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 [25, 14]). 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.
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