Passage of dynamical horizons to equilibrium

4.3 Passage of dynamical horizons to equilibrium

In physical situations, such as a gravitational collapse or black hole mergers, one expects the dynamical horizon to approach equilibrium at late times and become isolated. Because of back scattering, generically the approach is only asymptotic. However, the back scattering is generally quite weak and in simulations, within numerical errors, equilibrium is reached at finite times. The passage to equilibrium can be studied in detail in Vaidya solutions discussed in Section 2.2.2. Moreover, in spherically symmetric examples such as these solutions, exact equilibrium can be reached at a finite time (see right panel of Figure 4

). The question then arises: In these situations, do various notions introduced on dynamical horizons go over smoothly to those introduced on WIHs? This issue has been analyzed only in a preliminary fashion [53 , 30

]. In this section we will summarize the known results.

First, if the dynamical horizon is a FOTH, as the flux of matter and shear across $H$ tends to zero, $H$ 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 $ℓa = ^τa + ^ra$ on $H$ diverges, while $na = τ^a − ^ra$ 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 $𝒬 = H ∪ Δ$ be a $Ck+1$ 3-manifold (with $k ≥ 2$ ), topologically $𝕊2 × ℝ$ as in the second Penrose diagram of Figure 4 . Let the space-time metric $g ab$ in a neighborhood of $𝒬$ be $k C$ . The part $H$ of $𝒬$ is assumed to have the structure of a DH and the part $Δ$ of a non-expanding horizon. Finally, the pull-back $qab$ of $gab$ to $𝒬$ is assumed to admit an axial Killing field $ϕa$ . Then we have:

The angular momentum $J ϕ$ and the mass $M$ defined in the two regimes agree on the boundary between $H$ and $Δ$ .
The vector field $ta0$ defined on $H$ and used in the definition of mass matches with a preferred vector field $ta0$ used to define mass on $Δ$ .

This agreement provides an independent support in favor of the strategy used to introduce the notion of mass in the two regimes.