10.4 The expression of Liu and Yau

10.4.1 The Liu–Yau definition

Let (𝒮, q ) ab 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 (𝒮, qab) into the flat three-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 [338Jump To The Next Citation Point] by

∮ ( ) E (𝒮) := --1-- k0 − √k2--−-l2 d𝒮. (10.17 ) LY 8πG 𝒮
However, this is precisely Kijowski’s ‘free energy’ given by Eq. (10.15View 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 two-surface: a t Qa = 0). The mean curvature mass of [11Jump To The Next Citation Point, 12Jump To The Next Citation Point] is precisely ELY (𝒮) (see also Section 11.3.4).

Isolating the gauge invariant part of the SO (1,1) connection one-form, Liu and Yau defined a quasi-local angular momentum as follows [338Jump To The Next Citation Point]. Let α be the solution of the Poisson equation ab 2q δaδbα = Im (f) on 𝒮, whose source is just the field strength of Aa (see Eq. (4.3View Equation)). This α is globally well defined on 𝒮 and is unique up to addition of a constant. Then, define γa := Aa − 𝜀abδbα on the domain of the connection one-form A a, which is easily seen to be closed. Assuming the space and time orientability of the spacetime, Aa is globally defined on 2 𝒮 ≈ S, and hence, by 1 2 H (S ) = 0 the one-form γa is exact: γa = δaγ for some globally defined real function γ on 𝒮. This function is unique up to an additive constant. Therefore, Aa = 𝜀abδbα + δaγ, where the first term is gauge invariant, while the second represents the gauge content of Aa. Then for any rotation Killing vector 0i K of the flat three-space Liu and Yau define the quasi-local angular momentum by

∮ ( 0i) --1-- −1( 0i 0a) b JLY 𝒮, K := 8 πG φ ∗ K Πi 𝜀a (δbα )d𝒮. (10.18 ) 𝒮
Here φ : 𝒮 → ℝ3 is the embedding and Π0ai is the projection to the tangent planes of φ (𝒮 ) in ℝ3. Thus, in contrast to the Brown–York definition for the angular momentum (see Eqs. (10.4View Equation), (10.5View Equation), (10.6View Equation), (10.7View Equation), and (10.8View Equation)), in JLY (𝒮, K0i ) only the gauge invariant part 𝜀abδbα of the gauge potential Aa is used, and its generator vector field is the pullback to 𝒮 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].

10.4.2 The main properties of EKLY (𝒮 )

The most important property of the quasi-local energy (10.17View Equation) is its positivity. Namely [338Jump To The Next Citation Point], let Σ be a compact spacelike hypersurface with smooth boundary ∂Σ, 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 Σ. Then : ∑k EKLY (∂Σ ) = i=1 EKLY (𝒮i) is strictly positive unless the spacetime is flat along Σ. In this case ∂Σ 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, EKLY (𝒮i) is shown to be non-negative for all i = 1,...,k, and if EKLY (𝒮i) = 0 for some i, then the spacetime is flat along Σ and ∂Σ is connected. (In fact, since EKLY (∂ Σ) depends only on ∂Σ but is independent of the actual Σ, if the energy condition is satisfied on the domain of dependence D (Σ ), then E (∂Σ) = 0 KLY implies the flatness of the spacetime along every Cauchy surface for D (Σ ), i.e., the flatness of the whole domain of dependence as well.) A potential spinorial proof of the positivity of EKLY (𝒮i) 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 Σ would have to satisfy a nonlinear boundary condition.

If 𝒮 is an apparent horizon, i.e., l = ±k, then EKLY (𝒮 ) is just the integral of 0 k ∕(8πG ). Then, by the Minkowski inequality for the convex surfaces in the flat three-space (see, e.g., [519]) one has

∮ ∘ --------- 1 0 1 ∘ ------------ Area(𝒮 ) EKLY (𝒮) = 8πG-- k d𝒮 ≥ 8πG-- 16π Area (𝒮) = 2 -16πG2--, 𝒮

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 (4.8View Equation). In particular, for the event horizon of the Schwarzschild black hole it is 2m ∕G. (For a more detailed discussion, and, in particular, the interpretation of E (𝒮) KLY in the spherically-symmetric context, see [400Jump To The Next Citation Point].) E (𝒮) KLY was calculated for small spheres both in nonvacuum and vacuum, and for large spheres near the future null infinity in [575Jump To The Next Citation Point]. In the leading order in nonvacuum we get the expected result 4π 3 a b 3-r Tabtt (see Eq. (4.9View Equation)), but in vacuum, in addition to the expected Bel–Robinson ‘energy’, there are extra terms in the leading r5 order. As could be expected, at null infinity EKLY (𝒮 ) reproduces the Bondi energy.

However, EKLY (𝒮 ) can be positive even if 𝒮 is in the Minkowski spacetime. In fact, for a given intrinsic metric qab on 𝒮 (with positive scalar curvature) 𝒮 can be embedded into the flat ℝ3; this embedding is unique, and the trace of the extrinsic curvature k0 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, Codazzi–Mainardi, and Ricci equations) form a system of six equations for the six components of the two extrinsic curvatures kab and lab and the two components of the SO (1, 1) gauge potential Ae. Thus, even if we impose a gauge condition for the connection one-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 two-surfaces in the Minkowski spacetime are given in [401], for which EKLY (𝒮 ) > 0. Moreover, it is shown in [361] that the Kijowski–Liu–Yau energy for a closed two-surface 𝒮 in Minkowski spacetime strictly positive unless 𝒮 lies in a spacelike hyperplane. On the applicability of EKLY (𝒮) in the formulation and potential proof of Thorne’s hoop conjecture see Section 13.2.2.

10.4.3 Generalizations of the original construction

In the definition of EKLY (𝒮 ) one of the assumptions is the positivity of the scalar curvature of the intrinsic metric on the two-surface 𝒮. 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 E (𝒮 ) KLY 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. (2.7View Equation)). 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 EKLY (𝒮 ) in asymptotically anti-de Sitter spacetimes it seems natural to choose the reference configuration by embedding 𝒮 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., R > − 2κ2, can be isometrically embedded in a unique way into the hyperbolic space ℍ3 2 −κ with constant sectional curvature − κ2, and hence, this embedding can be (and in fact is) used to define the reference configuration. Let 0 k denote the mean curvature of 𝒮 in this embedding, where the hyperbolic space 3 ℍ −κ2 is thought of as a spacelike hypersurface with constant negative curvature in the Minkowski spacetime ℝ1,3. Then the main result is that, assuming that the mean curvature vector Qaab of 𝒮 in the spacetime is spacelike, there exists a function W a : 𝒮 → ℝ1,3, depending only on the length ------- |Qaab| = √ k2 − l2 of the mean curvature vector and the embedding of 𝒮 into 1,3 ℝ, such that the four integrals

∫ ( 0 √ --2---2) a k − k − l W d𝒮 (10.19 ) 𝒮
form a future-pointing nonspacelike vector in ℝ1,3. The functions W a, a-= 0,...,3, are solutions of a parabolic equation and are related to the norm of the Killing spinors on ℍ3−κ2. If κ → 0 then W a tend to the components of a constant vector field. Expression (10.19View Equation) can be interpreted as a comparison theorem for the total mean curvature of 𝒮 in the physical spacetime and in the hyperboloid ℍ3 2 ⊂ ℝ1,3 −κ. A similar result is proven in the Riemannian case, i.e., when 𝒮 is considered to be the boundary of a compact Riemannian three-manifold (Σ, h ) ab, and in (10.19View Equation) the length of the mean curvature vector is replaced by the mean curvature k of 𝒮 in Σ. Comparing (10.19View Equation) with the expression of the Bondi–Sachs energy-momentum (4.14View Equation) or with Eq. (6.2View Equation), the integrals can also be interpreted as the components of a quasi-local energy-momentum four-vector.

The proof of the nonspacelike nature of (10.19View Equation) is based on a Witten type argumentation, in which ‘the mass with respect to a Dirac spinor ϕ0 on 𝒮’ takes the form of an integral of 0 √ --2---2 (k − k − l ) weighted by the norm of ϕ0. Thus, the norm of ϕ0 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 𝒮 be the boundary of some spacelike hypersurface Σ on which the intrinsic scalar curvature is positive, let us isometrically embed 𝒮 into the Euclidean three-space, and let ϕ0 be the pull back to 𝒮 of a constant spinor field. Suppose that the dominant energy condition is satisfied on Σ, and consider the solution ϕ of the Witten equation on Σ with one of the chiral boundary conditions Π ±(ϕ − ϕ0 ) = 0, where Π± are the projections to the space of the right/left handed Dirac spinors, built from the projections π±A B 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 𝒮. In contrast to Brown–York type expressions, this mass, associated with the two-spheres of radius r in the t = const. hypersurfaces in Schwarzschild spacetime, is an increasing function of the radial coordinate, and tends to the ADM mass. In general, however, this limit is E − |P | ADM ADM, rather than the expected ADM mass. This construction is generalized in [581] by embedding 𝒮 into some 3 ℍ− κ2 instead of 3 ℝ. A modified version of these constructions is given in [582], which tends to the ADM energy and mass at spatial infinity. UpdateJump To The Next Update Information

Suggestion (11.12View Equation), due to Anco [11Jump To The Next Citation Point], can also be considered as a generalization of the Kijowski–Liu–Yau mass.

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