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10.4 The expression of Liu and Yau

10.4.1 The Liu-Yau definition

Let (S, q ) ab be a spacelike topological 2-sphere in spacetime such that the metric has positive scalar curvature. Then by the embedding theorem there is a unique isometric embedding of (S, qab) into the flat 3-space, and this embedding is unique. Let k0 be the trace of the extrinsic curvature of S in this embedding, which is completely determined by qab and is necessarily positive. Let k and l be the trace of the extrinsic curvatures of S in the physical spacetime corresponding to the outward pointing unit spacelike and future pointing timelike normals, respectively. Then Liu and Yau define their quasi-local energy in [253Jump To The Next Citation Point] by

gf ( ) E (S) := -1--- k0- V~ k2---l2- dS. (81) LY 8pG S
However, this is precisely Kijowski’s ‘free energy’ given by Equation (79View Equation), ELY(S) = FK(S), and hence we denote this by EKLY(S). Obviously, this is well-defined only if, in addition to the usual convexity condition R > 0 for the intrinsic metric, k2 > l2 also holds, i.e. the mean curvature vector Qa is spacelike or null. If k > 0 then EKLY(S) > EBY(S, ta), where the equality holds for ta corresponding to the quasi-local rest frame (in the sense that it is orthogonal to the mean curvature vector of the 2-surface: a tQa = 0).

Isolating the gauge invariant part of the SO(1, 1) connection 1-form Liu and Yau defined a quasi-local angular momentum as follows [253Jump To The Next Citation Point]. Let a be the solution of the Poisson equation 2qabdadba = Im(f ) on S, whose source is just the field strength of Aa (see Equation (22View Equation)). This a is globally well-defined on S and is unique up to addition of a constant. Then define b ga := Aa - ea dba on the domain of the connection 1-form Aa, which is easily seen to be closed. Assuming the space and time orientability of the spacetime, A a is globally defined on S ~~ S2, and hence by H1(S2) = 0 the 1-form ga is exact: ga = dag for some globally defined real function g on S. This function is unique up to an additive constant. Therefore, Aa = eabdba + dag, where the first term is gauge invariant, while the second represents the gauge content of Aa. Then for any rotation Killing vector K0i of the flat 3-space Liu and Yau define the quasi-local angular momentum by

gf J (S, K0i):= --1-- f- 1(K0iTT0a)e b (d a) dS. (82) LY 8pG S * i a b
Here 3 f : S --> R is the embedding and 0a TT i is the projection to the tangent planes of f(S) in 3 R. Thus, in contrast to the Brown-York definition for the angular momentum (see Equations (68View Equation, 69View Equation, 70View Equation, 71View Equation, 72View Equation)), in JLY(S, K0i) only the gauge invariant part daa of the gauge potential Aa is used, and its generator vector field is the pull-back to S of the rotation Killing vector of the flat 3-space.

10.4.2 The main properties of EKLY(S)

UpdateJump To The Next Update Information The most important property of the quasi-local energy definition (81View Equation) is its positivity. Namely [253], let S be a compact spacelike hypersurface with smooth boundary @S, consisting of finitely many connected components S1, …, Sk such that each of them has positive intrinsic curvature. Suppose that the matter fields satisfy the dominant energy condition on S. Then EKLY(@S) := sum k EKLY(Si) i=1 is strictly positive unless the spacetime is flat along S. In this case @S is necessarily connected. The proof is based on the use of Jang’s equation [217], by means of which the general case can be reduced to the results of Shi and Tam in the time-symmetric case [341], stated in Section 10.1.7 (see also [399]). This positivity result is generalized in [254]: It is shown that EKLY(Si) is non-negative for all i = 1,...,k, and if EKLY(Si) = 0 for some i then the spacetime is flat along S and @S is connected. (In fact, since EKLY(@S) depends only on @S but is independent of the actual S, if the energy condition is satisfied on the domain of dependence D(S) then EKLY(@S) = 0 implies the flatness of the spacetime along every Cauchy surface for D(S), i.e. the flatness of the whole domain of dependence too.)

If S is an apparent horizon, i.e. l = ± k, then EKLY(S) is just the integral of k0/(8pG). Then by the Minkowski inequality for the convex surfaces in the flat 3-space (see for example [380]) one has

V ~ --------- 1 gf 1 V~ ------------ Area(S) EKLY (S) = ----- k0 dS > ----- 16p Area(S) = 2 ------2-, 8pG S 8pG 16pG

i.e. it is not less than twice the irreducible mass of the horizon. For round spheres EKLY(S) coincides with EE(S), and hence it does not reduce to the standard round sphere expression (27View Equation). In particular, for the event horizon of the Schwarzschild black hole it is 2m/G. Although the strict mathematical analysis is still lacking, EKLY probably reproduces the correct large sphere limits in asymptotically flat spacetime (ADM and Bondi-Sachs energies), because the difference between the Brown-York, Epp, and Kijowski-Liu-Yau definitions disappear asymptotically.

However, EKLY(S) can be positive even if S is in the Minkowski spacetime. In fact, for given intrinsic metric qab on S (with positive scalar curvature) S can be embedded into the flat 3 R; this embedding is unique, and the trace of the extrinsic curvature 0 k is determined by qab. On the other hand, the isometric embedding of S in the Minkowski spacetime is not unique: The equations of the embedding (i.e. the Gauss, the Codazzi-Mainardi, and the Ricci equations) form a system of six equations for the six components of the two extrinsic curvatures k ab and l ab and the two components of the SO(1, 1) gauge potential Ae. Thus, even if we impose a gauge condition for the connection 1-form Ae, we have only six equations for the seven unknown quantities, leaving enough freedom to deform S (with given, fixed intrinsic metric) in the Minkowski spacetime to get positive Kijowski-Liu-Yau energy. Indeed, specific 2-surfaces in the Minkowski spacetime are given in [292] for which E (S) > 0 KLY.

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