Inside
of the Horizon of a Black Hole 
What if you cross the horizon? (continuing Schwarzschild Geometry, and Approaching the Horizon)
As we have noticed, Schwarzschild Geometry was not good for treating motions across the horizon, and within the horizon. For treating such problems, we have to devise a new frame (in a standard textbook, Eddington-Finkelstein coordinates, or Kruskal-Szekeres coordinates are used). Taylor and Wheeler strategy is as follows. Let us define the rain-frame, for the test particle coming from infinity (initial velocity is 0; if this condition is changed, some of the following results do not hold) and (along a radial direction) falling into the black hole, like rain drops on the earth.
(1) First, let us notice the relationship between the r-coordinate distance and the same distance measured by the free-float system of the particle (raindrop, and diver are other names). Suppose the observer on the particle measures the distance dr_rain between adjacent shells (i.e., the particle is still outside of the horizon). Remember that this distance is, according to Schwarzschild metric, longer than the coordinate distance; i.e.,
dr_shell = dr/√(1 - 2M/r).
However, since the Lorentz contraction holds on the particle (local inertial system), this distance, measured by the observer moving with velocity v (remember we have made the velocity of light 1, so v is a fraction of unity) relative to the shells, must be multiplied by the factor √(1 - vv), and recall that, according to Schwarzschild metric,
v = dr_shell/dt_shell = - √(2M/r).
Thus we obtain the result:
dr_rain = √(1 - vv)×dr/√(1 - 2M/r) = √(1- 2M/r)×dr/√(1 - 2M/r) = dr.
That is, we may continue to use r-coordinate in the rain-frame too!
(2) Next, we notice that the value of r-coordinate can be obtained even within the horizon. Recall that r-value is obtained as "reduced circumference", the circumference divided by 2π. Since we cannot have concentric shells within the horizon, you may wonder how we measure the circumfenrence in question. But we can measure as in the following figure (adapted from Taylor and Wheeler, Figure 2 on B-8).
(3) Since the time coordinate was most problematic in Schwarzschild coordinates for treating motions across and within the horizon, we replace it by the proper time of the test particle. This is called t_rain. Then, velocity can be expressed by
dr/dt_rain.
(4) Finally, we have to define the metric for the rain-frame. Since this part is most complicated, only the final result is shown here (see Taylor and Wheeler, B-13; briefly, this is obtained from Schwarzschild metric by appropriate substitutions).
Recall that this is the timelike version, for telling the time separation between two events connected by the worldline of the test particle (raindrop, diver); the spacelike version can also be obtained as before. In short, the rain-frame is obtained by replacing the time-coordinate of Schwarzschild Geometry by t_rain. But it can cover the outside of the horizon (e.g., dt_shell and dr_shell can be calculated by the new metric) as well as the inside, without artificial singularities. An easy analogy is that the North-pole can be recovered from Mercator projection (there, a line!), if you change the source of projection or method of mapping. Coordinates are a method of mapping, and you can choose whatever method, depending on your purpose.
With this preparation, let us return to the motion of the test particle (raindrop or diver) across the horizon. Does it move faster than light within the horizon? In the following figure (adapted from Taylor and Wheeler, Figure 6 on B-18), the trajectory of the particle is compared to two light rays.
What this means is that the diver can receive information (say E-mail, or even an express package!) from outsiders, even after he/she crosses the horizon; though, of course, he/she can never reply.
You should notice that the lightcone structure is retained in general relativity, and even within the horizon, although even light cannot go out of the horizon. These features are illustrated (figure adapted from Taylor and Wheeler, Figure 5 on B-15). Lightcones in the figure show instantaneous states of light-flash emitted by the diver. We talked about aritificial singularities arising from our choice of coordinates, a moment ago. However, a black hole is not an artifact, not a mere singularity arising from the choice of coordinates. The diver can experience its reality; or so the Einstein equations predict.
We previously presented an image of a shell-observer's view of the sky. Then what would be the visual experience of the diver, close to the black hole (moving fast toward the hole, but still alive)? Taylor and Wheeler provide materials for this in chapter 5; based on them, we imagine something like this! If you wonder why, read chapter 5.
References
Misner, Thorn, and Wheeler (1973) Gravitation, 826-835.
Taylor, Edwin F. and Wheeler, J. A. (2000) Exploring Black Holes, Addison Wesley Longman, 2000.
Thorn, Kip S. (1994) Black Holes and Time Warps, Papermac, 1995. [邦訳1997]
Last modified March 30, 2003. (c) Soshichi Uchii
