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6.1 The Hawking energy

6.1.1 The definition

Studying the perturbation of the dust-filled k = -1 Friedmann-Robertson-Walker spacetimes, Hawking found that

V~ --------- Area(S) ( 1 gf ) EH (S) := ------2- 1 + --- rr'dS = V~ -16pG---- 2p S Area(S) 1 gf ( ) = ------2---- ss'+ ss' - y2 - y2'+ 2f11 + 2/\ dS (38) 16pG 4p S
behaves as an appropriate notion of energy surrounded by the spacelike topological 2-sphere S [171Jump To The Next Citation Point]. Here we used the Gauss-Bonnet theorem and the GHP form of Equations (22View Equation, 23View Equation) for F to express ' rr 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, EH can be introduced in this way. Starting with the rough idea that the mass-energy surrounded by a spacelike 2-sphere S should be the measure of bending of the ingoing and outgoing light rays orthogonal to S, and recalling that under a boost gauge transformation a a l '--> al, a -1 a n '--> a n the convergences r and r' transform as r '--> ar and r''--> a -1r', respectively, the energy must have the form gf C + D S rr'dS, where the unspecified parameters C and D can be determined in some special situations. For metric 2-spheres of radius r in the Minkowski spacetime, for which r = - 1/r and ' r = 1/2r, we expect zero energy, thus D = C/2(p). For the event horizon of a Schwarzschild black hole with mass parameter m, for which ' r = 0 = r, we expect m/G, which can be expressed by the area of S. Thus C2 = Area(S)/(16pG2), and hence we arrive at Equation (38View Equation).

6.1.2 The Hawking energy for spheres

Obviously, for round spheres EH reduces to the standard expression (26View Equation). 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) r = 0, the Hawking energy of its spacelike spherical cross sections S is V~ ------------------ Area(S)/(16pG2). In particular, for the event horizon of a Kerr-Newman black hole it is just the familiar irreducible mass V~ ---2---2-----V ~ --2----2----2 2m - e + 2m m - e - a /(2G).

For a small sphere of radius r with centre p (- M in non-vacuum spacetimes it is 4p 3-r3Tabtatb, while in vacuum it is 452G-r5Tabcdtatbtctd, where Tab is the energy-momentum tensor and Tabcd is the Bel-Robinson tensor at p [204Jump To The Next Citation Point]. 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 EH must be higher than 3 r. Then even a simple dimensional analysis shows that the number of the derivatives of the metric in the coefficient of the rk order term in the power series expansion of EH is (k - 1). 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 5 r, and its coefficient must be a quadratic expression of the curvature tensor. It is remarkable that for small spheres E H 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 EH should be interpreted as energy rather than as mass: For small spheres in a pp-wave spacetime EH 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 (38View Equation) it is easy to see that at future null infinity EH tends to the Bondi-Sachs energy. A detailed discussion of the asymptotic properties of EH near null infinity, both for radiative and stationary spacetimes is given in [338Jump To The Next Citation Point340Jump To The Next Citation Point]. Similarly, calculating E H for large spheres near spatial infinity in an asymptotically flat spacelike hypersurface, one can show that it tends to the ADM energy.

6.1.3 Positivity and monotonicity properties

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 S the integral gf S rr'dS could be less than -2p. Indeed, in flat spacetime EH is proportional to gf (ss'+ ss') dS S by the Gauss equation. For topologically spherical 2-surfaces in the t = const. spacelike hyperplane of Minkowski spacetime ' ss is real and non-positive, and it is zero precisely for metric spheres, while for 2-surfaces in the r = const. timelike cylinder ss' is real and non-negative, and it is zero precisely for metric spheres10. If, however, S 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 EH behaves nicely [111Jump To The Next Citation Point]: S will be called round enough if it is a submanifold of a spacelike hypersurface S, and if among the 2-dimensional surfaces in S which enclose the same volume as S does, S has the smallest area. Then it is proven by Christodoulou and Yau [111] that if S is round enough in a maximal spacelike slice S 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 EH [171]. These calculations were refined by Eardley [131]. Starting with a weakly future convex 2-surface S and using the boost gauge freedom, he introduced a special family Sr of spacelike 2-surfaces in the outgoing null hypersurface N, where r will be the luminosity distance along the outgoing null generators. He showed that EH(Sr) is non-decreasing with r, provided the dominant energy condition holds on N. Similarly, for weakly past convex S and the analogous family of surfaces in the ingoing null hypersurface EH(Sr) is non-increasing. Eardley also considered a special spacelike hypersurface, filled by a family of 2-surfaces, for which EH(Sr) is non-decreasing. By relaxing the normalization condition l na = 1 a for the two null normals to lna = exp(f) a for some f : S --> R, 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 [182Jump To The Next Citation Point]. (For the more general GHP formalism in which lana is not fixed, see [312Jump To The Next Citation Point].) 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 S is a weakly future and past convex 2-surface, then qa := 2Qa/(Q Qb) = - [1/(2r)]la - [1/(2r')]na b is an outward directed spacelike normal to S. Here Qb is the trace of the extrinsic curvature tensor: a Qb := Q ab (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 St of 2-surfaces being Lie-dragged along its own inverse mean curvature vector qa. 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 a q on the 3-submanifold swept by St) exists, Frauendiener showed that the Hawking energy is non-decreasing along the vector field qa 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.

6.1.4 Two generalizations

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:

V~ --------- gf ( ) P a (S) = Area(S)--1- ss'+ ss' - y2 - y2'+ 2f11 + 2/\ W a dS, (39) H 16pG2 4p S
where W a, a = 0,...,3 --, are essentially the first four spherical harmonics:
0 1 z + z 2 1 z- z 3 1 - zz W = 1, W = ------, W = --------, W = -------. (40) 1 + zz i1 + zz 1 + zz
Here z and z are the standard complex stereographic coordinates on S ~~ S2.

Hawking considered the extension of the definition of EH(S) to higher genus 2-surfaces also by the second expression in Equation (38View Equation). Then in the expression analogous to the first one in Equation (38View Equation) the genus of S appears.

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