B Accurate Visualization of GR - Science Clic

[frost_zero]

If I had a nickel every time someone depicted curvature of space-time as a marble on rubber membrane I would have a lot of nickels however that description is not accurate; A channel called science clic has the most accurate visualization of bend of space-time I have ever seen however if anyone knows a more accurate version then do suggest it.

 

[PeroK]

however if anyone knows a more accurate version then do suggest it.

If we are talking about the spacetime outside a spherical mass $M$, such as the Earth, then most accurate of all is:
$$ds^2 = -\bigg (1 - \frac{2M}{r} \bigg )dt^2 + \bigg (1 - \frac{2M}{r} \bigg )^{-1}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$

 

[Ibix]

There's quite a confusion of ideas there. It's not really clear to me what the grid lines that are being drawn are supposed to represent - they appear to be based off the worldline of the apple (perhaps raindrop coordinates?). But if that's the case they're not showing curvature of spacetime so much as path curvature on a static background. And showing the gridlines changing over time seems misleading to me unless you're clear they're paths of an arbitrary object, rather than a visualisation of spacetime.

A better version of the falling apple is our own @A.T.'s video:I believe he derived it from Lewis Carroll Epstein's book Relativity Visualised, which means the approach dates back to 1985 at least.

Fundamentally, the problem is that visualisations of curved spacetime that are both honest and intuitive are really hard to do.

 

[vanhees71]

The only accurate visualization of curved spacetime I know is to just look out at the real universe.

 

[Ibix]

It's not really clear to me what the grid lines that are being drawn are supposed to represent

Also note the sleight of hand in having the apple hit the Earth and stopping the animation there. What would happen to his gridlines if the apple were in an open orbit, apparently passing "through" the center of the Earth (actually, displaced in the suppressed third spatial direction) and heading back out to infinity?

I think Epstein and A.T. have diagrams for that, actually, but they're somewhat more complex than the video I posted above.

 

[A.T.]

If I had a nickel every time someone depicted curvature of space-time as a marble on rubber membrane I would have a lot of nickels however that description is not accurate; A channel called science clic has the most accurate visualization of bend of space-time I have ever seen however if anyone knows a more accurate version then do suggest it.

We have discussed that video already here. The rubber membrane is bad, but the video above has similar issues:
https://www.physicsforums.com/threa...-what-is-holding-us-down.1004751/post-6513955

 

[pervect]

It's not a "visualization", but one can regard GR as drawing space-time diagrams on something that is curved, for example a sphere (the 2d surface of a globe), rather than a plane (such as the usual flat sheet of paper). AT's visulalizations are of this general type, for instance.

I'm afraid I haven't looked at the video overmuch, but their approach seems similar from the little bit I did look at it. I haven't read the thread that was mentioned though.

There are the following issues and limitations. 2d space-time diagrams can handle only one spatial dimension and one time dimension. So, one is unfortunately limited to a very simple world, where there is only one spatial dimension (and time). While it is possible to draw a 3 dimensional space-time diagrams, it's more difficult to find a curved 3d surface on which to draw them. One could, for example, imagine the curved 3d surface of a 4d hypersphere, but I'm not sure this is helpful - I would rather do the math than try to visualize that personally.

What is meant by a curved surface isn't really precise without some math, but a sphere is a good example, being one of the simplest and hopefully being familiar. A basic introduction to curvature in a mathematically useful way is pretty complex, unfortunately. In particular , it's hard to explain why a cylinder is not curved in the sense that I'm talking about, while a sphere is.

To get any mileage out of this approach, one needs to know how to draw and interpret space-time diagrams. Additionally, some knowledge about spherical trignometry is helpful. In spherical trignometry, one draws the spherical triangles with great circles. Great circles on a sphere are more or less the equivalent of a 'straight line' in Euclidean geometry, the great circle being mostly the path of shortest length connecting two points that lies entirely on the sphere (the 2d surface of the 3d globe). Antipodal points complicate this slightly.

 

[robphy]

I think that this may be a promising approach to approximate what is going on:

https://www.spacetimetravel.org/sectormodels1/sectormodels1.html
Sector models - A toolkit for teaching general relativity: I. Curved spaces and spacetimes
Corvin Zahn, Ute Kraus, May 1, 2014

Sector models—A toolkit for teaching general relativity: I. Curved spaces and spacetimes
C Zahn and U Kraus 2014 Eur. J. Phys. 35 055020
https://doi.org/10.1088/0143-0807/35/5/055020
( also https://arxiv.org/abs/1405.0323 )

You can use the above to find the continuation articles: II and III .

The central idea of this introduction to general relativity is the use of so-called sector models. Sector models describe curved spaces the Regge calculus way by subdivision into blocks with euclidean geometry. This procedure is similar to the approximation of a curved surface by flat triangles. We outline a workshop for high school and undergraduate students that introduces the notion of curved space by means of sector models of black holes. We further describe the extension to sector models of curved spacetimes. The spacetime models are suitable for learners with a basic knowledge of special relativity. The teaching materials presented in this paper are available online for teaching purposes at www.spacetimetravel.org.

They were the hosts of the conference I attended
https://www.physicsforums.com/threa...-as-a-challenge-for-physics-education.966053/

Since these sectors are piecewise-flat, I think it should be possible to use my light-clock diamonds with them.

 

[A.T.]

While it is possible to draw a 3 dimensional space-time diagrams, it's more difficult to find a curved 3d surface on which to draw them.

That is exactly the problem with this video and similar distorted 3D grid visualizations: They are claimed to be more accurate, because they show more dimensions than 2D surfaces embedded in 3D. But that actually prevents them from showing the key geometric aspects (intrinsic curvature, geodesics) correctly.

As I wrote previously:

You cannot correctly show the curved 3D space around a mass with a distorted 3D-grid that is embedded in non-curved 3D space (the illustration). The shown distorted 3D-grid still encompasses the same total volume as would a non-distorted 3D-grid with the same outer boundary. But in actual curved 3D-space around a mass there is more spatial volume enclosed than in flat space of the same outer boundary.

So basically the 2D sheet with a dent or hill is actually a better representation of the spatial geometry than those distorted grids, because it has more surface area than a flat sheet would have. But spatial geometry doesn't explain gravity.