Since is boost-gauge dependent and Eq. (10.20) 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 and/or a ‘renormalization’ of Eq. (10.20) (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 isometrically into the Minkowski spacetime in an appropriate way.

Thus, suppose that there is an isometric embedding , and let us fix a constant future-pointing unit timelike vector field in . This defines a global orthonormal frame field in the normal bundle of by requiring , and let us denote the mean extrinsic curvature vector of this embedding by . Then, supposing that the mean extrinsic curvature vector of in the physical spacetime is spacelike, there is a uniquely-determined global orthonormal frame field in the normal bundle of such that . This fixes the boost gauge in , and, in addition, makes it possible to identify the normal bundle of in and the normal bundle of in via the identification , . This, together with the natural identification of the tangent bundle of and the tangent bundle of yields a natural identification of the Lorentzian vector bundles over in and over in . Therefore, any vector (and tensor) field on yields a vector (tensor) field on . In particular, if , then is a tangent of , and hence, there is a uniquely determined tangent of such that . Consequently can be identified with the vector field on . Similarly, the connection one-form on the normal bundle (in the boost gauge ) can be pulled back along to a one-form on . Then, denoting by and the mean curvature of and in the direction and , respectively, Wang and Yau [544] define the quasi-local energy with respect to the pair by

Here is assumed only to be isometrically embeddable into and that has spacelike mean curvature vector in . Note that this energy still depends on the pair .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 : should have a convex shadow in the direction , must be the boundary of some spacelike hypersurface in on which the Dirichlet boundary value problem for the Jang equation can be solved with the time function 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 [544]; for the geometrical background see [543] and Section 4.1.3.) Then it is shown that if the dominant energy condition holds and has a spacelike mean curvature vector, then for the admissible pairs the quasi-local energy (10.21) is non-negative. Therefore, if the set of the admissible pairs is not empty (e.g., when the scalar curvature of is positive), then the infimum of among all admissible pairs is non-negative, and is called the quasi-local mass. If this infimum is achieved by the pair , i.e., by an embedding and a timelike , then 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 is vanishing, then the domain of dependence of the spacelike hypersurface with boundary 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 , and hence, for the event horizon it gives . 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 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].

Living Rev. Relativity 12, (2009), 4
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