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coordinate—a coordinate singularity but nothing more. An ant running along the circle would just move on after each lap, entirely unimpressed by the jump of the angle. The same happens with a calendar: The circle here is the motion of Earth around the sun, and the coordinates—months, days, hours—are just chosen in a more complicated way than a single angle, which would actually have sufficed. Here, too, a jump occurs, namely the one from 2400 hours on December 31 to zero hours on January 1. New Year’s Eve is nothing but a coordinate singularity—such a meaningless phenomenon that it seems questionable, certainly from a mathematical standpoint and surely from more practical ones as well, to squander large sums of money on this occasion.

A further important property of the horizon can already be seen from some of the preceding remarks: The horizon is not a surface in the usual sense, not a spatial surface surrounding a voluminous object. After all, the horizon in a Penrose diagram of figure 23 is defined by a 45-degree line, a light ray! No massive object, nor the surface of one, can stay on this line in space-time. It is just as impossible to move up to that line—to land on a surface, as it were—and then move back into the exterior. Once at the horizon, there is no escape: As a material object, one falls into the interior, and soon thereafter into the singularity. Even light can at best linger forever on the boundary and just avoid the crash into the singularity, but it can never get back to the exterior—at least according to classical general relativity.

NAKED SINGULARITIES AND COSMIC CENSORSHIP: LAW AND DECENCY

As shown in figure 23, the singularity of a black hole, formed by collapse, is a point in time, not in space: a horizontal rather than vertical line in the Penrose diagram. There are other solutions of general relativity, rotationally symmetrical as well, where already the left boundary of the diagram, or at least a part of it, is singular. It can then happen that a part of this singular piece is not covered by a horizon; sometimes there is no shrouding horizon at all in spite of the singularity, as shown by the example in figure 24. This possibility of so-called naked singularities has upset some relativists so much that, hoping that general relativity is “modest” enough, they introduced the concept of cosmic censorship. Such disrobed singularities are supposed to arise from gravitational collapse only under conditions so special that they play no role in the universe, making them only mathematical possibilities far removed from reality.

Of course, this conjecture has a serious basis: If an uncovered singularity exists, light and much more can reach us from there. From the viewpoint of general relativity, which fails at the singularity and cannot tell us anything about it, literally anything can happen. But when whatever is going on there can influence the rest of space-time, science loses all predictivity.

Something similar, strictly speaking, occurs at the big bang singularity. But here one is so used to the notion of a “beginning,” laying the basis for our existence, that one accepts, in fact expects from it, a strong influence on the world. That would be required for an interpretation as a beginning; for when nothing existed before but something does afterward, the singularity must simply leave an extremely dominant impression on the rest of space-time. This transition from nothing to something cannot be explained by relativity, but once it is accepted, at least the further development is computable. The big bang may be undefined by general relativity, but it happened in the past and is already over. The revolution of a naked singularity, by contrast, lives on, possibly forever. At each moment, the unpredictable can hit us, a far less acceptable violation of the rules of determinism. Differences between the types of singularities show up in the Penrose diagram: The big bang singularity in the space-time picture of figure 25 is not the same as a naked singularity such as may form in gravitational collapse