6.2 The Geroch energy

6.2.1 The definition

Suppose that the two-surface 𝒮 for which EH is defined is embedded in the spacelike hypersurface Σ. Let χab be the extrinsic curvature of Σ in M and kab the extrinsic curvature of 𝒮 in Σ. (In Section 4.1.2 we denote the latter by νab.) Then ′ ab 2 ab 2 8 ρρ = (χabq ) − (kabq ), by means of which

∘ --------( ∮ ∮ ) Area-(𝒮-) --1- ( ab)2 -1-- ( ab)2 EH (𝒮) = 16πG2 1 − 16 π kabq d𝒮 + 16π χabq d𝒮 ≥ ∘ --------( ∮ 𝒮 ) 𝒮 Area-(𝒮-) --1- ( ab)2 ≥ 16πG2 1 − 16 π kabq d𝒮 = ∘ ---------∮ 𝒮 -1-- Area-(𝒮-) ( 𝒮 ( ab)2) = 16π 16πG2 2 R − kabq d𝒮 =: EG (𝒮 ). (6.4 ) 𝒮
In the last step we use the Gauss–Bonnet theorem for 𝒮 ≈ S2. EG (𝒮 ) is known as the Geroch energy [207Jump To The Next Citation Point]. Thus, it is not greater than the Hawking energy, and, in contrast to EH, it depends not only on the two-surface 𝒮, but on the hypersurface Σ as well.

The calculation of the small sphere limit of the Geroch energy was saved by observing [275] that, by Eq. (6.4View Equation), the difference of the Hawking and the Geroch energies is proportional to ∘ -------- Area (𝒮 )× ∮ 𝒮 (χabqab)2d𝒮. Since, however, χabqab – for the family of small spheres 𝒮r – does not tend to zero in the r → 0 limit, in general, this difference is 𝒪 (r3). It is zero if Σ is spanned by spacelike geodesics orthogonal to ta at p. Thus, for general Σ, the Geroch energy does not give the expected 4π 3 a b 3 rTabt t result. Similarly, in vacuum, the Geroch energy deviates from the Bel–Robinson energy in 5 r order even if Σ is geodesic at p.

Since E (𝒮 ) ≥ E (𝒮) H G and since the Hawking energy tends to the ADM energy, the large sphere limit of EG (𝒮) in an asymptotically flat Σ cannot be greater than the ADM energy. In fact, it is also precisely the ADM energy [207Jump To The Next Citation Point].

For a definition of Geroch’s energy as a quasi-local energy oparator in loop quantum gravity, see [565Jump To The Next Citation Point].

6.2.2 Monotonicity properties

The Geroch energy has interesting positivity and monotonicity properties along a special flow in Σ [207Jump To The Next Citation Point, 291Jump To The Next Citation Point]. This flow is the inverse mean curvature flow defined as follows. Let t : Σ → ℝ be a smooth function such that

  1. its level surfaces, 𝒮 := {q ∈ Σ | t(q) = t} t, are homeomorphic to S2,
  2. there is a point p ∈ Σ such that the surfaces 𝒮t are shrinking to p in the limit t → − ∞, and
  3. they form a foliation of Σ − {p}.

Let n be the lapse function of this foliation, i.e., if va is the outward directed unit normal to 𝒮t in Σ, then nvaDat = 1. Denoting the integral on the right-hand side in Eq. (6.4View Equation) by W t, we can calculate its derivative with respect to t. In general this derivative does not seem to have any remarkable properties. If, however, the foliation is chosen in a special way, namely if the lapse is just the inverse mean curvature of the foliation, n = 1∕k where k := kabqab, and furthermore Σ is maximal (i.e., χ = 0) and the energy density of the matter is non-negative, then, as shown by Geroch [207], Wt ≥ 0 holds. Jang and Wald [291] modified the foliation slightly, such that t ∈ [0,∞ ), and the surface 𝒮 0 was assumed to be future marginally trapped (i.e., ρ = 0 and ′ ρ ≥ 0). Then they showed that, under the conditions above, ∘ --------- ∘ --------- Area(𝒮0)W0 ≤ Area(𝒮t)Wt. Since EG (𝒮t) tends to the ADM energy as t → ∞, these considerations were intended to argue that the ADM energy should be non-negative (at least for maximal Σ) and not less than ∘ ------------------ Area(𝒮0)∕(16 πG2 ) (at least for time-symmetric Σ), respectively. Later Jang [289Jump To The Next Citation Point] showed that, if a certain quasi-linear elliptic differential equation for a function w on a hypersurface Σ admits a solution (with given asymptotic behavior), then w defines a mapping between the data set (Σ,hab,χab) on Σ and a maximal data set (Σ, ¯hab, ¯χab) (i.e., for which χ¯ab¯hab = 0) such that the corresponding ADM energies coincide. Then Jang shows that a slightly modified version of the Geroch energy is monotonic (and tends to the ADM energy) with respect to a new, modified version of the inverse mean curvature foliation of (Σ, ¯hab).

The existence and the properties of the original inverse-mean–curvature foliation of (Σ,h ) ab above were proven and clarified by Huisken and Ilmanen [278Jump To The Next Citation Point, 279Jump To The Next Citation Point], giving the first complete proof of the Riemannian Penrose inequality, and, as proven by Schoen and Yau [444], Jang’s quasi-linear elliptic equation admits a global solution.

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