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 :)
Answer
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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