Chapter 12: The Principle of Computational Equivalence

Section 4: The Validity of the Principle

Time and gravity

General relativity implies that time can be affected by gravitational fields—and that for example a process in a lower gravitational field will seem to be going faster if it is looked at by an observer in a higher gravitational field. (Related phenomena associated with motion in special relativity are more difficult to interpret in a static way.) But presumably there are effects that prevent infinite speedups. For if, say, energy were coming from a process at a constant rate, then an infinite speedup would lead to infinite energy density, and thus presumably to infinite gravitational fields that would change the system.

At least formally, general relativity does nevertheless suggest infinite transformations of time in various cases. For example, to a distant observer, an object falling into a black hole will seem to take an infinite time to cross the event horizon—even though to the object itself only a finite time will seem to have passed. One might have thought that this would imply in reverse that to an observer moving with the object the whole infinite future of the outside universe would in effect seem to go by in a finite time. But in the simplest case of a non-rotating black hole (Schwarzschild metric), it turns out that an object will always hit the singularity at the center before this can happen. In a rotating but perfectly spherical black hole (Kerr metric), the situation is nevertheless different, and in this case the whole infinite future of the outside universe can indeed in principle be seen in the finite time between crossing the outer and inner event horizons. But for the reasons mentioned above, this very fact presumably implies instability, and the whole effect disappears if there is any deviation from perfect spherical symmetry.

Even without general relativity there are already issues with time and gravity. For example, it was shown in 1990 that close encounters in a system of 5 idealized point masses can lead to infinite accelerations which cause one mass to be able to go infinitely far in a finite time.

From Stephen Wolfram: A New Kind of Science [citation]