Proximity of black hole horizons: Lessons from Vaidya spacetime

In dynamical spacetimes apparent and event horizons do not coincide. In this paper we propose a geometrical measure of the distance between those horizons and investigate it for the case of the Vaidya spacetime. We show that it is well defined, physically meaningful, and has the expected behavior in the near-equilibrium limit. We consider its implications for our understanding of black hole physics.

Figure 1

A schematic showing the various horizons for a spherically symmetric spacetime with the angular dimensions suppressed. Horizontal location measures the areal radius of the associated spherical shell while the direction of increasing time is roughly vertical outside the event horizon but tipping horizontal-and-to-the-left inside. The shaded gray region is a shell of infalling null dust, gray dashed lines represent “outgoing” radial null rays, and the apparent horizon is that associated with a spherically symmetric foliation of the spacetime.

Figure 2

Event and apparent horizons for three transitions from $m_1$ to $m_2=2m_1$. For ease of visual comparison (particularly for $L=1$ and $1/100$), the vertical axis is shown in terms of $v/m_1$ rather than $V$. As in Fig. 1, the apparent horizon expansion is caused by the infalling matter, here represented by the shading (with density proportional to shading intensity). For small $L$ this also appears to be the case for event horizons. In contrast for large $L$, event horizon expansion is curtailed by the arrival of the matter shells.

Figure 3

A spacelike surface $\Sigma$ in Minkowski space along with some of its timelike geodesic normals and a conjugate point. The normal vectors are assumed to be past-oriented (as they are in our problem). By the theorem, the maximum proper time from $q_1$ to $\Sigma$ is measured along $\gamma$. However, $q_2$ is beyond the conjugate point $x$ and so the maximum proper time from $q_2$ to $\Sigma$ may or may not be along $\gamma$.

Figure 4

The geometric horizon and its timelike geodesic normals for $L=1$. The lower part of the figure is an enlargement of the boxed region in the upper part of the figure. Nearby geodesics intersect outside the event horizon, suggesting the existence of conjugate points there. The dashed part of the geometric horizon has $K_{(-u)}<0$ (past oriented-normal).

Figure 5

The timelike geodesics normal to geometric horizons for three representative values of the scale parameter $L$. Again the shading shows the concentration of infalling matter and the vertical axis is $v/m_1$ rather than $V$. To reduce clutter, the geodesics are cut off at the event horizon.

Figure 6

Proper time lengths of normal geodesics versus departure point from the geometric horizon for seven values of the scale parameter $L$. Starting from the left-most and sharpest peak and ending with the right-most and flattest peak they are $L={10}^{-3}$, $L={10}^{-2}$, $L={10}^{-1}$, $L=1$, $L={10}^1$ and $L={10}^2$, $L={10}^3$. For $V>V_{\mathrm{peak}}$ geodesics intersect the event horizon and their length between the horizons is show. For $V<V_{\mathrm{peak}}$ there is no intersection with the event horizon. Instead the total length of the geodesic from $V=-\infty$ to $V_f$ is plotted.

Figure 7

Two shells of dust fall into a black hole causing two distinct phases of expansion. As in the previous figure the top graph shows horizons and timelike normal geodesics while the bottom graph is of the proper time between horizons measured along those geodesics.