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I know that time "stops" at the speed of light, but at what gravity does time "stop"...??

To clarify: From my knowledge of special relativity, if an outside observer were to watch someone move at the speed of light (which is impossible, but bear with me for the example's sake), the time for the person moving at the speed of light would "stop" (from the outside observer's point of view). Knowing this, I wonder if this is also somehow applicable in general relativity (in my opinion, there should be some sort of equivalence between the effects of moving at the speed of light (in our example) and a large enough gravitational field. I also think that, because the speed of light is finite, so should the gravitational strength required to "stop" time using gravity be finite). So my question is: how strong does a gravitational field have to be to "stop" time for an object experiencing the gravity (from the perspective of an outside observer, of course)?? Please show the math behind the answer, if possible. Thank you!!

Question Date: 2011-05-31
Answer 1:

You are correct in your assumptions, and there is indeed a "gravitational equivalent". Special relativity tells us that the speed of light is the "universal speed limit". One consequence of this statement: Were an observer able to move at the speed of light, the passage of time would seem to stop. In order to consider the "gravitational equivalent", we need general relativity.

To answer your question, we need a black hole. A black hole is formed when the mass inside of a spherical volume causes such an extreme gravitational pull that not even light can escape from within that volume. The radius of this sphere is called the Schwarzschild radius. If you were to imagine yourself as a beam of light traveling tangentially to this sphere, you would be in a perfect orbit to circle the center of the black hole at the speed of light, seemingly forever. This only holds if the black hole is not rotating and is uncharged, though similar cases may exist in those situations, as well.

The Schwarzschild radius (r_s) of an object is:
r_s = 2 G m / c2
where G is the gravitational constant, m is the mass of the object and c is the speed of light.

Interestingly, the AdS/CFT correspondence (a result of string theory research) is showing a remarkable similarity between black holes and superconductivity, though how that relates to your question may not be well understood at the moment.

Answer 2:

In order to stop time, a gravitational field must be infinitely strong. The passage of time goes to zero in the limit of gravitational field strength going to infinity, in much the same way as the mass or momentum of an object becomes infinite as you reach the speed of light.

Incidentally, this is exactly what happens at the very center of a black hole, as per our current understanding of relativity (quantum mechanics doesn't jive with this, so we know at least one theory has to be wrong, however). Time literally comes to an end at the center of a black hole, just as time literally begins at the center of a hypothetical white hole (or, at the less-hypothetical Big Bang ~14 billion years ago). Of course, you can't actually SEE time coming to an end in a normal black hole, however, because it's concealed inside of the event horizon, but a naked singularity, if they exist (basically a black hole without an event horizon) would enable you to observe such a thing.

The mistake you're making about the finite speed of light is that although yes, information does propagate at a finite speed in the universe, the amount of effort to get something up to that speed requires infinity. Thus, pushing relativistic time dilation to zero also requires infinity. Therefore, and not surprisingly, doing the same thing with gravity, too, requires infinity.

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