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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?

Qmechanic
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Vedansh Bodkhe
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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 . . .

m4r35n357
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  • 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. – Prahar Mar 08 '22 at 09:55
  • Then they should be ashamed ;) If they are not "reading too much into such pictures", then what sense are they "using" it? – m4r35n357 Mar 08 '22 at 09:57
  • 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. – Prahar Mar 08 '22 at 09:58
  • My point is that newbies, by definition, do not understand the limitations of those tools! – m4r35n357 Mar 08 '22 at 09:59
  • I'm not talking about newbies. You mentioned in your answer that "Nobody who understands General Relativity uses it." and I'm pointing out that this is false. People who understand GR do use it to communicate with each other. I never said anything about using them to teach newbies. – Prahar Mar 08 '22 at 10:00
  • I'm sure the OP will appreciate that distinction ;) – m4r35n357 Mar 08 '22 at 10:01
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    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. – Andrew Mar 08 '22 at 15:15
  • @Prahar I am now intrigued. OK I am just a learner here but I haven't heard of any positive applications for the rubber sheet. Perhaps you could give a brief pointer. I trust you are not referring to https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm's_paraboloid. – m4r35n357 Mar 09 '22 at 09:50
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    @m4r35n357 - Figure 2 of this paper - http://202.38.64.11/~jmy/documents/publications/Bardeen1972apj178_347.pdf 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. – Prahar Mar 09 '22 at 11:38
  • @Prahar thanks for the link! My problem with this is that, as I understand it, the embedding is a spacelike surface (t = constant, from the diagram), so I don't see how one can draw any conclusions regarding motion! – m4r35n357 Mar 09 '22 at 13:00
  • You can't - at least not in that diagram. As I said earlier, the important thing is to understand the limitations of these sort of diagrams. The goal of that particular figure is to give an explanation for the near horizon geometry at a particular instant of time and as a function of $a$ (a parameter in the Kerr metric). Each diagram has its own particular applications and should not be extrapolated beyond it! A good physicist must learn what conclusions one can and cannot draw from any given drawing. – Prahar Mar 09 '22 at 13:05
  • Motion of particles can be described by drawing geodesics in that same diagram. If you want to show time evolution of the geometry itself you can convert it to a movie and that works too. – Prahar Mar 09 '22 at 13:07
  • The rubber sheet diagram (embedding diagram) is not the only type of drawings people use in GR to depict spacetimes. For instance, Penrose-Carter diagrams are used a lot to represent the geometries of spacetimes as well. That, too has its limitations. A Penrose-Carter diagram accurately depicts causal relationships, but not distances. Those diagrams never have any equations and yet they are useful to gain intuition. – Prahar Mar 09 '22 at 13:09
  • Spacelike geodesics, yes. (In fact I have got so far as to simulate Kerr geodesics, but not to use the simulations to do any significant investigations: https://www.youtube.com/playlist?list=PLvGnzGhIWTGRuIsHzDLdeWS4GSZ9CIY-b) – m4r35n357 Mar 09 '22 at 13:11
  • Of course, I am not arguing against Penrose diagrams! – m4r35n357 Mar 09 '22 at 13:16