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

∘ --------- Area (𝒮) ( 1 ∮ ) EH (𝒮) := ------2- 1 + --- ρ ρ′d𝒮 = ∘ -16πG---- 2 π 𝒮 Area (𝒮) 1 ∮ ( ) = ------2---- σ σ′ + σ¯¯σ′ − ψ2 − ¯ψ2′ + 2ϕ11 + 2Λ d𝒮 (6.1 ) 16πG 4 π 𝒮
behaves as an appropriate notion of energy surrounded by the spacelike topological two-sphere 𝒮 [236Jump To The Next Citation Point]. Here we used the Gauss–Bonnet theorem and the GHP form of Eqs. (4.3View Equation) and (4.4View Equation) for F to express ′ ρρ 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, EH can be introduced in this way. Starting with the rough idea that the mass-energy surrounded by a spacelike two-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 ↦→ αl, a − 1 a n ↦→ α n the convergences ρ and ρ′ transform as ρ ↦→ α ρ and ρ′ ↦→ α −1ρ′, respectively, the energy must have the form ∮ C + D 𝒮 ρρ′d𝒮, where the unspecified parameters C and D can be determined in some special situations. For metric two-spheres of radius r in the Minkowski spacetime, for which ρ = − 1∕r and ′ ρ = 1∕2r, we expect zero energy, thus, D = C∕ (2 π). For the event horizon of a Schwarzschild black hole with mass parameter m, for which ′ ρ = 0 = ρ, we expect m∕G, which can be expressed by the area of 𝒮. Thus, C2 = Area (𝒮)∕(16πG2 ), and hence, we arrive at Eq. (6.1View Equation).

6.1.2 Hawking energy for spheres

Obviously, for round spheres, EH reduces to the standard expression (4.7View 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) ρ = 0, the Hawking energy of its spacelike spherical cross sections 𝒮 is ∘ ------------------ Area (𝒮)∕(16πG2 ). In particular, for the event horizon of a Kerr–Newman black hole it is just the familiar irreducible mass ∘ ---2---2------√---2----2----2 2m − e + 2m m − e − a ∕ (2G ). For more general surfaces Hawking energy is calculated numerically in [272].

For a small sphere of radius r with center p ∈ M in nonvacuum spacetimes it is 4π3 r3Tabtatb, while in vacuum it is -2-r5Tabcdtatbtctd 45G, where Tab is the energy-momentum tensor and Tabcd is the Bel–Robinson tensor at p [275Jump To The Next Citation Point]. 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 EH must be higher than r3. 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 r5, and its coefficient must be a quadratic expression of the curvature tensor. It is remarkable that for small spheres EH is positive definite both in nonvacuum (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 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. (6.1View 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 E H near null infinity both for radiative and stationary spacetimes is given in [455Jump To The Next Citation Point, 457Jump To The Next Citation Point]. Similarly, calculating EH 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, Hawking energy may be negative, even in Minkowski spacetime. Geometrically this should be clear, since for an appropriately general (e.g., concave) two-surface 𝒮, the integral ∮ ρρ′d𝒮 𝒮 could be less than − 2π. Indeed, in flat spacetime EH is proportional to ∮ ′ ′ 𝒮(σ σ + ¯σ ¯σ)d𝒮 by the Gauss equation. For topologically-spherical two-surfaces in the t = const. spacelike hyperplane of Minkowski spacetime σσ′ is real and nonpositive, and it is zero precisely for metric spheres, while for two-surfaces in the r = const. timelike cylinder σσ ′ is real and non-negative, and it is zero precisely for metric spheres.9 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 [143Jump To The Next Citation Point]: 𝒮 will be called round enough if it is a submanifold of a spacelike hypersurface Σ, and if among the two-dimensional surfaces in Σ, which enclose the same volume as 𝒮 does, 𝒮 has the smallest area. It is proven by Christodoulou and Yau [143] that if 𝒮 is round enough in a maximal spacelike slice Σ 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 EH [236]. These calculations were refined by Eardley [176]. Starting with a weakly future convex two-surface 𝒮 and using the boost gauge freedom, he introduced a special family 𝒮r of spacelike two-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 nondecreasing 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 nonincreasing. Eardley also considered a special spacelike hypersurface, filled by a family of two-surfaces, for which E (𝒮 ) H r is nondecreasing. By relaxing the normalization condition a lan = 1 for the two null normals to a lan = exp(f) 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 neighborhood of a mean convex two-surface by special mean convex two-surfaces [247Jump To The Next Citation Point]. (For the more general GHP formalism in which lana is not fixed, see [425Jump To The Next Citation Point].) 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 𝒮 is a weakly future and past convex two-surface, then a a b a ′ a q := 2Q ∕(QbQ ) = − [1∕(2ρ)]l − [1∕(2ρ )]n 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 two-surface, Frauendiener gives an argument for the construction of a one-parameter family 𝒮t of two-surfaces being Lie-dragged along its own inverse mean curvature vector qa. Assuming that such a family of surfaces (and hence, the vector field a q on the three-submanifold swept by 𝒮t) exists, Frauendiener showed that the Hawking energy is nondecreasing along the vector field qa 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 [565Jump To The Next Citation Point].

6.1.4 Two generalizations

UpdateJump To The Next Update Information

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:

∘ --------- a Area (𝒮 ) 1 ∮ ( ) PH-(𝒮) = ------2---- σ σ′ + ¯σ ¯σ′ − ψ2 − ψ¯2 ′ + 2ϕ11 + 2Λ W ad 𝒮, (6.2 ) 16πG 4π 𝒮
where W a, a-= 0,...,3, are essentially the first four spherical harmonics:
0 1 -ζ +-ζ¯ 2 1--ζ −-¯ζ- 3 1-−-ζ-¯ζ W = 1, W = 1 + ζ¯ζ, W = i1 + ζ¯ζ, W = 1 + ζ ¯ζ. (6.3 )
Here ζ and ¯ ζ are the standard complex stereographic coordinates on 2 𝒮 ≈ S.

Hawking considered the extension of the definition of EH (𝒮 ) to higher genus two-surfaces as well by the second expression in Eq. (6.1View Equation). Then, in the expression analogous to the first one in Eq. (6.1View Equation), the genus of 𝒮 appears. For recent generalizations of the Hawking energy for two-surfaces foliating the stationary and dynamical untrapped hypersurfaces, see [527Jump To The Next Citation Point, 528Jump To The Next Citation Point] and Section 11.3.4.


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