Once Before Time - Martin Bojowald [98]
The intuitive basis of a Penrose diagram is the following: As often is the case in astrophysics, a concrete investigation of black holes is a scattering problem. A black hole does not emit its own light, and the aura of Hawking radiation emerging from its neighborhood is usually too weak to permit observations. Instead, one can recognize a black hole by light emitted from other stars that passes near it and is deflected into our observational instruments (or away from them). As with the moon, one can “see” a black hole and determine its properties by the scattered radiation, even though this comes with additional complications arising from the great distance and the special properties of scattering. (The black hole, after all, does not have a distinct surface but only a horizon.)
On the basis of this scattering problem we can visualize black holes in two-dimensional representations. To that end, we draw light rays in a space-time diagram (figure 20), containing the radius and time. Time in relativity is usually drawn upward; we thus choose the vertical direction as that of time. The radius can then change to the left or the right, depending on whether we approach the center (leftward) or move away (rightward). But the radius, in contrast to time, is always positive: It tells us the distance to the center, but not the direction. We have to add a boundary to our diagram, in the form of a vertical line. On this line, the radius takes the value zero; we are at all times in the center of spherical symmetry.
What happens when we approach this line along a physical trajectory? This is not a boundary of space-time analogous to the big bang singularity; at this place nothing unusual happens. The line merely represents a special position, since it sits in the center of symmetry. When we, or light rays, approach the center, it is simply penetrated, as in figure 20. The center is not a physical object and thus cannot influence matter or light. When something passes through, just the radius as the distance to that point changes: It first shrinks, then vanishes, and finally grows back. The radius never becomes negative as required, but it flips its direction of change from decreasing to increasing. In the two-dimensional space-time diagram, this appears as a “reflection” of the light ray at the vertical line; coming from the right, it runs back to the right after having touched the boundary. Nevertheless, there is no material such as a mirror. The “reflection” is merely a consequence of our intuitive two-dimensional drawing of actually four-dimensional events by ignoring the spatial angles.
20. A rotationally symmetrical space-time can be represented by a plane on which time and the radial distance from a center function as coordinates, bounded as a result of the positivity of the radius. Light rays run along straight lines, inclined by 45 degrees. When they hit the boundary, they are reflected in this representation, for they traverse the center in such a way that their distance to it first shrinks, then vanishes, and finally grows back. Left: Space-time diagram with the boundary at left as the rotational center in time changing vertically, and with the trajectory of a light ray moving through the center. Right: Spatial diagram of the same situation, with the rotational center at midpoint and the trajectory of a light ray at different times. The radius—the distance of light from the center—first decreases, then vanishes at the center (the left boundary of the space-time diagram), and finally grows after passing through the center.
With this,