### 10.4 The expression of Liu and Yau

#### 10.4.1 The Liu-Yau definition

Let 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 into the flat 3-space, and this embedding is unique. Let be the trace of the extrinsic curvature of in this embedding, which is completely determined by and is necessarily positive. Let and be the trace of the extrinsic curvatures of 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 [253] by

However, this is precisely Kijowski’s ‘free energy’ given by Equation (79), , and hence we denote this by . Obviously, this is well-defined only if, in addition to the usual convexity condition for the intrinsic metric, also holds, i.e. the mean curvature vector is spacelike or null. If then , where the equality holds for corresponding to the quasi-local rest frame (in the sense that it is orthogonal to the mean curvature vector of the 2-surface: ).

Isolating the gauge invariant part of the connection 1-form Liu and Yau defined a quasi-local angular momentum as follows [253]. Let be the solution of the Poisson equation on , whose source is just the field strength of (see Equation (22)). This is globally well-defined on and is unique up to addition of a constant. Then define on the domain of the connection 1-form , which is easily seen to be closed. Assuming the space and time orientability of the spacetime, is globally defined on , and hence by the 1-form is exact: for some globally defined real function on . This function is unique up to an additive constant. Therefore, , where the first term is gauge invariant, while the second represents the gauge content of . Then for any rotation Killing vector of the flat 3-space Liu and Yau define the quasi-local angular momentum by

Here is the embedding and is the projection to the tangent planes of in . Thus, in contrast to the Brown-York definition for the angular momentum (see Equations (68, 69, 70, 71, 72)), in only the gauge invariant part of the gauge potential is used, and its generator vector field is the pull-back to of the rotation Killing vector of the flat 3-space.

#### 10.4.2 The main properties of

The most important property of the quasi-local energy definition (81) is its positivity. Namely [253], let be a compact spacelike hypersurface with smooth boundary , consisting of finitely many connected components , …, such that each of them has positive intrinsic curvature. Suppose that the matter fields satisfy the dominant energy condition on . Then is strictly positive unless the spacetime is flat along . In this case 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 is non-negative for all , and if for some then the spacetime is flat along and is connected. (In fact, since depends only on but is independent of the actual , if the energy condition is satisfied on the domain of dependence then implies the flatness of the spacetime along every Cauchy surface for , i.e. the flatness of the whole domain of dependence too.)

If is an apparent horizon, i.e. , then is just the integral of . Then by the Minkowski inequality for the convex surfaces in the flat 3-space (see for example [380]) one has

i.e. it is not less than twice the irreducible mass of the horizon. For round spheres coincides with , and hence it does not reduce to the standard round sphere expression (27). In particular, for the event horizon of the Schwarzschild black hole it is . Although the strict mathematical analysis is still lacking, 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, can be positive even if is in the Minkowski spacetime. In fact, for given intrinsic metric on (with positive scalar curvature) can be embedded into the flat ; this embedding is unique, and the trace of the extrinsic curvature is determined by . On the other hand, the isometric embedding of 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 and and the two components of the gauge potential . Thus, even if we impose a gauge condition for the connection 1-form , we have only six equations for the seven unknown quantities, leaving enough freedom to deform (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 .