Studying the perturbation of the dust-filled Friedmann–Robertson–Walker spacetimes, Hawking
found that
Hawking energy has the following clear physical interpretation even in a general spacetime, and, in fact,
can be introduced in this way. Starting with the rough idea that the mass-energy surrounded by a
spacelike two-sphere
should be the measure of bending of the ingoing and outgoing light rays
orthogonal to
, and recalling that under a boost gauge transformation
,
the
convergences
and
transform as
and
, respectively, the energy must have the
form
, where the unspecified parameters
and
can be determined in some
special situations. For metric two-spheres of radius
in the Minkowski spacetime, for which
and
, we expect zero energy, thus,
. For the event horizon of a
Schwarzschild black hole with mass parameter
, for which
, we expect
, which
can be expressed by the area of
. Thus,
, and hence, we arrive at
Eq. (6.1
).
Obviously, for round spheres, reduces to the standard expression (4.7
). This implies, in particular,
that the Hawking energy is not monotonic in general, since for a Killing horizon (e.g., for a stationary event
horizon)
, the Hawking energy of its spacelike spherical cross sections
is
.
In particular, for the event horizon of a Kerr–Newman black hole it is just the familiar irreducible mass
. For more general surfaces Hawking energy is calculated
numerically in [272].
For a small sphere of radius with center
in nonvacuum spacetimes it is
, while
in vacuum it is
, where
is the energy-momentum tensor and
is the
Bel–Robinson tensor at
[275
]. The first result shows that in the lowest order the gravitational ‘field’
does not have a contribution to Hawking energy, that is due exclusively to the matter fields.
Thus, in vacuum the leading order of
must be higher than
. Then, even a simple
dimensional analysis shows that the number of the derivatives of the metric in the coefficient of the
-order term in the power series expansion of
is
. However, there are no
tensorial quantities built from the metric and its derivatives such that the total number of the
derivatives involved would be three. Therefore, in vacuum, the leading term is necessarily of
order
, and its coefficient must be a quadratic expression of the curvature tensor. It is
remarkable that for small spheres
is positive definite both in nonvacuum (provided the
matter fields satisfy, for example, the dominant energy condition) and vacuum. This shows, in
particular, that
should be interpreted as energy rather than as mass: For small spheres in a
pp-wave spacetime
is positive, while, as we saw for matter fields in Section 2.2.3, a mass
expression could be expected to be zero. (We will see in Sections 8.2.3 and 13.5 that, for the
Dougan–Mason energy-momentum, the vanishing of the mass characterizes the pp-wave metrics
completely.)
Using the second expression in Eq. (6.1) it is easy to see that at future null infinity
tends to the
Bondi–Sachs energy. A detailed discussion of the asymptotic properties of
near null infinity both for
radiative and stationary spacetimes is given in [455
, 457
]. Similarly, calculating
for large spheres near
spatial infinity in an asymptotically flat spacelike hypersurface, one can show that it tends to the ADM
energy.
In general, Hawking energy may be negative, even in Minkowski spacetime. Geometrically this should be
clear, since for an appropriately general (e.g., concave) two-surface , the integral
could be
less than
. Indeed, in flat spacetime
is proportional to
by the Gauss
equation. For topologically-spherical two-surfaces in the
spacelike hyperplane of Minkowski
spacetime
is real and nonpositive, and it is zero precisely for metric spheres, while for two-surfaces in
the
timelike cylinder
is real and non-negative, and it is zero precisely for metric
spheres.9
If, however,
is ‘round enough’ (not to be confused with the round spheres in Section 4.2.1), which is
some form of a convexity condition, then
behaves nicely [143
]:
will be called round enough if it is
a submanifold of a spacelike hypersurface
, and if among the two-dimensional surfaces in
, which
enclose the same volume as
does,
has the smallest area. It is proven by Christodoulou and
Yau [143] that if
is round enough in a maximal spacelike slice
on which the energy density of the
matter fields is non-negative (for example, if the dominant energy condition is satisfied), then the Hawking
energy is non-negative.
Although Hawking energy is not monotonic in general, it has interesting monotonicity properties for
special families of two-surfaces. Hawking considered one-parameter families of spacelike two-surfaces
foliating the outgoing and the ingoing null hypersurfaces, and calculated the change of [236]. These
calculations were refined by Eardley [176]. Starting with a weakly future convex two-surface
and using
the boost gauge freedom, he introduced a special family
of spacelike two-surfaces in the outgoing null
hypersurface
, where
will be the luminosity distance along the outgoing null generators. He
showed that
is nondecreasing with
, provided the dominant energy condition holds
on
. Similarly, for weakly past convex
and the analogous family of surfaces in the
ingoing null hypersurface
is nonincreasing. Eardley also considered a special spacelike
hypersurface, filled by a family of two-surfaces, for which
is nondecreasing. By relaxing
the normalization condition
for the two null normals to
for some
, Hayward obtained a flexible enough formalism to introduce a double-null foliation (see
Section 11.2 below) of a whole neighborhood of a mean convex two-surface by special mean
convex two-surfaces [247
]. (For the more general GHP formalism in which
is not fixed,
see [425
].) Assuming that the dominant energy condition holds, he showed that the Hawking
energy of these two-surfaces is nondecreasing in the outgoing, and nonincreasing in the ingoing
direction.
In contrast to the special foliations of the null hypersurfaces above, Frauendiener defined a special
spacelike vector field, the inverse mean curvature vector in the spacetime [194]. If is a weakly future
and past convex two-surface, then
is an outward-directed
spacelike normal to
. Here
is the trace of the extrinsic curvature tensor:
(see
Section 4.1.2). Starting with a single weakly future and past convex two-surface, Frauendiener gives an
argument for the construction of a one-parameter family
of two-surfaces being Lie-dragged along its
own inverse mean curvature vector
. Assuming that such a family of surfaces (and hence, the
vector field
on the three-submanifold swept by
) exists, Frauendiener showed that the
Hawking energy is nondecreasing along the vector field
if the dominant energy condition is
satisfied. This family of surfaces would be analogous to the solution of the geodesic equation,
where the initial point and direction at that point specify the whole solution, at least locally.
However, it is known (Frauendiener, private communication) that the corresponding flow is
based on a system of parabolic equations such that it does not admit a well-posed initial value
formulation.10
Motivated by this result, Malec, Mars, and Simon [351] considered the inverse mean curvature flow of
Geroch on spacelike hypersurfaces (see Section 6.2.2). They showed that if the dominant energy condition
and certain additional (essentially technical) assumptions hold, then the Hawking energy is monotonic.
These two results are the natural adaptations for the Hawking energy of the corresponding results known
for some time for the Geroch energy, aiming to prove the Penrose inequality. (We return to this latter issue
in Section 13.2, only for a very brief summary.) The necessary conditions on flows of two-surfaces on null,
as well as spacelike, hypersurfaces ensuring the monotonicity of the Hawking energy are investigated
in [114]. The monotonicity property of the Hawking energy under another geometric flows is discussed
in [89].
For a discussion of the relationship between Hawking energy and other expressions (e.g.,
the Bartnik mass and the Brown–York energy), see [460]. For the first attempts to introduce
quasi-local energy oparators, in particular the Hawking energy oparator, in loop quantum gravity,
see [565].
Hawking defined not only energy, but spatial momentum as well, completely analogously to how the spatial components of Bondi–Sachs energy-momentum are related to Bondi energy:
where Hawking considered the extension of the definition of to higher genus two-surfaces as
well by the second expression in Eq. (6.1
). Then, in the expression analogous to the first one
in Eq. (6.1
), the genus of
appears. For recent generalizations of the Hawking energy for
two-surfaces foliating the stationary and dynamical untrapped hypersurfaces, see [527
, 528
] and
Section 11.3.4.
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Living Rev. Relativity 12, (2009), 4
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