Once Before Time - Martin Bojowald [102]
This behavior is relevant not only for an imaginary experiment in the vicinity of a black hole, but also for astronomical observations. Light is nothing but such a regularly emitted signal. If stars or hot, glowing gas exist in the neighborhood of a horizon, as is the case in the center of the Milky Way, one can notice delays in the periodicity in the form of changing colors: The delay implies a reduction of the frequency, and so one speaks of redshift (red light being located at the lower-frequency end of visible light). Such effects are indeed planned to be made use of in future observations to measure the neighborhoods of black hole horizons.
The horizon and its meaning have been understood only rather recently, long after Schwarzschild found his mathematical solution. Grasping its meaning is made difficult by the fact that infinities arise even at the horizon but turn out not to be disastrous for space-time. What diverges here is the redshift as it is measured by an exterior observer: The periodic signal of a source falling in, when it reaches the horizon shortly before disappearing, is infinitely redshifted. Viewed from the outside, such a signal no longer shows any temporal change. While outside, far from the black hole and under weak gravity, time progresses normally, processes near the horizon appear delayed. Light itself is influenced near the horizon, largely disappearing from the visible range of the electromagnetic spectrum when it reaches an exterior observer and so, as it were, fading. Here, infinities arise only in the perceptions of an observer; they do not refer to direct physical properties of a material or space-time object. Such infinities are acceptable, but they do require care.
All this gives rise to difficulties when one attempts to describe the whole of space-time by coordinates, similarly to the use of latitude and longitude as a standardized specification of positions on Earth. In curved spaces, it is not always easy to choose globally defined coordinates; even on the spherical space of Earth’s surface there are, strictly speaking, problems. At the poles, after all, geographical longitude loses meaning: all longitude lines intersect. Which longitude should one then attribute to the poles? Such points, where some of the chosen coordinates lose their meaning, are called coordinate singularities. Mathematically they appear as singular, and they can, if one is not careful, lead to infinities in calculations. But physically, nothing unreasonable is happening at such a point. They are just coordinate singularities, not strict, unqualified singularities.
By the way, the North Pole is often, and misleadingly, used as a comparison with a physical singularity, leading some to assert that it is just as meaningless to speak of “before the big bang” (or, in our present context, “beyond the black hole”) as it is meaningless to speak of “above the North Pole” (or “north of the North Pole”) if one is tied to Earth’s surface. However, this objection is misleading, because the North Pole is just a coordinate singularity, while the big bang singularity and that of a black hole are real, physical ones: There, curvature grows unboundedly, and with it perceptible properties such as temperature or tidal forces. There, one is literally torn apart; at the North Pole one would at worst freeze to death.3 (And even that happens only because the North Pole has arbitrarily been located in the Arctic, or because one has used Earth’s rotation axis for orientation when defining geographical latitude and longitude.)
A simple example for a coordinate singularity can be found on a circle. A single angle suffices as coordinate, but there is always a point where the angle must jump from 360 degrees back to zero degrees. This happens because the circle is cyclic, while we must use nonperiodic numbers as coordinates. It comes to the abrupt jump of the