Let be a spacelike topological two-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 three-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 [338] by

However, this is precisely Kijowski’s ‘free energy’ given by Eq. (10.15), , 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 two-surface: ). The mean curvature mass of [11, 12] is precisely (see also Section 11.3.4).Isolating the gauge invariant part of the connection one-form, Liu and Yau defined a quasi-local angular momentum as follows [338]. Let be the solution of the Poisson equation on , whose source is just the field strength of (see Eq. (4.3)). This is globally well defined on and is unique up to addition of a constant. Then, define on the domain of the connection one-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 one-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 three-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 Eqs. (10.4), (10.5), (10.6), (10.7), and (10.8)), in only the gauge invariant part of the gauge potential is used, and its generator vector field is the pullback to of the rotation Killing vector of the flat three-space.For a definition of the Kijowski–Liu–Yau energy as a quasi-local energy oparator in loop quantum gravity, see [565].

The most important property of the quasi-local energy (10.17) is its positivity. Namely [338], 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 [289], by means of which the general case can be reduced to the results of Shi and Tam in the time-symmetric case [458], stated in Section 10.1.7 (see also [566]). This positivity result is generalized in [339], Namely, is shown to be 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 as well.) A potential spinorial proof of the positivity of is suggested in [12]. This is based on the use of the Nester–Witten 2–form and a Witten type argumentation. However, the spinor field solving the Witten equation on the spacelike hypersurface would have to satisfy a nonlinear boundary condition.

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 three-space (see, e.g., [519]) 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 (4.8). In particular, for the event horizon of the Schwarzschild black hole it is . (For a more detailed discussion, and, in particular, the interpretation of in the spherically-symmetric context, see [400].) was calculated for small spheres both in nonvacuum and vacuum, and for large spheres near the future null infinity in [575]. In the leading order in nonvacuum we get the expected result (see Eq. (4.9)), but in vacuum, in addition to the expected Bel–Robinson ‘energy’, there are extra terms in the leading order. As could be expected, at null infinity reproduces the Bondi energy.

However, can be positive even if is in the Minkowski spacetime. In fact, for a 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, Codazzi–Mainardi, and 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 one-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 two-surfaces in the Minkowski spacetime are given in [401], for which . Moreover, it is shown in [361] that the Kijowski–Liu–Yau energy for a closed two-surface in Minkowski spacetime strictly positive unless lies in a spacelike hyperplane. On the applicability of in the formulation and potential proof of Thorne’s hoop conjecture see Section 13.2.2.

In the definition of one of the assumptions is the positivity of the scalar curvature of the intrinsic metric on the two-surface . Thus, it is natural to ask if this condition can be relaxed and whether or not the quasi-local mass can be associated with a wider class of surfaces. Moreover, though in certain circumstances behaves as energy (see [400, 575]), it is the (renormalized) integral of the length of the mean curvature vector, i.e., it is analogous to mass (compare with Eq. (2.7)). Hence, it is natural to ask if a energy-momentum four-vector can be introduced in this way. In addition, in the calculation of the large sphere limit of in asymptotically anti-de Sitter spacetimes it seems natural to choose the reference configuration by embedding into a hyperbolic rather than Euclidean three-space. These issues motivate the following generalization [542] of the Kijowski–Liu–Yau expression.

One of the key ideas is that two-surfaces with spherical topology and scalar curvature that are bounded from below by a negative constant, i.e., , can be isometrically embedded in a unique way into the hyperbolic space with constant sectional curvature , and hence, this embedding can be (and in fact is) used to define the reference configuration. Let denote the mean curvature of in this embedding, where the hyperbolic space is thought of as a spacelike hypersurface with constant negative curvature in the Minkowski spacetime . Then the main result is that, assuming that the mean curvature vector of in the spacetime is spacelike, there exists a function , depending only on the length of the mean curvature vector and the embedding of into , such that the four integrals

form a future-pointing nonspacelike vector in . The functions , , are solutions of a parabolic equation and are related to the norm of the Killing spinors on . If then tend to the components of a constant vector field. Expression (10.19) can be interpreted as a comparison theorem for the total mean curvature of in the physical spacetime and in the hyperboloid . A similar result is proven in the Riemannian case, i.e., when is considered to be the boundary of a compact Riemannian three-manifold , and in (10.19) the length of the mean curvature vector is replaced by the mean curvature of in . Comparing (10.19) with the expression of the Bondi–Sachs energy-momentum (4.14) or with Eq. (6.2), the integrals can also be interpreted as the components of a quasi-local energy-momentum four-vector.The proof of the nonspacelike nature of (10.19) is based on a Witten type argumentation, in which ‘the mass with respect to a Dirac spinor on ’ takes the form of an integral of weighted by the norm of . Thus, the norm of appears to be a nontrivial lapse function. The suggestion of [580] for a quasi-local mass-like quantity is based on an analogous expression. Let be the boundary of some spacelike hypersurface on which the intrinsic scalar curvature is positive, let us isometrically embed into the Euclidean three-space, and let be the pull back to of a constant spinor field. Suppose that the dominant energy condition is satisfied on , and consider the solution of the Witten equation on with one of the chiral boundary conditions , where are the projections to the space of the right/left handed Dirac spinors, built from the projections of Section 4.1.7. Then, by the Sen–Witten identity, a positive definite boundary expression is introduced, and interpreted as the ‘quasi-local mass’ associated with . In contrast to Brown–York type expressions, this mass, associated with the two-spheres of radius in the hypersurfaces in Schwarzschild spacetime, is an increasing function of the radial coordinate, and tends to the ADM mass. In general, however, this limit is , rather than the expected ADM mass. This construction is generalized in [581] by embedding into some instead of . A modified version of these constructions is given in [582], which tends to the ADM energy and mass at spatial infinity. Update

Suggestion (11.12), due to Anco [11], can also be considered as a generalization of the Kijowski–Liu–Yau mass.

Living Rev. Relativity 12, (2009), 4
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