Studying the perturbation of the dust-filled Friedmann–Robertson–Walker spacetimes, Hawking found that

behaves as an appropriate notion of energy surrounded by the spacelike topological two-sphere [236]. Here we used the Gauss–Bonnet theorem and the GHP form of Eqs. (4.3) and (4.4) for to express by the curvature components and the shears. Thus, Hawking energy is genuinely quasi-local.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 , , are essentially the first four spherical harmonics: Here and are the standard complex stereographic coordinates on .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.

Living Rev. Relativity 12, (2009), 4
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