bridge to traditional total-energy definitions at infinity.

Abstract

We present a detailed examination of the variational principle for metric general relativity as applied to a ``quasilocal'' spacetime region $\M$ (that is, a region that is both spatially and temporally bounded). Our analysis relies on the Hamiltonian formulation of general relativity, and thereby assumes a foliation of $\M$ into spacelike hypersurfaces $Σ$. We allow for near complete generality in the choice of foliation. Using a field--theoretic generalization of Hamilton--Jacobi theory, we define the quasilocal stress-energy-momentum of the gravitational field by varying the action with respect to the metric on the boundary $\partial\M$. The gravitational stress-energy-momentum is defined for a two--surface $B$ spanned by a spacelike hypersurface in spacetime. We examine the behavior of the gravitational stress-energy-momentum under boosts of the spanning hypersurface. The boost relations are derived from the geometrical and invariance properties of the gravitational action and Hamiltonian. Finally, we present several new examples of quasilocal energy--momentum, including a novel discussion of quasilocal energy--momentum in the large-sphere limit towards spatial infinity.

Figures (5)

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• Figure 2: Penrose diagram for the exterior region of the Schwarzschild geometry.
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