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 [157] 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.