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

3.3.1 Non-locality of the gravitational energy-momentum and angular momentum

One reaction to the non-tensorial 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 the previous Section 3.2.4 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 non-tensorial 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 non-local geometric object, connecting the fibres 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 non-local. In fact, although the connection coefficients at a given point can be taken 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 pull-back 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 Sections [142Jump To The Next Citation Point266Jump To The Next Citation Point352Jump To The Next Citation Point353Jump To The Next Citation Point], and the expression of the pseudotensors by their superpotentials are the pull-backs of the Sparling equation [345Jump To The Next Citation Point130266Jump To The Next Citation Point]. In addition, Chang, Nester, and Chen [104Jump 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 11.3.4). 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 [324213]: Since 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), 2-surfaces (e.g. the area of a closed 2-surface) etc. determined by some body or physical fields. 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.

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 hypersurfaces S with boundary (interpreted as Cauchy surfaces for globally hyperbolic domains D), and
  3. the closed, orientable spacelike 2-surfaces S (interpreted as the boundary @S of Cauchy surfaces for globally hyperbolic domains).

A typical example fo 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 2-surface, as in Equation (12View Equation), or the expression QS [K] for the matter fields given by Equation (5View Equation). An example for Type 2 might be the integral of the Bel-Robinson ‘momentum’ on the hypersurface S:

integral ES[qa] := qdTdefgtetf-1egabc. (18) S 3!
This quantity is analogous to the integral a ES[q ] for the matter fields given by Equation (6View Equation) (though, by the remarks on the Bel-Robinson ‘energy’ in Section 3.1.1, its physical dimension cannot be of energy). If qa is a future pointing nonspacelike vector then ES[qa] > 0. Obviously, if such a quantity were independent of the actual hypersurface S, then it could also be rewritten as a charge integral on the boundary @S. The gravitational Hamiltonian provides an interesting example for the mixture of Type 2 and 3 expressions, because the form of the Hamiltonian is the 3-surface integral of the constraints on S and a charge integral on its boundary @S, thus if the constraints are satisfied then the Hamiltonian reduces to a charge integral. Finally, an example for Type 1 might be
ED := inf {ES[t] |S is a Cauchy surface for D}, (19)
the infimum of the ‘quasi-local Bel-Robinson energies’, where the infimum is taken on the set of all the Cauchy surfaces S for D with given boundary @S. (The infimum always exists because the Bel-Robinson ‘energy density’ Tabcdtatbtctd is non-negative.) Quasi-locality in any of these three senses agrees with the quasi-locality of Haag and Kastler [168Jump To The Next Citation Point169]. The specific quasi-local energy-momentum constructions provide further examples both for charge-integral-type expressions and 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 the main idea is that of Regge and Teitelboim [319Jump 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. 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 2-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 2-surface integrals at infinity (and, as we will see in Section 7.1.1, both the energy-momentum and angular momentum of the source in the linear approximation and the gravitational mass in the Newtonian theory of gravity can also be written as 2-surface integrals), the 2-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 2-surface observables, then three things must be specified:

  1. an appropriate general 2-surface integral (e.g. the integral of a superpotential 2-form in the Lagrangian approaches or a boundary term in the Hamiltonian approaches),
  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 2-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 2-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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