Once Before Time - Martin Bojowald [101]
Although this may appear sneaky, the singularity does make itself known earlier. In its neighborhood—understood in the temporal sense, thus before it forms—space-time curvature is very large, even though it becomes infinite only at the singularity itself. Strong curvature, according to the descriptions in chapter 1, means that the strength and direction of the gravitational force vary greatly even at nearby points. Every extended object, such as a space capsule, is subject to stretching forces when its sides experience different forces tugging its parts in opposite directions. Such forces are called tidal forces, by analogy with the well-known, if weaker, effect on Earth, where space-time curvature caused by the joint gravity of the earth and the moon raises the tides. From these forces, one would notice the singularity long before one fell in; objects would be torn apart early in their approach.
Dreadful is the appearance of the singularity for an observer who has already fallen into the black hole and can no longer escape. How is the black hole experienced by an observer far outside? Here, it is important that the space-time portion containing a black hole is divided into two different regions. There is the triangular part underneath the horizontal singularity line in figure 23 from which nothing can escape, and there is the rest of space-time where we would stand as external observers. Since not even light can escape—it can, after all, only move along the 45-degree lines in the interior, all ending at the singularity—one must consider the dividing line as the boundary of the black hole. This is the storied horizon.
The horizon has several important properties. First, it is, except for its endpoint in the distant future, rather far away from the singularity, for heavy black holes, very far away. A black hole of one million solar masses, as it can easily occur in galactic centers, would be as large as the sun. If the total solar mass were to collapse to a black hole, the distance from the horizon to the singularity—the so-called Schwarzschild radius—would be one-millionth of the current solar radius of about 700,000 kilometers. For common stars, the ratio of their current radius to their Schwarzschild radius is similar. For white dwarfs, the Schwarzschild radius is about one ten-thousandth of their radius, and thus comparably higher than for the sun, as a sign of their stronger compression.
Extremely dense neutron stars have a radius just above the Schwarzschild radius determined by their mass. They stand at the brink of collapse into a black hole; were they just slightly denser, their surface would come to lie within the horizon and then be pulled into the singularity. For even heavier black holes, their radius increases proportionally to the mass. In such cases, the horizon is so far outside the high curvature region near the singularity that the space-time curvature around the horizon is quite weak. Quantum gravity is not usually required in this context to understand all the properties—but sometimes one needs the quantum theory of matter, as in the Hawking effect.
Low curvature also means that one would initially not notice one’s own passage through the horizon, justifying the use of this term. As a horizon on Earth appears as a distinct border but then, on approach and during passage, turns out to be an immaterial perception, traversing a black hole horizon is initially not accompanied by any threatening or foreboding signs. In the near neighborhood of an infalling observer, space-time appears no different than the exterior space. An outside observer would, however, notice the horizon because signals emitted by the infalling observer will need more and more time to reach a stationary observer outside. Time, after all, proceeds more slowly in a region of strong gravity than it does under weak gravity; time for an observer approaching the horizon passes more and more slowly compared to time as experienced by