In non-gravitational 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 2-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 non-trivial, let us consider the expressions based on the linkage
integral (16).
does not satisfy the first part of Requirement 1. 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 2-surface, but on the way in which
is extended off the 2-surface. Therefore,
is ‘less quasi-local’ than
or
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 ) has the form of the charge integral of a superpotential:
, i.e. it has a Lagrangian interpretation. The corresponding conserved
Komar-current is 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
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 behaviour of the quasi-local quantities.
One such list for the energy-momentum and mass, based mostly on [131, 111
] and the
properties of the quasi-local energy-momentum of the matter fields of Section 2.2, might be the
following:
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 [131, 111
]. Usually,
is expected to be monotonic in some appropriate sense [111
]. For example,
if
for some achronal (and hence spacelike or null) hypersurface
in which
is a spacelike
closed 2-surface and the dominant energy condition is satisfied on
, then
seems to be a reasonable expectation [131
]. (But see also the next 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
and
are two connected 2-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 of the second 2-surface, respectively.
Thus, even if we consider the disjoint union
to surround a single physical system,
then 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 contradicts 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 |
|
2.2 |
For 2-surfaces with zero quasi-local mass the Pauli-Lubanski spin should be proportional to
the (null) energy-momentum 4-vector |
2.3 |
|
2.4 |
|
2.5 |
For small spheres the anti-self-dual part of |
Since there is no generally accepted definition for the angular momentum at null infinity, we cannot expect anything definite there in non-stationary spacetimes. Similarly, there are inequivalent suggestions for the centre-of-mass at the spatial infinity (see Sections 3.2.2 and 3.2.4).
As Eardley noted in [131], 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 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 non-stationary systems the concepts
of the event and apparent horizons deviate. Thus the causal argument above does not seem
possible to be formulated in the hypersurface
of Section 4.3.2. Actually, the root of the
non-monotonicity is the fact that the quasi-local energy is a 2-surface observable in the sense of
Expectation 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 2-surface along some special spacetime vector
field.
On the other hand, in the literature sometimes the positivity and the monotonicity requirements are
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 2-surfaces, then the energy may be positive definite and not monotonic.
The standard round sphere energy expression (26) in the closed Friedmann-Robertson-Walker
spacetime, or, more generally, the Dougan-Mason energy-momentum (see Section 8.2.3) are such
examples.
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