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If I throw a stone on space, in a place where gravity is equal zero, will the stone move forward forever, because no air, so no friction?

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Technically space isn't a perfect vacuum.. There's a bit of gases and dust and whatnot even in the empty regions. So, after an extremely long time, it will stop. Though before then it is more likely it encounters a gravity-laced region or something; we don't have any sufficiently large empty regions. – Manishearth♦ May 4 '12 at 2:52
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@Manishearth: Hey, left out interactions with the CMB! – dmckee♦ May 4 '12 at 4:38

3 Answers

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According to Newton's first law, yes. The velocity of any object will remain constant if no forces affect it. That holds in any Inertial frame of reference (if you are accelerating by yourself, then the stone will be accelerating relative to you, even if no forces act upon it).

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This is the same thing i was thinking – Zignd May 6 '12 at 1:05
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From the perspective of General Relativity, assuming we can ignore interactions with intergalatic gas and the CMB then a thrown stone follows a curve called a geodesic. In general geodesics go on forever so your stone will be moving forever, just as Lev said in his answer.

However there are circumstances in which geodesic curves appear not to go on forever. I say "appear" because future theories of quantum gravity will probably change things, but at the moment we think that a geodesic that leads into a static black hole will just end when it hits the singularity at the center of the black hole. This idea is called geodesic incompleteness.

So the answer to your question is that the stone will almost certainly go on forever, unless it hits a black hole. Even then it would have to be a static black hole because for charged and rotating black holes the stone could miss the singularity and emerge again (into a different universe, but that's another story!).

Later: Oops, I've just seen Logan's comment and you did say "where gravity is equal to zero" so my comments about black holes don't apply. Still, I think the idea of geodesic incompleteness is interesting enough to warrant a mention.

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I didn't -1, but this answer has issues even if it is technically correct. For me, black holes are incompatible with the OP's statement that "gravity is equal zero" (admittedly poorly phrased, but the idea is to ignore gravity, i.e. work in Minkowski Spacetime). Answering the question in a GR context is also tricky because of strange things like closed timelike geodesics, making both "forward" and "forever" difficult to define. But the basic idea is that "this is mostly still true in general relativity once we make sense of what that even means" which I think you mostly captured. – Logan Maingi May 4 '12 at 6:19
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Actually the stone is NOT moving just and YOU are not moving as well. You just put the stone on a different reference frame and then ... it is not moving any longer. An ant on that stone would see YOU moving as you see the stone moving. If the question is : would that stone return to YOUR reference frame then the answer is : why would it do that? The stone has no memory and hence no potential willing to get back to your reference frame.

The main thing is that Einstein thinking was that anything that occur should be described independantly from the observer point of view which is the complete opposite of quantum mechanics which does not consider the measured object but the measure itself (the observed point of view) which caracterize the object which exist just relatively to the oserver.

A lot of particles would hit the stone but statistically it would be hit equally from all direction and momentum so this may not affect the stone rest.

Expansion of the universe may affect it but that's another story (after some time it won't be longer a stone).

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