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Let's suppose there is a very massive object and a small object that are 1 lightyear apart.

The massive object is large enough that the gravitational force pulling the small object is easily noticeable.

Suppose (never mind how) that through some freaky event the massive object suddenly disappears or is suddenly transported to some part of the universe too far away to have noticeable gravitational effect on the small object.

Since nothing travels faster than light, does it take 1 year or more for the small object to "figure out" that the massive object is gone and to stop accelerating towards where it used to be? Or does the magical disappearance of the massive object have immediately observable effects on the small object?

To put it more concisely, is there a delay in the effect of gravitational force?

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Yes.

$c$ is the highest possible speed for light/any information to travel. So for a person 1 light year away, he wouldn't even realise that the object has disappeared, until the light carrying that information has travelled there.

You can also think about this another way. Gravitational waves (which researchers today are trying very hard to detect) can only travel at the speed of light. When the object vanishes it causes disturbances in the space-time continuum , but these disturbances are also clocked at the highest possible speed of $c$.

So there is a time delay.

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  • $\begingroup$ Perhaps you should also explain why (other than by the information argument) gravitational waves can only travel at the speed of light. $\endgroup$ – Řídící Feb 12 '14 at 20:26
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The situation is too ill-defined for an answer.

The problem there is that in general relativity, you do have general conservation laws that follow from the Einstein field equation. In the asymptotically flat case, you have conservation of a global ADM mass, and in all cases there is a local covariant conservation law that requires the stress-energy to be divergenceless. Stess-energy must move locally; it can't just disappear at one place and move to another.

That means that if the very massive object simply winks out of existence or is teleported to far reaches of the universe, then we can't use general relativity to predict what happens, because such an event blatantly violates it already. Thus:

Or does the magical disappearance of the massive object have immediately observable effects on the small object?

Newtonian gravity doesn't care about conservation of mass, but general relativity does, or at least has roughly analogous laws that prohibit this type of situation. If one starts invoking magical disappearances or teleportations, then there is no reason for the same magical event to not make the spacetime instantly flat either. It doesn't make any sense to apply a theory to a situation that directly contradicts it, so the question of what happens becomes ill-defined.

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There is a sense in which you can make the matter of the very massive object disappear by collapsing it into a black hole, since (isolated) black holes are technically vacuum solutions of GTR, and it gets around limitation of a divergenceless stress-energy because singularities are not part of spacetime. But in that case the other body could still orbit it in just the same way.

Perhaps one could drop the matter into a wormhole. I don't know how that would work out in general, but at least for the flat "Lorentzian wormholes" investigated in Visser's monograph, the mass you transport in this manner gets added to the wormhole mouth. So you'll have a very heavy wormhole end, which the other body would still accelerate toward it.

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    $\begingroup$ I think the spirit of his question is, "if the massive object were to be moved, would the gravitational force felt by the small test mass immediately reflect this movement?" I see no reason that magic needs to happen. This could be a continuous movement. $\endgroup$ – Brian Moths Feb 12 '14 at 20:38
  • $\begingroup$ @NowIGetToLearnWhatAHeadIs: The "suddenly transported to some part of the universe too far away to have noticeable gravitational effect" and further invocations of "magic" to me suggest a very discontinuous disappearance/teleportation event, rather than simply moving the object. If it was simply moved, I wouldn't call it sudden, at least, because it wouldn't get "too far away" fast. $\endgroup$ – Stan Liou Feb 12 '14 at 20:42
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    $\begingroup$ Just blow up the large object into two pieces, with enough energy to send them off (orthogonally to the line between the original objects) at a good fraction of $c$. (I believe the technical term is "rapid change in the quadrupole moment," but for popular science, "blowing up" will do.) It's not quite magic teleportation, but it's a pretty fair approximation. $\endgroup$ – Ilmari Karonen Feb 12 '14 at 21:26

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