Studying the perturbation of the dust-filled Friedmann-Robertson-Walker spacetimes, Hawking
found that
The 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 2-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 2-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
Equation (38
).
Obviously, for round spheres reduces to the standard expression (26
). 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 a small sphere of radius with centre
in non-vacuum spacetimes it is
,
while in vacuum it is
, where
is the energy-momentum tensor and
is the
Bel-Robinson tensor at
[204
]. The first result shows that in the lowest order the gravitational ‘field’
does not have a contribution to the 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 non-vacuum (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 this for the 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 Equation (38) 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 [338
, 340
]. 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 the Hawking energy may be negative, even in the Minkowski spacetime. Geometrically this
should be clear, since for an appropriately general (e.g. concave) 2-surface the integral
could be less than
. Indeed, in flat spacetime
is proportional to
by the
Gauss equation. For topologically spherical 2-surfaces in the
spacelike hyperplane of Minkowski
spacetime
is real and non-positive, and it is zero precisely for metric spheres, while for 2-surfaces in
the
timelike cylinder
is real and non-negative, and it is zero precisely for metric
spheres10.
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 [111
]:
will be called round enough if it is
a submanifold of a spacelike hypersurface
, and if among the 2-dimensional surfaces in
which
enclose the same volume as
does,
has the smallest area. Then it is proven by Christodoulou and
Yau [111] 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 the Hawking energy is not monotonic in general, it has interesting monotonicity properties for
special families of 2-surfaces. Hawking considered one-parameter families of spacelike 2-surfaces foliating the
outgoing and the ingoing null hypersurfaces, and calculated the change of
[171]. These calculations
were refined by Eardley [131]. Starting with a weakly future convex 2-surface
and using the boost
gauge freedom, he introduced a special family
of spacelike 2-surfaces in the outgoing null
hypersurface
, where
will be the luminosity distance along the outgoing null generators. He
showed that
is non-decreasing 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 non-increasing. Eardley also considered a special spacelike
hypersurface, filled by a family of 2-surfaces, for which
is non-decreasing. 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 neighbourhood of a mean convex 2-surface by special mean
convex 2-surfaces [182
]. (For the more general GHP formalism in which
is not fixed,
see [312
].) Assuming that the dominant energy condition holds, he showed that the Hawking
energy of these 2-surfaces is non-decreasing in the outgoing, and non-increasing 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 [145]. If is a weakly future
and past convex 2-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 2-surface, Frauendiener gives an
argument for the construction of a one-parameter family
of 2-surfaces being Lie-dragged along its own
inverse mean curvature vector
. Hence this family of surfaces would be analogous to the
solution of the geodesic equation, where the initial point and direction in that point specify the
whole solution, at least locally. Assuming that such a family of surfaces (and hence the vector
field
on the 3-submanifold swept by
) exists, Frauendiener showed that the Hawking
energy is non-decreasing along the vector field
if the dominant energy condition is satisfied.
However, no investigation has been made to prove the existence of such a family of surfaces.
Motivated by this result, Malec, Mars, and Simon [261] considered spacelike hypersurfaces with an
inverse mean curvature flow of Geroch thereon (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.
Hawking defined not only energy, but spatial momentum as well, completely analogously to how the spatial components of the Bondi-Sachs energy-momentum are related to the Bondi energy:
where Hawking considered the extension of the definition of to higher genus 2-surfaces also by the
second expression in Equation (38
). Then in the expression analogous to the first one in Equation (38
) the
genus of
appears.
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