On lists of criteria of reasonableness of the quasi-local quantities

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:

To see that these conditions are nontrivial, let us consider the expressions based on the linkage integral (3.8

). $L [K ] 𝒮$ does not satisfy the first part of Requirement 1. In fact, it depends on the derivative of the normal components of $a K$ in the direction orthogonal to $𝒮$ for any value of the parameter $α$ . Thus it depends not only on the geometry of $𝒮$ and the vector field $Ka$ given on the two-surface, but on the way in which $Ka$ is extended off the two-surface. Therefore, $L 𝒮[K ]$ is ‘less quasi-local’ than $A 𝒮[ω]$ or $H𝒮 [λ, ¯μ]$ introduced in Sections 7.2.1 and 7.2.2, respectively.

We will see that the Hawking energy satisfies Requirement 1, but not Requirements 2 and 3. The Komar integral (i.e., the linkage for $α = 0$ ) has the form of the charge integral of a superpotential: $∮ 8πGK 𝒮 [K ] := ∇ [aKb ]1𝜀abcd 𝒮 2$ , i.e., it has a Lagrangian interpretation. The corresponding conserved Komar-current is defined by $a 8 πGK C [K ]$ $: a b [a b] = G bK + ∇b ∇ K$ . However, its flux integral on some compact spacelike hypersurface with boundary $𝒮 := ∂Σ$ cannot be a Hamiltonian on the ADM phase space in general. In fact, it is

4.3.2 Pragmatic criteria

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 [162

, 131

] and the properties of the quasi-local energy-momentum of the matter fields of Section 2.2, might be the following:

1.1	The quasi-local energy-momentum $P a 𝒮$ must be a future-pointing nonspacelike vector (assuming that the matter fields satisfy the dominant energy condition on some $Σ$ for which $𝒮 = ∂Σ$ , and maybe some form of the convexity of $𝒮$ should be required) (‘positivity’).

1.2	$a P 𝒮$ must be zero iff $D (Σ)$ is flat, and null iff $D (Σ )$ has a pp-wave geometry with pure radiation (‘rigidity’).

1.3	$P a𝒮$ must give the correct weak field limit.

1.4	$P a𝒮$ must reproduce the ADM, Bondi–Sachs and Abbott–Deser energy-momenta in the appropriate limits (‘correct large-sphere behavior’).

1.5	For small spheres $P a 𝒮$ must give the expected results (‘correct small sphere behavior’): $4 3 ab 3πr T tb$ in nonvacuum and $5 abcd kr T tbtctd$ in vacuum for some positive constant $k$ and the Bel–Robinson tensor $Tabcd$ .
1.6	For round spheres $Pa 𝒮$ must yield the ‘standard’ round-sphere expression.

1.7	For marginally trapped surfaces the quasi-local mass $m 𝒮$ must be the irreducible mass $∘ --------------2- Area(𝒮 )∕16πG$ .

For a different view on the positivity of the quasi-local energy see [358

]. 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 [162

, 131

]. Usually, $m 𝒮$ is expected to be monotonic in some appropriate sense [131

]. For example, if $𝒮1 = ∂Σ$ for some achronal (and hence spacelike or null) hypersurface $Σ$ in which $𝒮2$ is a spacelike closed two-surface and the dominant energy condition is satisfied on $Σ$ , then $m 𝒮1 ≥ m𝒮2$ seems to be a reasonable expectation [162

]. (See also Section 4.3.3.) On the other hand, 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 $𝒮1$ and $𝒮2$ 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 $𝒮1 ∪ 𝒮2$ 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 3.1.1, 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 $ab J𝒮$ may be expected to satisfy the following:

2.1	$J ab 𝒮$ 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 $P a 𝒮$ .

2.3	$ab J 𝒮$ must give the correct weak field limit.

2.4	$ab J 𝒮-$ must reproduce the generally-accepted spatial angular momentum at spatial infinity, and in stationary spacetimes it should reduce to the ‘standard’ expression at the null infinity as well (‘correct large-sphere behavior’).

2.5	For small spheres the anti-self-dual part of $ab J 𝒮$ , defined with respect to the center of the small sphere (the ‘vertex’ in Section 4.2.2) is expected to give $4 3 c D (A B )D ′ 3πr Tcdt(r𝜀 t )$ in nonvacuum and $′ Cr5Tcdeftctdte(r𝜀F (AtB )F )$ in vacuum for some constant $C$ (‘correct small sphere behavior’).

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

4.3.3 Incompatibility of certain ‘natural’ expectations

As Eardley noted in [162

], 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 $ΩM,m$ 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 $Σ$ . 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.

On the other hand, 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.

4.3 On lists of criteria of reasonableness of the quasi-local quantities

4.3.1 General expectations

4.3.2 Pragmatic criteria

4.3.3 Incompatibility of certain ‘natural’ expectations