Is there a way to visualize curvature of time? I mean curvature of space is both mathematically and physically is comprehensible. I have doubt about curvature of time.
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1$\begingroup$ More on visualizing GR. $\endgroup$– Qmechanic ♦Commented Oct 7, 2022 at 9:01
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$\begingroup$ Hi! I am new to GR. I get that a space-time curve can be parameterized by proper time. Can you expand on what you mean by "curvature of time" in context of a space-time curve? $\endgroup$– GalenCommented Oct 7, 2022 at 18:56
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$\begingroup$ The following paper may be useful: Rickard M. Jonsson, "Visualizing curved spacetime" $\endgroup$– RuslanCommented Oct 7, 2022 at 23:50
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$\begingroup$ The radial curvature is visualized with Flamm's paraboloid, while the temporal curvature can be visualized in a similar way, see flamm.yukterez.net where the blue curve is the spatial curvature and the red one the temporal. $\endgroup$– YukterezCommented Oct 10, 2022 at 3:58
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3 Answers
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If you are thinking of spacetime as a product of a three-dimensional space $S$ with one-dimensional time $T$, then time is necessarily flat, because all one-dimensional manifolds are flat. So no, there is no way to visualize the curvature of time separately from the curvature of space, because there is no such thing as the curvature of time separately from the curvature of space.
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$\begingroup$ Why do you say that all 1D manifolds are flat? $\endgroup$– basicsCommented Oct 7, 2022 at 12:46
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3$\begingroup$ @basics: Because it is true. The space of two-forms on a one-dimensional manifold is zero-dimensional. $\endgroup$– WillOCommented Oct 7, 2022 at 12:48
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$\begingroup$ You're right. I was thinking at lines with curvature, but since they're lines they are flat. $\endgroup$– basicsCommented Oct 7, 2022 at 19:42
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$\begingroup$ @basics A curved line is isometric to a straight line. $\endgroup$– SandejoCommented Oct 8, 2022 at 2:00
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Here's one simple visualization. Imagine that the surface of the earth is spacetime. Time points north. There is one space dimension, which points east-west.
Imagine two objects with worldlines that point north. Such worldlines are geodesics, meaning the objects are unaccelerated. At the "equator" time, they are at rest with respect to each other, in the sense that their spatial separation is (instantaneously) not changing with respect to time. As we follow the worldlines farther and farther north, we find that their spatial separation begins to decrease! This effect arises purely due to curvature, and the interpretation is that gravity is pulling these objects together.
Spacetime isn't really a sphere, but hopefully this gives you an idea for what it means for spacetime to be curved.
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$\begingroup$ I would expect gravity in SurfaceWorld to pull objects along lines of latitude. What you have described sounds more like space itself collapsing...a Big Crunch scenario. $\endgroup$ Commented Oct 7, 2022 at 18:29
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$\begingroup$ There's no distinction between space collapsing and everything being pulled together. They just correspond to different coordinates. For example you can describe a black hole such that "space is contracting" (e.g. en.wikipedia.org/wiki/Lemaitre_coordinates ). But yes, spacetime being globally a sphere isn't realistic. You could still imagine the same picture with just a section of a sphere. $\endgroup$– StenCommented Oct 7, 2022 at 19:33
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Space and time are both concepts strongly related to an Euclidean vision of the world around us: in the proper sense they do not curve singularly. It's spacetime that curves and doing so it determines the properties and characteristics of the universe.
I think it's important to stress this point in order to visualize correctly the curvature of spacetime. Other than that, I haven't really understood your question.
