First, if the dynamical horizon is a FOTH, as the
flux of matter and shear across tends to zero,
becomes null and furthermore a non-expanding
horizon. By a suitable choice of null normals, it can be made
weakly isolated. Conditions under which it would also become an
isolated horizon are not well-understood. Fortunately, however, the
final expressions of angular momentum and horizon mass refer only
to that structure which is already available on non-expanding
horizons (although, as we saw in Section 4.1, the underlying Hamiltonian framework
does require the horizon to be weakly isolated [26
, 15]). Therefore, it is
meaningful to ask if the angular momentum and mass defined on the
DHs match with those defined on the non-expanding horizons. In the
case when the approach to equilibrium is only asymptotic, it is
rather straightforward to show that the answer is in the
affirmative.
In the case when the transition occurs at a
finite time, the situation is somewhat subtle. First, we now have
to deal with both regimes and the structures available in the two
regimes are entirely different. Second, since the intrinsic metric
becomes degenerate in the transition from the dynamical to isolated
regimes, limits are rather delicate. In particular, the null vector
field on
diverges, while
tends to zero at the boundary. A priori therefore,
it is not at all clear that angular momentum and mass would join
smoothly if the transition occurs at a finite time. However, a
detailed analysis shows that the two sets of notions in fact
agree.
More precisely, one has the following results.
Let be a
3-manifold (with
), topologically
as in the
second Penrose diagram of Figure 4
. Let the space-time
metric
in a neighborhood of
be
. The part
of
is assumed to have the structure of a DH and the
part
of a non-expanding horizon. Finally, the pull-back
of
to
is assumed to admit an axial
Killing field
. Then we have:
This agreement provides an independent support in favor of the strategy used to introduce the notion of mass in the two regimes.