Once Before Time - Martin Bojowald [100]
Significant things happen when we consider a black hole. Bringing these aspects out clearly is the true strength of Penrose diagrams. One could expect the vertical line of the center to become an actual boundary of space-time: the central singularity. This could be one possibility, indeed realized for some solutions of general relativity. But those solutions play hardly any role in astrophysics, for a black hole as it arises in the gravitational collapse of a star leads to a different diagram. As a mathematical analysis shows, the singularity does continue the vertical line of the center, but it does so not in a vertical but, as in figure 23, in a horizontal direction! It is not timelike—not a fixed point at which time would change—but spacelike, a part of space at a fixed time. After all, the vertical position, determining time, does not change along the singularity. For this reason, our initial expectation that the singularity is a point in space was incorrect. Such a point in space, visible at all times, would be vertical in our diagram, just like the center line. Instead, being horizontal, the singularity occupies several spatial points, but it does so at a fixed time: at a fixed point in time.
23. Space-time diagram of a black hole, with the dash-dotted singularity as additional boundary. Not all light rays can now escape to a safe distance, to the upper right boundary. There is one light ray (drawn as coming in from the bottom right) that can just barely escape, marking the horizon by its final segment from the center line. Light rays starting in any direction from within the horizon (in the diagram, above the final segment of the ray that barely escapes) always reach the singularity, even if they move away from the rotational center. Light rays starting outside the horizon and moving away from the center do escape to a safe distance.
Collapse into a point is expected to be drawn extended in a space-time picture rather than pointlike, because the point persists in time. The singularity as the end state of collapse is indeed an extended line in the Penrose diagram—but since the process is combined with a space-time transformation deeply inside the black hole, the extension shows itself in space and the pointlike nature in time, and not the other way around.
In general relativity, the new, horizontal line really does represent a boundary: Space-time curvature here is infinitely high, and so is the density of infalling matter that led to this black hole. Here is the point (in time) where matter completely collapses. The equations of general relativity can tell us something about the form of the singularity—namely, that this is a point in time—but they do not lead us beyond it. One cannot use them to explore the possibility of space-time above the horizontal line, since the infinities make all equations lose their mathematical sense. For the collapsing matter, or sub-sequently infalling matter including unfortunate or sacrificial observers, one can only say that they will reach the boundary after some finite time and then, within this theory, cease to exist. Whether this limit of theoretical existence is also a real limit of the world is, however, a different question that is answerable only with a more general theory such as, perhaps, quantum gravity.
Owing to the singularity’s point-in-time nature, it is not visible as a bright spot to an observer approaching it. Were this the case, light would have to start from that point and reach the observer. But all light would travel from the singularity upward in the diagram, into the region where general relativity does not provide us with any space-time to propagate light. The singularity appears entirely different from all other astrophysical objects, and not only because of its extremely