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 (3.15).
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
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 |
|
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 |
|
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 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.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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