3.3 The necessity of quasi-locality for observables in general relativity

3.3.1 Nonlocality of the gravitational energy-momentum and angular momentum

One reaction to the nontensorial nature of the gravitational energy-momentum density expressions was to consider the whole problem ill defined and the gravitational energy-momentum meaningless. However, the successes discussed in Section 3.2 show that the global gravitational energy-momenta and angular momenta are useful notions, and hence, it could also be useful to introduce them even if the spacetime is not asymptotically flat. Furthermore, the nontensorial nature of an object does not imply that it is meaningless. For example, the Christoffel symbols are not tensorial, but they do have geometric, and hence physical content, namely the linear connection. Indeed, the connection is a nonlocal geometric object, connecting the fibers of the vector bundle over different points of the base manifold. Hence, any expression of the connection coefficients, in particular the gravitational energy-momentum or angular momentum, must also be nonlocal. In fact, although the connection coefficients at a given point can be taken to zero by an appropriate coordinate/gauge transformation, they cannot be transformed to zero on an open domain unless the connection is flat.

Furthermore, the superpotential of many of the classical pseudotensors (e.g., of the Einstein, Bergmann, Møller’s tetrad, Landau–Lifshitz pseudotensors), being linear in the connection coefficients, can be recovered as the pullback to the spacetime manifold of various forms of a single geometric object on the linear frame bundle, namely of the Nester–Witten 2-form, along various local cross sections [192Jump To The Next Citation Point, 358Jump To The Next Citation Point, 486Jump To The Next Citation Point, 487Jump To The Next Citation Point], and the expression of the pseudotensors by their superpotentials are the pullbacks of the Sparling equation [476Jump To The Next Citation Point, 175, 358Jump To The Next Citation Point]. In addition, Chang, Nester, and Chen [131Jump To The Next Citation Point] found a natural quasi-local Hamiltonian interpretation of each of the pseudotensorial expressions in the metric formulation of the theory (see Section 11.3.5). Therefore, the pseudotensors appear to have been ‘rehabilitated’, and the gravitational energy-momentum and angular momentum are necessarily associated with extended subsets of the spacetime.

This fact is a particular consequence of a more general phenomenon [76, 439, 284]: Since (in the absence of any non-dynamical geometric background) the physical spacetime is the isomorphism class of the pairs (M, gab) (instead of a single such pair), it is meaningless to speak about the ‘value of a scalar or vector field at a point p ∈ M’. What could have meaning are the quantities associated with curves (the length of a curve, or the holonomy along a closed curve), two-surfaces (e.g., the area of a closed two-surface) etc. determined by some body or physical fields. In addition, as Torre showed [523] (see also [524]), in spatially-closed vacuum spacetimes there can be no nontrivial observable, built as spatial integrals of local functions of the canonical variables and their finitely many derivatives. Thus, if we want to associate energy-momentum and angular momentum not only to the whole (necessarily asymptotically flat) spacetime, then these quantities must be associated with extended but finite subsets of the spacetime, i.e., must be quasi-local.

The results of Friedrich and Nagy [202Jump To The Next Citation Point] show that under appropriate boundary conditions the initial boundary value problem for the vacuum Einstein equations, written into a first-order symmetric hyperbolic form, has a unique solution. Thus, there is a solid mathematical basis for the investigations of the evolution of subsystems of the universe, and hence, it is natural to ask about the observables, and in particular the conserved quantities, of their dynamics.

3.3.2 Domains for quasi-local quantities

The quasi-local quantities (usually the integral of some local expression of the field variables) are associated with a certain type of subset of spacetime. In four dimensions there are three natural candidates:

  1. the globally hyperbolic domains D ⊂ M with compact closure,
  2. the compact spacelike (in fact, acausal) hypersurfaces Σ with boundary (interpreted as Cauchy surfaces for globally hyperbolic domains D), and
  3. the closed, orientable spacelike two-surfaces 𝒮 (interpreted as the boundary ∂ Σ of Cauchy surfaces for globally hyperbolic domains).

A typical example of type 3 is any charge integral expression: The quasi-local quantity is the integral of some superpotential 2-form built from the data given on the two-surface, as in Eq. (3.10View Equation), or the expression Q 𝒮[K ] for the matter fields given by (2.5View Equation). An example of type 2 might be the integral of the Bel–Robinson ‘momentum’ on the hypersurface Σ:

∫ a d ef-1 g E Σ[ξ ] := ξ Tdefgtt 3!𝜀 abc. (3.17 ) Σ
This quantity is analogous to the integral E Σ[ξa] for the matter fields given by Eq. (2.6View Equation) (though, by the remarks on the Bel–Robinson ‘energy’ in Section 3.1.2, its physical dimension cannot be of energy). If ξa is a future-pointing nonspacelike vector then E Σ[ξa] ≥ 0. Obviously, if such a quantity were independent of the actual hypersurface Σ, then it could also be rewritten as a charge integral on the boundary ∂ Σ. The gravitational Hamiltonian provides an interesting example for the mixture of type 2 and 3 expressions, because the form of the Hamiltonian is the three-surface integral of the constraints on Σ and a charge integral on its boundary ∂Σ, and thus, if the constraints are satisfied then the Hamiltonian reduces to a charge integral. Finally, an example of type 1 might be
E := inf {E [t] | Σ is a Cauchy surface for D} , (3.18 ) D Σ
the infimum of the ‘quasi-local Bel–Robinson energies’, where the infimum is taken on the set of all the Cauchy surfaces Σ for D with given boundary ∂Σ. (The infimum always exists because the Bel–Robinson ‘energy density’ Tabcdtatbtctd is non-negative.) Quasi-locality in any of these three senses is compatible with the quasi-locality of Haag and Kastler [231Jump To The Next Citation Point, 232]. The specific quasi-local energy-momentum constructions provide further examples both for charge-integral–type expressions and for those based on spacelike hypersurfaces.

3.3.3 Strategies to construct quasi-local quantities

There are two natural ways of finding the quasi-local energy-momentum and angular momentum. The first is to follow some systematic procedure, while the second is the ‘quasi-localization’ of the global energy-momentum and angular momentum expressions. One of the two systematic procedures could be called the Lagrangian approach: The quasi-local quantities are integrals of some superpotential derived from the Lagrangian via a Noether-type analysis. The advantage of this approach could be its manifest Lorentz-covariance. On the other hand, since the Noether current is determined only through the Noether identity, which contains only the divergence of the current itself, the Noether current and its superpotential is not uniquely determined. In addition (as in any approach), a gauge reduction (for example in the form of a background metric or reference configuration) and a choice for the ‘translations’ and ‘boost-rotations’ should be made.

The other systematic procedure might be called the Hamiltonian approach: At the end of a fully quasi-local (covariant or not) Hamiltonian analysis we would have a Hamiltonian, and its value on the constraint surface in the phase space yields the expected quantities. Here one of the main ideas is that of Regge and Teitelboim [433Jump To The Next Citation Point], that the Hamiltonian must reproduce the correct field equations as the flows of the Hamiltonian vector fields, and hence, in particular, the correct Hamiltonian must be functionally differentiable with respect to the canonical variables. This differentiability may restrict the possible ‘translations’ and ‘boost-rotations’ too. Another idea is the expectation, based on the study of the quasi-local Hamiltonian dynamics of a single scalar field, that the boundary terms appearing in the calculation of the Poisson brackets of two Hamiltonians (the ‘Poisson boundary terms’), represent the infinitesimal flow of energy-momentum and angular momentum between the physical system and the rest of the universe [502Jump To The Next Citation Point]. Therefore, these boundary terms must be gauge invariant in every sense. This requirement restricts the potential boundary terms in the Hamiltonian as well as the boundary conditions for the canonical variables and the lapse and shift. However, if we are not interested in the structure of the quasi-local phase space, then, as a short cut, we can use the Hamilton–Jacobi method to define the quasi-local quantities. The resulting expression is a two-surface integral. Nevertheless, just as in the Lagrangian approach, this general expression is not uniquely determined, because the action can be modified by adding an (almost freely chosen) boundary term to it. Furthermore, the ‘translations’ and ‘boost-rotations’ are still to be specified.

On the other hand, at least from a pragmatic point of view, the most natural strategy to introduce the quasi-local quantities would be some ‘quasi-localization’ of those expressions that gave the global energy-momentum and angular momentum of asymptotically flat spacetimes. Therefore, respecting both strategies, it is also legitimate to consider the Winicour–Tamburino-type (linkage) integrals and the charge integrals of the curvature.

Since the global energy-momentum and angular momentum of asymptotically flat spacetimes can be written as two-surface integrals at infinity (and, as we saw in Section 3.1.1 that the mass of the source in Newtonian theory, and as we will see in Section 7.1.1 that both the energy-momentum and angular momentum of the source in the linearized Einstein theory can also be written as two-surface integrals), the two-surface observables can be expected to have special significance. Thus, to summarize, if we want to define reasonable quasi-local energy-momentum and angular momentum as two-surface observables, then three things must be specified:

  1. an appropriate general two-surface integral (e.g., in the Lagrangian approaches the integral of a superpotential 2-form, or in the Hamiltonian approaches a boundary term together with the boundary conditions for the canonical variables),
  2. a gauge choice (in the form of a distinguished coordinate system in the pseudotensorial approaches, or a background metric/connection in the background field approaches or a distinguished tetrad field in the tetrad approach), and
  3. a definition for the ‘quasi-symmetries’ of the two-surface (i.e., the ‘generator vector fields’ of the quasi-local quantities in the Lagrangian, and the lapse and the shift in the Hamiltonian approaches, respectively, which, in the case of timelike ‘generator vector fields’, can also be interpreted as a fleet of observers on the two-surface).

In certain approaches the definition of the ‘quasi-symmetries’ is linked to the gauge choice, for example by using the Killing symmetries of the flat background metric.

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