10.5 The expression of Wang and Yau

The new quasi-local energy (in fact, energy-momentum) expression of Wang and Yau [544Jump To The Next Citation Point] (and for a review, see also [540]) is based on the ‘renormalized’ form of the ‘natural’ Hamiltonian
∫ ∮ H [K ] = --1-- KaGab -1𝜀bcde − -1--- Ka (⊥𝜀abQccb + Aa) d𝒮. (10.20 ) 8πG Σ 3! 8πG ∂Σ
(See also Eq. (11.11View Equation), and compare with Eq. (8.1View Equation): apart from the SO (1,3) gauge-dependent terms, this boundary expression is just the two-surface integral of the Nester–Witten 2-form.) Thus, while the expressions based on Eq. (10.17View Equation) are analogous to Eq. (2.7View Equation), i.e., the two-surface integrals of locally-defined mass density, the expressions based on Eq. (10.20View Equation) are analogous to Eq. (2.5View Equation) (or rather Eq. (2.8View Equation)), i.e., the charge integrals ‘indexed’ by a vector field Ka.

Since Ae is boost-gauge dependent and Eq. (10.20View Equation) in itself does not yield, e.g., the correct ADM energy in asymptotically flat spacetime, a boost gauge and a restriction on the vector field a K and/or a ‘renormalization’ of Eq. (10.20View Equation) (in the form of an appropriate reference term) must be given. Wang and Yau suggest that one determine these by embedding the spacelike two-surface 𝒮 isometrically into the Minkowski spacetime in an appropriate way.

Thus, suppose that there is an isometric embedding 1,3 i : 𝒮 → ℝ, and let us fix a constant future-pointing unit timelike vector field T a in ℝ1,3. This Ta defines a global orthonormal frame field {0ta,0va} in the normal bundle of i(𝒮 ) ⊂ ℝ1,3 by requiring 0vaTa = 0, and let us denote the mean extrinsic curvature vector of this embedding by 0Qaab --. Then, supposing that the mean extrinsic curvature vector a Q ab of 𝒮 in the physical spacetime is spacelike, there is a uniquely-determined global orthonormal frame field a a { ¯t,¯v } in the normal bundle of 𝒮 ⊂ M such that Qaab ¯tb = 0Qaab0ta. This fixes the boost gauge in N 𝒮, and, in addition, makes it possible to identify the normal bundle of 𝒮 in M and the normal bundle of i(𝒮 ) in ℝ1,3 via the identification 0ta ↦→ t¯a, 0va ↦→ v¯a. This, together with the natural identification of the tangent bundle T 𝒮 of 𝒮 and the tangent bundle Ti(𝒮) of i(𝒮 ) yields a natural identification of the Lorentzian vector bundles over 𝒮 in M and over i(𝒮) in ℝ1,3. Therefore, any vector (and tensor) field on i(𝒮 ) yields a vector (tensor) field on 𝒮. In particular, if T a = N ta + N a 0 0 0, then a 0N is a tangent of i(𝒮 ), and hence, there is a uniquely determined tangent a 0N of 𝒮 such that 0N a = i∗(0N a). Consequently T a can be identified with the vector field 0N ¯ta + 0N a on 𝒮. Similarly, the connection one-form 0Aa- on the normal bundle (in the boost gauge {0ta,0va}) can be pulled back along i to a one-form 0Aa on 𝒮. Then, denoting by 0k and ¯ k the mean curvature of i(𝒮 ) and 𝒮 in the direction a 0v and a ¯v, respectively, Wang and Yau [544Jump To The Next Citation Point] define the quasi-local energy with respect to the pair a (i,T ) by

∮ ( a) -1--- (( ¯ ) ( ) e) EWY 𝒮; i,T := 8πG 0k − k 0N − 0Ae − ¯Ae 0N d 𝒮. (10.21 ) 𝒮
Here 𝒮 is assumed only to be isometrically embeddable into ℝ1,3 and that 𝒮 has spacelike mean curvature vector in M. Note that this energy still depends on the pair (i,Ta).

To prove, e.g., the positivity of this energy, or to ensure that in flat spacetime the energy be zero, further conditions must be satisfied. Wang and Yau formulate these conditions in the notion of admissible pairs a (i,T ): i(𝒮) should have a convex shadow in the direction a T-, i(𝒮) must be the boundary of some spacelike hypersurface in ℝ1,3 on which the Dirichlet boundary value problem for the Jang equation can be solved with the time function τ discussed in Section 4.1.3, and the connection 1-form and the mean curvature in a certain gauge must satisfy an inequality. (For the precise definition of the admissible pairs see [544Jump To The Next Citation Point]; for the geometrical background see [543] and Section 4.1.3.) Then it is shown that if the dominant energy condition holds and 𝒮 has a spacelike mean curvature vector, then for the admissible pairs the quasi-local energy (10.21View Equation) is non-negative. Therefore, if the set of the admissible pairs is not empty (e.g., when the scalar curvature of (𝒮, qab) is positive), then the infimum mWY (𝒮) of a EWY (𝒮; i,T ) among all admissible pairs is non-negative, and is called the quasi-local mass. If this infimum is achieved by the pair a (i,T ), i.e., by an embedding i and a timelike Ta, then P a := mWY (𝒮 )Ta is called the quasi-local energy-momentum, which is then future pointing and timelike. It is still an open question that if the quasi-local mass mWY (𝒮) is vanishing, then the domain of dependence D (Σ) of the spacelike hypersurface Σ with boundary 𝒮 can be curved (e.g., a pp-wave geometry with pure radiation) or not. If not, then the quasi-local energy-momentum would be expected to be null.

The quasi-local energy-momentum associated with any two-surface in Minkowski spacetime with a convex shadow in some direction is clearly zero. The mass has been calculated for round spheres in the Schwarzschild spacetime. It is ∘ ----------- r(1 − 1 − (2m ∕r))∕G, and hence, for the event horizon it gives 2m ∕G. mWY has been calculated for large spheres and it has the expected limits at the spatial and null infinities [545, 142]. Also, it has the correct small sphere limit both in nonvacuum and vacuum [544]. Upper and lower estimates of the Wang–Yau energy are derived, and its critical points are investigated in [362] and [363], respectively. On the applicability of EWY (𝒮) in the formulation and potential proof of Thorne’s hoop conjecture see Section 13.2.2. A recent review of the results in connection with the Wang–Yau energy see [541].

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