In nongravitational physics the notions of conserved quantities are connected with symmetries of the system, and they are introduced through some systematic procedure in the Lagrangian and/or Hamiltonian formalism. In general relativity the total energy-momentum and angular momentum are two-surface observables, thus, we concentrate on them even at the quasi-local level. These facts motivate our three a priori expectations:

- The quasi-local quantities that are two-surface observables should depend only on the two-surface data, but they cannot depend, e.g., on the way that the various geometric structures on are extended off the two-surface. There seems to be no a priori reason why the two-surface would have to be restricted to spherical topology. Thus, in the ideal case, the general construction of the quasi-local energy-momentum and angular momentum should work for any closed orientable spacelike two-surface.
- It is desirable to derive the quasi-local energy-momentum and angular momentum as the charge integral (Lagrangian interpretation) and/or as the value of the Hamiltonian on the constraint surface in the phase space (Hamiltonian interpretation). If they are introduced in some other way, they should have a Lagrangian and/or Hamiltonian interpretation.
- These quantities should correspond to the ‘quasi-symmetries’ of the two-surface, which quasi-symmetries are special spacetime vector fields on the two-surface. In particular, the quasi-local energy-momentum should be expected to be in the dual of the space of the ‘quasi-translations’, and the angular momentum in the dual of the space of the ‘quasi-rotations’.

To see that these conditions are nontrivial, let us consider the expressions based on the linkage integral (3.15). does not satisfy the first part of our first requirement. In fact, it depends on the derivative of the normal components of in the direction orthogonal to for any value of the parameter . Thus, it depends not only on the geometry of and the vector field given on the two-surface, but on the way in which is extended off the two-surface. Therefore, is ‘less quasi-local’ than or that will be introduced in Sections 7.2.1 and 7.2.2, respectively.

We will see that the Hawking energy satisfies our first requirement, but not the second and the third ones. The Komar integral (i.e., half of the linkage for ) has the form of the charge integral of a superpotential, , i.e., it has a Lagrangian interpretation. The corresponding conserved Komar-current was defined by . However, its flux integral on some compact spacelike hypersurface with boundary cannot be a Hamiltonian on the ADM phase space in general. In fact, it is

Here and are, respectively, the Hamiltonian and momentum constraints of the vacuum theory, is the future-directed unit normal to , is the outward-directed unit normal to in , and and are the lapse and shift part of , respectively, defined by . Thus, is a well-defined function of the configuration and velocity variables and , respectively. However, since the velocity cannot be expressed by the canonical variables (see e.g. [558, 63]), can be written as a function on the ADM phase space only if the boundary conditions at ensure the vanishing of the integral of .

Since in certain special situations there are generally accepted definitions for the energy-momentum and angular momentum, it seems reasonable to expect that in these situations the quasi-local quantities reduce to them. One half of the pragmatic criteria is just this expectation, and the other is a list of some a priori requirements on the behavior of the quasi-local quantities.

One such list for the energy-momentum and mass, based mostly on [176, 143] and the properties of the quasi-local energy-momentum of the matter fields of Section 2.2, might be the following:

For a different view on the positivity of the quasi-local energy see [391]. Item 1.7 is motivated by the expectation that the quasi-local mass associated with the apparent horizon of a black hole (i.e., the outermost marginally-trapped surface in a spacelike slice) be just the irreducible mass [176, 143]. Update

Usually, is expected to be monotonically increasing in some appropriate sense [143]. For example, if for some achronal (and hence spacelike or null) hypersurface in which is a spacelike closed two-surface and the dominant energy condition is satisfied on , then seems to be a reasonable expectation [176]. (However, see also Section 4.3.3.) A further, and, in fact, a related issue is the (post) Newtonian limit of the quasi-local mass expressions. In item 1.4 we expected, in particular, that the quasi-local mass tends to the ADM mass at spatial infinity. However, near spatial infinity the radiation and the dynamics of the fields and the geometry die off rapidly. Hence, in vacuum asymptotically flat spacetimes in the asymptotic regime the gravitational ‘field’ approaches the Newtonian one, and hence its contribution to the total energy of the system is similar to that of the negative definite binding energy [400, 199]. Therefore, it seems natural to expect that the quasi-local mass tends to the ADM mass as a monotonically decreasing function (see also sections 3.1.1 and 12.3.3).

In contrast to the energy-momentum and angular momentum of the matter fields on the Minkowski spacetime, the additivity of the energy-momentum (and angular momentum) is not expected. In fact, if and are two connected two-surfaces, then, for example, the corresponding quasi-local energy-momenta would belong to different vector spaces, namely to the dual of the space of the quasi-translations of the first and second two-surface, respectively. Thus, even if we consider the disjoint union to surround a single physical system, we can add the energy-momentum of the first to that of the second only if there is some physically/geometrically distinguished rule defining an isomorphism between the different vector spaces of the quasi-translations. Such an isomorphism would be provided for example by some naturally-chosen globally-defined flat background. However, as we discussed in Section 3.1.2, general relativity itself does not provide any background. The use of such a background would contradict the complete diffeomorphism invariance of the theory. Nevertheless, the quasi-local mass and the length of the quasi-local Pauli–Lubanski spin of different surfaces can be compared, because they are scalar quantities.

Similarly, any reasonable quasi-local angular momentum expression may be expected to satisfy the following:

2.1 |
must give zero for round spheres. |

2.2 |
For two-surfaces with zero quasi-local mass, the Pauli–Lubanski spin should be proportional to the (null) energy-momentum four-vector . |

2.3 |
must give the correct weak field limit. |

2.4 |
must reproduce the generally-accepted spatial angular momentum at spatial infinity, and in stationary and in axi-symmetric spacetimes it should reduce to the ‘standard’ expressions at the null infinity as well (‘correct large-sphere behaviour’). |

2.5 |
For small spheres the anti-self-dual part of , defined with respect to the center of the small sphere (the ‘vertex’ in Section 4.2.2) is expected to give in nonvacuum and in vacuum for some constant (‘correct small sphere behaviour’). |

Since there is no generally accepted definition for the angular momentum at null infinity, we cannot expect anything definite there in nonstationary, non-axi-symmetric spacetimes. Similarly, there are inequivalent suggestions for the center-of-mass at spatial infinity (see Sections 3.2.2 and 3.2.4).

As Eardley noted in [176], probably no quasi-local energy definition exists, which would satisfy all of his criteria. In fact, it is easy to see that this is the case. Namely, any quasi-local energy definition, which reduces to the ‘standard’ expression for round spheres cannot be monotonic, as the closed Friedmann–Robertson–Walker or the spacetimes show explicitly. The points where the monotonicity breaks down are the extremal (maximal or minimal) surfaces, which represent an event horizon in the spacetime. Thus, one may argue that since the event horizon hides a portion of spacetime, we cannot know the details of the physical state of the matter + gravity system behind the horizon. Hence, in particular, the monotonicity of the quasi-local mass may be expected to break down at the event horizon. However, although for stationary systems (or at the moment of time symmetry of a time-symmetric system) the event horizon corresponds to an apparent horizon (or to an extremal surface, respectively), for general nonstationary systems the concepts of the event and apparent horizons deviate. Thus, it does not seem possible to formulate the causal argument of Section 4.3.2 in the hypersurface . Actually, the root of the nonmonotonicity is the fact that the quasi-local energy is a two-surface observable in the sense of requirement 1 in Section 4.3.1 above. This does not mean, of course, that in certain restricted situations the monotonicity (‘local monotonicity’) could not be proven. This local monotonicity may be based, for example, on Lie dragging of the two-surface along some special spacetime vector field. Update

If the quasi-local mass should, in fact, tend to the ADM mass as a monotonically decrasing function in the asymptotic region of asymptotically flat spacetimes, then neither item 1.6 nor 1.7 can be expected to hold. In fact, if the dominant energy condition is satisfied, then the standard round-sphere Misner–Sharp energy is a monotonically increasing or constant (rather than strictly decreasing) function of the area radius . For example, the Misner–Sharp energy in the Schwarzschild spacetime is the constant function . The Schwarzschild solution provides a conterexample to item 1.7, too: Since both its ADM mass and the irreducible mass of the black hole are , any quasi-local mass function of the radius which is strictly decreasing for large and coincides with them at infinity and on the horizon, respectively, would have to take its maximal value on some two-surface outside the horizon. However, it does not seem why such a gemetrically, and hence physically distinguished two-surface should exist.

In the literature the positivity and monotonicity requirements are sometimes confused, and there is an ‘argument’ that the quasi-local gravitational energy cannot be positive definite, because the total energy of the closed universes must be zero. However, this argument is based on the implicit assumption that the quasi-local energy is associated with a compact three-dimensional domain, which, together with the positive definiteness requirement would, in fact, imply the monotonicity and a positive total energy for the closed universe. If, on the other hand, the quasi-local energy-momentum is associated with two-surfaces, then the energy may be positive definite and not monotonic. The standard round sphere energy expression (4.7) in the closed Friedmann–Robertson–Walker spacetime, or, more generally, the Dougan–Mason energy-momentum (see Section 8.2.3) are such examples.

Living Rev. Relativity 12, (2009), 4
http://www.livingreviews.org/lrr-2009-4 |
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