I am trying to visualize the curvature of space-time. In almost all of the Yt videos on the topic, it's shown as depression in the space-time fabric. But what does the dimension into which space-time curves represent? If some mass creates more curvature in the fabric, what does it show?
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The visualization of curvature that is shown in the popular accounts of general relativity is completely wrong.
In particular, such visualizations show what is called extrinsic curvature, i.e., how a manifold is curved with respect to the ambient manifold that the given manifold is embedded in. For example, a circle of radius $R$ that lives on a flat plane is extrinsically curved w.r.t. its embedding in the flat plane (with a radius of curvature $R$).
However, the curvature that general relativity deals with is what is called intrinsic curvature of a manifold. It doesn't concern with what is the ambient manifold in which the given manifold is embedded or even whether such an embedding exists. The intrinsic curvature of a manifold, very roughly speaking, characterizes how lines on a manifold that originate as parallel lines diverge from being parallel. For example, if you consider two nearby great circles on a $2-$sphere, they are parallel at the equator of the sphere but as they meet at the pole, they certainly have diverged from being parallel. This intrinsic curvature is mathematically represented by what is called a Riemann curvature tensor. And as I said, this is the curvature that general relativity is concerned with when it describes (true) gravity as curvature of spacetime.
Trying to visualize gravity as the extrinsic curvature of spacetime is wrong, first and foremost, because that's not in alignment with what general relativity tells us. But furthermore, it's not just a white lie that we can tell children so that they can build an intuition until they get to learn the real deal. That is because it is simply misleading. For example, to the best of our knowledge, our spacetime is not embedded in some higher dimensional spacetime. And so, the popular visualization runs into trouble as soon as someone asks, "OK, so I see that this rubber sheet is curved because it curves into this third dimension, what does our spacetime curve into?". Another way in which it is deeply misleading is that it builds a wrong intuition about what counts as curved because what might be extrinsically curved might be completely flat intrinsically and thus would count as flat spacetime as far as gravity is concerned. For example, a cylinder or a circle is extrinsically curved when it's embedded in a higher dimensional flat space but they are intrinsically flat.
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Unfortunately that "model" is complete and utter rubbish. You cannot learn anything from it, so please do not try!
Nobody who understands General Relativity uses it. It does not represent any equations or allow any calculations.
It needs to go away, now. Unfortunately (again) the internet will preserve it forever . . .
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