Does space curvature automatically imply extra dimensions? 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 :)
 A: 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.
A: 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".
A: General Relativity says nothing about extra dimensions and they are not needed to model the theory.  To avoid abstractions, I may have a model to more easily conceptualize the curvature of space-time by a super massive object.  The huge mass actually causes a compression of the space-time 4D grid.
What does compression of space-time grid mean?  Well first of all, no mass-energy is static, ie motionless, in space-time.  Even if you have 3 masses that are relatively motionless with respect to each other in 3D space, the clock is still ticking for each of them, so everything is moving in time even if they are motionless in 3D.
In flat (uncompressed) space-time the clocks of said 3 objects would tick at the same rate.  However, in a compressed space-time grid caused by the effect of a huge mass, the clock would tick slower.  Any mass-energy object would accelerate into the compressed space-time grid where the clock ticks slower.
This is not so difficult to understand why.  I think we all would rush to a place where the clocks tick slower:)  Another way for the clock to tick slower relative another object is for that object to move at relativistic speeds close to the speed of light with respect to the other object.  But, that is described by the special theory of relativity.
A: You're probably taking the rubber sheet analogy too far. There is nothing in Einstein's equations that predict the existence of a higher dimension through which space can curve. While in the analogy they show 2D space curving in a 3D world (and the wormhole analogy where you fold a paper and take a shortcut through a tunnel in what appears to be a higher dimension) there's so far NO evidence that our spacetime requires a higher dimension to curve. It can be easier for the human mind to understand this concept using analogies like space curving like a trampoline, however in reality that's just not how general relativity works.
Spacetime is not a fabric, it is no material. Space is just an illusion, time is just an illusion therefore spacetime is just an illusion and a good way of simplifying the concept of general relativity to the public. In fact there is no 'outside' of the universe where spacetime can curve. The universe is all that exists.
As said, math is the language of the universe. Stuff like general relativity and complex physics can NEVER be explained in English or any spoken language with 100% accuracy. General relativity can only be explained 100% accurate with pure mathematics. Here's a better and far more accurate way of explaining spacetime curving in presence of mass/energy:
Gravity does not only pull in objects with mass, it also pulls in massless objects such as light. This is what Einstein found using his equivalence principle that he derived which states that you can't distinguish between acceleration and being in a gravitational field. So if a light beam is travelling close to a massive object, because gravity pulls light in with same acceleration as any other body, the light beam's path would be slightly curved. That curvature depends on the gravitational strength of the planet/star. So that curvature results in light's path being longer than if it just travelled in a straight line. However since light always takes the shortest path between two points, the fact that light didn't in the gravitational field and instead took the longer curved path would mean the curved path IS the shortest path. There's no such thing as a straight line in a gravitational field, if you intend to go in a straight line at a constant velocity you'd be automatically going in a curved path! So this is what people mean by space curving, if space curves then the distance between two points change! Therefore even the shortest path between two points would change (for gravity it would increase).
And what about time? Well this goes into the fact that the speed of light is constant in all reference frames. However since the light appears to take a longer curved path in presence of gravity, time itself must change for the speed of light to remain constant. This is special relativity. Speed=Distance/Time. If speed is constant and distance increases by let's say 5, then time must also increase by 5. Then the equation would look like Speed=Distance5/Time5. So time dilates in a gravitational field. So time dilation is proportional to space increase. This is what people mean by "Gravity is just a curvature of space and time".  This is general relativity in a paragraph, simplified with no math but extremely accurate!
Now with this explanation you see that you don't need any higher dimension for this to work! All you need is that the shortest distance between two points must increase! And that speed of light is constant in all reference frames! So please STOP thinking of spacetime as a fabric curving in a hyper-space since there's NO such thing as hyper-space. And don't let analogies fool you, they're just analogies and not what the truth really is.
A: Your intuition is correct. The intrinsic curvature is indeed equivalent to embedding in higher dimensions. Unfortunately most people don't understand the difference between physics and math. The math can describe the intrinsic curvature without referring to higher dimensions. However, from the physics standpoint, the mere ability to calculate things is different from understanding what actually is happening in the physical reality. And if space is curved, then one of two tings is happening. Either (1) there are invisible higher dimensions, in which the curved space is embedded, or (2) space is curved in time while the 4-dimensional spacetime is flat. Because higher dimensions have not been observed, the latter option must be correct. What constraints this puts on General Relativity is a different question. If you feel you truly understand something, never surrender to what others tell you. If pioneers believed what everyone else says, Earth would still be flat.
