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By Root 724 0

we already have the first ingredients of Penrose diagrams: a vertical line symbolizing the behavior in time of the center, and light rays. Since light in empty space always has the same velocity, as a space-time angle unchanged by the conformal mapping, we fix the direction of light rays to be always 45 degrees, to the left or to the right depending on whether light moves from large radii to small ones or the other way around. One can imagine these rays as coming from infinity and moving back to it after having traversed the center. On the right-hand side, no boundary lines are necessary. However, boundedness comes from the mentioned “completion” of space-time to a finite region, and so one often draws an empty space-time as an isosceles triangle, with the vertical line of the center as one boundary and the finitized origins of incoming light rays and the targets of outgoing light rays as the two other sides, as in figure 21.

The result is a complete two-dimensional diagram of a space-time where only light rays propagate. (Having disregarded spatial angles, we can only draw light rays moving through the center in our diagram. This is sufficient for an understanding of space-time and its scattering properties.) Empty space, except for light rays, is certainly not very spectacular; but one can easily extend it by further elements to describe the spherical bodies of astrophysics. Then, the 45-degree lines show the scattering problem that represents astronomical observations.

Perhaps surprisingly, a compact object in the center, for instance the moon, does not present itself much differently from empty space. The object has a certain radius, so we must draw its trajectory in time somewhere to the right of the left boundary (see figure 22). The lunar radius is constant; the surface should have a fixed distance from the center at the left boundary. But distances, unlike angles, are not respected by the conformal mapping, and thus the surface is generally represented by a curved line in conformal diagrams. This avoids having the surface intersect the two borders to the right (except at the corners), which are supposed to be reached only by light rays.

21. Complete space-time diagram with two propagating light rays. Coming from the past, corresponding to an arbitrary distance away from the center to the bottom right, they approach the left vertical line, traverse the center, and move away from it into the future. This illustrates a scattering problem, with light shining from a light source at the bottom right and being measured by detectors at the top right. Both the light source and the detector move upward in time along the boundaries. The diagram visualizes a whole history of events of emission and detection. That the lines of sources and detectors intersect with the vertical boundary, seemingly emerging from or falling into the center, is an artifact of the completion procedure. As parallel lines in perspective drawings seem to intersect at their vanishing points, the points at which the boundary lines of a conformal completion intersect do not exist in reality but lie at infinity.

At the curved line, as with the surface of the moon, some part of the light falling in from the right is absorbed, and the rest is now reflected in a truly physical sense. The only difference from empty space is that some light rays stop—something we can ignore since lost rays become irrelevant for the scattering problem—while the rest are simply sent back to the right at a different position. From this viewpoint, all compact objects in astrophysics can visually hardly be distinguished from empty space, as shown by a comparison of figure 21 with the right-hand side of figure 22. Such a broad view may seem a disadvantage, given that our observations of all these objects show important differences. But the similarity of those images—in sharp contrast to that of a black hole, to which we turn now—illustrates in an impressive manner the black hole’s special character and its strong effects on space and time.

22. Space-time diagram of the moon