Yet, there are at least two general considerations that suggest that something special may happen on DHs. Consider a stellar collapse leading to the formation of a black hole. At the end of the process, one has a black hole and, from general physical considerations, one expects that the energy in the final black hole should equal the total matter plus gravitational energy that fell across the horizon. Thus, at least the total integrated flux across the horizon should be well defined. Indeed, it should equal the depletion of the energy in the asymptotic region, i.e., the difference between the ADM energy and the energy radiated across future null infinity. The second consideration involves the Penrose inequality  introduced in Section 1. Heuristically, the inequality leads us to think of the radius of a marginally trapped surface as a measure of the mass in its interior, whence one is led to conclude that the change in the area is due to influx of energy. Since a DH is foliated by marginally trapped surfaces, it is tempting to hope that something special may happen, enabling one to define the flux of energy and angular momentum across it. This hope is borne out.
In the discussion of DHs (Sections 3 and 4.2) we will use the following conventions (see Figure 5). The DH is denoted by and marginally trapped surfaces that foliate it are referred to as cross-sections. The unit, time-like normal to is denoted by with . The intrinsic metric and the extrinsic curvature of are denoted by and , respectively. is the derivative operator on compatible with , its Ricci tensor, and its scalar curvature. The unit space-like vector orthogonal to and tangent to is denoted by . Quantities intrinsic to are generally written with a tilde. Thus, the two-metric on is and the extrinsic curvature of is ; the derivative operator on is and its Ricci tensor is . Finally, we fix the rescaling freedom in the choice of null normals to cross-sections via and (so that ). To keep the discussion reasonably focused, we will not consider gauge fields with non-zero charges on the horizon. Inclusion of these fields is not difficult but introduces a number of subtleties and complications which are irrelevant for numerical relativity and astrophysics.
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