### 3.1 Preliminaries

The first law of black hole mechanics (1) tells us how the area of the black hole increases when it makes
a transition from an initial equilibrium state to a nearby equilibrium state. The question we want to
address is: Can one obtain an integral generalization to incorporate fully dynamical situations?
Attractive as this possibility seems, one immediately encounters a serious conceptual and technical
problem. For, the generalization requires, in particular, a precise notion of the flux of gravitational
energy across the horizon. Already at null infinity, the expression of the gravitational energy
flux is subtle: One needs the framework developed by Bondi, Sachs, Newman, Penrose, and
others to introduce a viable, gauge invariant expression of this flux [52, 32, 185]. In the strong
field regime, there is no satisfactory generalization of this framework and, beyond perturbation
theory, no viable, gauge invariant notion of the flux of gravitational energy across a general
surface.
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.