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Total newbie with basically no physics knowledge here :) I would welcome any correction to the steps of my reasoning that lead to my question, which could easily turn out to be invalid :)

My current understanding is that General Relativity is definitely accepted by the vast majority of scientists, and that according to General Relativity spacetime is curved by the presence of mass.

I also seem to understand that not as many scientists (although probably still the majority) agree that extra dimensions of space exist.

But if we agree that spacetime can "curve", aren't we automatically saying that extra dimensions of space exist?

I mean, if I grab a stick and I bend it, it becomes curved with respect to our good old 3D space. The stick needs to be in a 3D space, with respect to which it can be straight or curved.

So, if the thing that gets curved is not an object in space but space itself, doesn't space need to be in "another" space in order to be curved?

I apologize in advance if I used inaccurate terms here, which I most likely did :)

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Small detail: in GR, space is not just bent by the presence of mass. Intrinsic curvature is also possible. – Danu Feb 17 '14 at 13:21
If you like this question you may also enjoy reading this post. – Qmechanic Feb 17 '14 at 13:38
@Qmechanic Definitely on-topic, thanks! – SantiBailors Feb 17 '14 at 13:50
Also a duplicate: What is the universe 'expanding' into? – John Rennie Feb 17 '14 at 13:53
up vote 12 down vote accepted

No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work. You're thinking of something called "extrinsic curvature". In fact, many examples of extrinsic curvature - including your example of a stick being bent - don't have intrinsic curvature at all. Let me try to be a bit clearer: imagine there is an ant who lives on your stick. As far as the ant is concerned, the world is one dimensional. Now, suppose we tell the ant that space is really 3D and his little 1D world is inside ("embedded in") that 3D space. There is absolutely no way the ant would be able to figure out if his stick was straight or bent the way you're describing. So, this isn't the sort of curvature that interests us in general relativity.

Basically, intrinsic curvature is just concerned with the geometric relationship between nearby points. It's entirely possible to think about this in terms of embedding space-time into some higher dimensional world, but you don't have to: it works just fine if you confine yourself to the observable four space-time dimensions. "Curved" in this sense is just a short hand way of saying "parallel lines don't do what they do in 'flat' (Euclidean or Minkowski) space / space-time".

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Thanks, nicely put. I don't think I will ever be able to wrap my mind around the idea of "curved with respect to nothing" - not even if I imagine to be the ant - but that's just because of my ignorance of and unfamiliarity with the subject and physics in general. I believe that the level of understanding I get from this answer is more or less the farthest I can ever go. – SantiBailors Feb 17 '14 at 16:16
@SantiBailors: Maybe this math.SE thread might help a little. – Ilmari Karonen Feb 18 '14 at 1:34

Nope, spacetime curvature says nothing about the dimensionality. Your intuition here is probably wrong because human imagination needs 'some dimension to bend into' in order for something to be curved (i.e. an embedding in a higher-dimensional space). This is just our lack of imagination showing, though.

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Thanks, apparently I needed an example, now I think I see what you mean. About imagination, I think it's not the human imagination that needs some dimensions to bend into; I actually think it's the human logic that needs that, until of course one acquires the necessary theoretical knowledge to expand the reach of his/her logic. In my opinion human imagination is not too limited and that's why there are so many views floating around that are completely lacking any proof or link with reality :) – SantiBailors Feb 17 '14 at 16:27
@SantiBailors On second thought, intuition is the appropriate word for what we naively lack as humans (when it comes to advanced physics). – Danu Feb 17 '14 at 16:55
Good point, actually intuition describes that better. We humans might have it just for "humans" things, like being sometimes able to predict people's behavior; fortunately a few humans have it for advanced physics / math and those are the geniuses who make these fields advance the most. – SantiBailors Feb 18 '14 at 16:20

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