Studying the perturbation of the dust-filled Friedmann-Robertson-Walker spacetimes, Hawking found that. Here we used the Gauss-Bonnet theorem and the GHP form of Equations (22, 23) for to express by the curvature components and the shears. Thus the Hawking energy is genuinely quasi-local.
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 . 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 : 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  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 . These calculations were refined by Eardley . 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 . (For the more general GHP formalism in which is not fixed, see .) 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 . 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  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:
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.
© Max Planck Society and the author(s)