From Hacker News
This is a far more interesting question than it might seem at first glance, and it deserves some attention because it tells us something fundamental and wonderful and just bloody awesome about the universe.
But I don't know how to tell the story succinctly. So I'm going to do that thing I do. I am very, very sorry. Please feel free to move on if this strikes you as tiresome.
Consider the Earth, and you on it. You're not floating freely, so clearly something's going on. We call that "gravity." We can call it, in the most generic sense, an interaction: you and the Earth are interacting somehow, and that's what's keeping you from floating freely.
We can then ask what the speed of that interaction is by putting it in these specific terms: How much time will elapse between your changing your position relative to the ground and your beginning to fall?
Yes, it's the Wile E. Coyote problem. Wile E. Coyote runs off a cliff, floats in mid-air long enough to hold up a sign that says "Help," then begins to fall.
Clearly that's an exaggeration. But just how much time does elapse, in real life, between stepping off a cliff and beginning to fall?
We can approach the problem naively by remembering that all propagating phenomena in the universe are limited by the speed of light. Given that fact, it makes sense to hypothesize that the time between the moment when Wile E. steps off the cliff and when he begins to fall will be equal to or more than the distance between him and the ground divided by the speed of light. It certainly can't be less, right?
We can then construct a set of very, very precise experiments with very fine tolerances — probably involving electromagnets and lasers or something — to test this hypothesis.
And then we can find that we're totally goddamn wrong.
To the absolute limit of our ability to measure it — and our ability to measure it is really good, since we used electromagnets and lasers and other expensive science things — when an object is dropped, it begins falling instantaneously. Not after a very small interval of time, but absolutely instantaneously. As in zero time elapses between dropping and falling.
This is fairly earthshaking, really. Because it implies that somehow a "signal" of some kind is getting from the ground to Wile E. faster than the speed of light. Which is supposed to be impossible.
I'm going to skip ahead a bit here, because I don't feel like explaining the entire theory of general relativity, and it won't be that useful in answering the question anyway. Suffice to say that no, no time elapses between dropping and falling, but at the same time no, no signal or interaction has to propagate upward from the ground to Wile E. in order to make him start falling. In fact, what's going on is that Wile E. Coyote is always falling, due to the curvature of spacetime created by the Earth. Whenever he's standing at the edge of the cliff, on the ground, the ground beneath his feet — paws? — is arresting his fall by, effectively, pushing up against him. The very instant that's removed, he starts falling.
So in that sense, gravity has no speed. Because it doesn't actually propagate through space. One way to look at it is to say the gravitational field fills space, so wherever you are, you're already being affected by it all the time. Another way is to say that gravitation essentially is space, so it affects you simply by virtue of existing. The two are essentially equivalent English translations of the equations that actually describe the phenomenon.
But okay, that's half the problem. The gravity of a static body fills space, or is space, and as such can't be meaningfully said to have a speed. But what about the gravity of a changing body? Like you said, what if "suddenly a black hole appeared?"
Well, the answer of course is that that never happens, ever. Gravitation doesn't suddenly anything; macroscopic things don't just appear out of nowhere, and teleportation is impossible. So we don't have to think about that … and in fact we couldn't get meaningful answers if we tried.
But things do move. The moon's moving relative to the surface of the Earth; we can tell, even apart from the fact that we can see it up there, because the moon is the major contributor to the tides, and the tides rise and fall. But what's the relationship between the moon's position in space and the tidal acceleration on the Earth? Are the two somehow always in perfect sync, or is there some lag? If so, how much, and in what direction?
That's actually a much harder question to answer than you might think. There was a now-infamous paper some years ago by a fellow named Tom Van Flandern (recently passed, God rest his soul) that asserted that the change in gravitational acceleration in a dynamical system actually propagates many times faster than the speed of light — at least twenty billion times faster than the speed of light — but not instantaneously. This got a lot of attention at the time. If the propagation speed of changes in spacetime geometry were equal to the speed of light, that'd be fine. If it were literally instantaneous, that'd also be fine, more or less, though our theory would need some tweaking. But faster than c but still finite? That was really hard to explain.
It turned out not to be a problem though. Because Van Flandern just made a mistake in his paper. See, the relationship between motion and gravitation is not as straightforward as it might seem. In fact — and I'm glossing over this now, because the maths are damn complicated — whenever a gravitating object moves inertially, the gravitational acceleration vector at a point removed actually points at where the object actually is at a given instant, as opposed to where the object's light is seen to be coming from at that instant. So in that sense, we're back to gravitation being instantaneous again!
But is it really? No. Because you see, if the inertially moving object were to come to a stop instantaneously, the acceleration vector would continue to point toward its future position for a time, as if it were still moving inertially, even though the object is actually somewhere else. The sum of effects that serve to cancel out aberration when everything moves inertially would break down, and the acceleration field would point toward empty space for however long it takes for the change in geometry to propagate through space at the speed of light from the gravitating object to the point in question.
Except things don't stop moving instantaneously. Things accelerate, and acceleration requires energy, and when you factor that in, the equations balance out again.
(If you feel up to the challenging of following a lot of advanced mathematics, here's the best paper I know on the subject.) - http://arxiv.org/abs/gr-qc/9909087v2
So what does that mean? It means that the "speed of gravity" is the speed of light … technically. Changes in the geometry of spacetime actually propagate at the speed of light, but the apparent effects of gravitation end up being instantaneous in all real-world dynamical systems, because things don't start or stop moving or gain or lose mass instantaneously for no reason. Once you factor in everything you need to in order to model a real system behaving in a realistic manner, you find that all the aberrations you might expect because of a finite speed of light end up canceling out, so gravity acts like it's instantaneous, even though the underlying phenomenon is most definitely not.
The universe is pretty damn cool, if you ask me.