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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2$\begingroup$ youtube.com/watch?v=wrwgIjBUYVc $\endgroup$– Stéphane RollandinCommented Mar 8, 2022 at 9:49
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2$\begingroup$ Spacetime doesn't curve into an extra dimension. It curves into itself. The models you've seen on YT are (rather poor) representations of the true picture. $\endgroup$– PraharCommented Mar 8, 2022 at 9:54
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2$\begingroup$ Does this answer your question? Bowling ball on a rubber sheet $\endgroup$– ohneValCommented Mar 8, 2022 at 10:34
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$\begingroup$ @Prahar I thought the "extra dimension" that the fabric curves down into was just the gravitational potential energy? $\endgroup$– electronpusherCommented Mar 8, 2022 at 14:11
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1$\begingroup$ Does this answer your question? Better explanation of the common general relativity illustration (stretched sheet of fabric) $\endgroup$– John RennieCommented Mar 8, 2022 at 15:10
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2 Answers
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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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$\begingroup$ @m4r35n357 I said it is "not just a white lie". $\endgroup$– user87745Commented Mar 8, 2022 at 9:59
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$\begingroup$ OK pardon me, doing too much ATM. Deleted. $\endgroup$ Commented Mar 8, 2022 at 10:01
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$\begingroup$ Can you please elaborate on the extrinsic curvature example (in 2nd para)? Thanks very much! $\endgroup$ Commented Mar 10, 2022 at 3:52
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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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$\begingroup$ Actually, those who understand GR properly actually do us it quite often as it is quite an informative and succinct way to represent geometries. The difference is that those who understand GR properly know not to read too much into such pictures. $\endgroup$– PraharCommented Mar 8, 2022 at 9:55
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$\begingroup$ Then they should be ashamed ;) If they are not "reading too much into such pictures", then what sense are they "using" it? $\endgroup$ Commented Mar 8, 2022 at 9:57
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$\begingroup$ I don't think so. I think it is perfectly acceptable to use crude tools for specific purposes as long as you understand the limitations of those tools. It is in the same vein that the images are used. $\endgroup$– PraharCommented Mar 8, 2022 at 9:58
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2$\begingroup$ To add to what @Prahar is saying, it's not enough for most working physicists to just know the equations. The equations obviously give you the rigorous and exact description of what is going on, but are also very complicated and abstract. A successful physicist also needs physical intuition, which comes from solving problems, building toy models, and yes, using crude visualizations that capture some important aspect of reality. It is part of a good physics education to learn how to use such tools appropriately, which aren't rigorous but are often essential for building intuition. $\endgroup$– AndrewCommented Mar 8, 2022 at 15:15
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1$\begingroup$ @m4r35n357 - Figure 2 of this paper - 202.38.64.11/~jmy/documents/publications/… describes the near horizon geometry of a Kerr black hole close to extremality. The equations in the paper describe all the details of course, but the diagram is a clean summary of all the results. $\endgroup$– PraharCommented Mar 9, 2022 at 11:38