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6.2 The Geroch energy

6.2.1 The definition

Suppose that the 2-surface S for which EH is defined is embedded in the spacelike hypersurface S. Let xab be the extrinsic curvature of S in M and kab the extrinsic curvature of S in S. (In Section 4.1.2 we denoted the latter by nab.) Then ' ab 2 ab 2 8rr = (xabq ) - (kabq ), by means of which

V~ ---------( gf gf ) Area(S)- -1-- ( ab)2 -1-- ( ab)2 EH (S) = 16pG2 1 - 16p kabq dS + 16p xabq dS > V~ ---------( gf S ) S Area(S)- -1-- ( ab)2 > 16pG2 1 - 16p kabq dS = V~ -------- gf S --1- Area(S)- ( S ( ab)2) = 16p 16pG2 2 R - kabq dS =: EG (S) . (41) S
In the last step we used the Gauss-Bonnet theorem for S ~~ S2. EG(S) is known as the Geroch energy [150Jump 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 2-surface S, but the hypersurface S as well.

The calculation of the small sphere limit of the Geroch energy was saved by observing [204] that, by Equation (41View Equation), the difference of the Hawking and the Geroch energies is proportional to V~ -------- Area(S) ×gf S(xabqab)2dS. Since, however, xabqab - 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 S is spanned by spacelike geodesics orthogonal to ta at p. Thus, for general S, the Geroch energy does not give the expected 4p 3 a b 3 r Tabt t result. Similarly, in vacuum the Geroch energy deviates from the Bel-Robinson energy in r5 order even if S is geodesic at p.

Since EH(S) > EG(S) and since the Hawking energy tends to the ADM energy, the large sphere limit of EG(S) in an asymptotically flat S cannot be greater than the ADM energy. In fact, it is also precisely the ADM energy [150Jump To The Next Citation Point].

6.2.2 Monotonicity properties

The Geroch energy has interesting positivity and monotonicity properties along a special flow in S [150Jump To The Next Citation Point219Jump To The Next Citation Point]. This flow is the so-called inverse mean curvature flow defined as follows. Let t : S --> R be a smooth function such that

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

Let n be the lapse function of this foliation, i.e. if va is the outward directed unit normal to St in S, then nvaDat = 1. Denoting the integral on the right hand side in Equation (41View Equation) by Wt, we can calculate its derivative with respect to t. In general this derivative does not seem to have any remarkable property. 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 ab k := kabq, furthermore S is maximal (i.e. x = 0) and the energy density of the matter is non-negative, then, as shown by Geroch [150], Wt > 0 holds. Jang and Wald [219] modified the foliation slightly such that t (- [0, oo ), and the surface S0 was assumed to be future marginally trapped (i.e. r = 0 and r'> 0). Then they showed that, under the conditions above, V~ --------- V~ --------- Area(S0) W0 < Area(St) Wt. Since EG(St) tends to the ADM energy as t-- > oo, these considerations were intended to argue that the ADM energy should be non-negative (at least for maximal S) and not less than V~ ------------------ Area(S0)/(16pG2) (at least for time-symmetric S), respectively. Later Jang [217Jump To The Next Citation Point] showed that if a certain quasi-linear elliptic differential equation for a function w on a hypersurface S admits a solution (with given asymptotic behaviour), then w defines a mapping between the data set (S,hab,xab) on S and a maximal data set (S, hab,xab) (i.e. for which xabhab = 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 (S, hab).

The existence and the properties of the original inverse mean curvature foliation of (S, hab) above were proven and clarified by Huisken and Ilmanen [207Jump To The Next Citation Point208Jump To The Next Citation Point], giving the first complete proof of the Riemannian Penrose inequality, and, as proved by Schoen and Yau [328], Jang’s quasi-linear elliptic equation admits a global solution.

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