Formation of a Black Hole

      The eternal black hole described by the static Schwarzschild geometry is an idealization. In nature, black holes are formed from the collapse of gravitating matter. The simpest model for black hole formation involves a collapsing thin spherical shell of massless matter. For example, a shell of photons, gravitons, or massless neutrinos with very small radial extension and total energy M provides an example.

       To construct the geometry, we begin with the empty space Penrose diagram with the infalling shell represented by an incoming light-like line

Fig. 1.1. Penrose diagram for Schwarzschild black hole, showing regions (top) and  curves of fixed radial position and constant time (bottom).

(see Figure 1.2). The particular value of Y ⁺ chosen for the trajectory is arbitrary since any two such values are related by a time translation. The infalling shell divides the Penrose diagram into two regions, A and B .The Region A is interior to the shell and represents the initial flat space-time

Fig. 1.2. Minkowski space Penrose diagram for radially infalling spherical shell of massless particles with energy M.

before the shell passes. Region B is the region outside the shell and must be modified in order to account for the gravitational field due to the mass M.

        In Newtonian physics the gravitational field exterior to a spherical mass distribution is uniquely that of a point mass located at the center of the distribution. Much the same is true in general relativity. In this case Birkoff's theorem tells us that the geometry outside the shell must be the Schwarzschild geometry. Accordingly , we consider the Penrose diagram for a black hole of mass M divided into regions A’ and B’ by an infalling massless shell as in Figure 1.3. Once again the particular value of Y + chosen for the trajectory is immaterial. Just as in Figure 1.8 where the Region B is unphysical, in Figure 1.3 the Region A’ is to be discarded. To form the full classical evolution the regions A of Figure 1.2 and B’ of Figure 1.3 must be glued together. However this must be done so that the “radius” of the local two sphere represented by the angular coordinates (θ, φ) is continuous. In other words, the mathematical identification of the boundaries of A and B’ must respect the continuity of the variable r.

Fig. 1.3. Penrose diagram of Schwarzschild black hole with radially infalling shell of massless particle with energy M.

Since in both cases r varies monotonically from r = ∞ at I− to r = 0, the identification is always possible. One of the two Penrose diagrams will have to undergo a deformation along the Y ^-- direction in order to make the identification smoothly, but this will not disturb the form of the light cones. Thus in Figure 1.4 we show the resulting Penrose diagram for the complete geometry. On Fig 1.4, a light-like surface H is shown as a dotted line. It is clear that any light ray or time like trajectory that originates to the upper left of H must end at the singularity and cannot escape to I⁺ (or t = ∞).This identifies H as the horizon. In Region B’ the horizon is identical to the surface H+ of Figure 1.7, that is it coincides with the future horizon of the final black hole geometry and is therefore found at r = 2MG. On the other hand, the horizon also extends into the Region A where the metric is just that of flat space-time. In this region the value of r on the horizon grows from an initial value r = 0 to the value r = 2MG at the shell.

      It is evident from this discussion that the horizon is a global and not a local concept. In the Region A no local quantity will distinguish the presence of the horizon whose occurence is due entirely to the future collapse of the shell.

      Consider next a distant observer located on a trajectory with r >> 2MG.The observer originates at past time-like infinity and eventually ends at future time-like infinity, as shown in Figure 1.11. The distant observer collects information that arrives at any instant from his backward light cone. Evidently such an observer never actually sees events on the

Fig. 1.4. Penrose diagram for collapsing shell of massless particles.

Fig. 1.5. Distant observer to collapsing spherical shell.

horizon. In this sense the horizon must be regarded as at the end of time. Any particle or wave which falls through the horizon is seen by the distant observer as asymptotically approaching the horizon as it is infinitely red shifted. At least that is the case classically.

     This basic description of black hole formation is much more general than might be guessed. It applies with very little modification to the collapse of all kinds of massive matter as well as to non-spherical distributions. In all cases the horizon is a light like surface which separates the space-time into an inner and an outer region .An y light ray which originates in the inner region can never reach future asymptotic infinity, or for that matter ever reach any point of the outer region. The events in the outer region can send light rays to I⁺ and time-like trajectories to t = ∞. The horizon, as we have seen, is a global concept whose location depends on all future events. I t is composed of a family of light rays or null geodesics, passing through each space-time point on the horizon. This is shown in Figure 1.6. Notice that null geodesics are vertical after the shell crosses the horizon and essentially at 45o prior to that crossing. These light rays are called the generators of the horizon.

Fig. 1.6. The horizon as family of null geodesics.