Visualize 2D Intrinsic Curvature of Spacetime (1s+1t) in 3D

[Martian2020]

TL;DR Summary

I want to see correct visualization of curvature of spacetime (1d space+time) in 3d for spherical object of uniform density. Kind of Minkowski diagram.

Via web search found https://www.physicsforums.com/threads/what-dimension-does-space-time-curve-in.852103/
Read it and watched two videos mentioned:



I understand we cannot perceive 5D ;-), so extrinsic visualization of maximum of 2D intrinsic curvature is possible. So time+1d space is all we can see, and it is fine for sphere due to symmetry. I know not all intrinsic can be seen as extrinsic (is my language correct here?), however I understand spacetime curvature resulting from spherical mass can be correctly embedded, right?

In linked videos general idea of object going "straight"/geodesic line in spacetime is clear. However, I would like to find/see more informative visualization of spacetime curvature of spherical body with uniform density where it is clearly seen:
1) change from "gravitational force formula" outside a sphere to "falling" inside.
2) going though center and "raising" to opposite side of the sphere; extending it to several cycles
3) like on Minkowski diagrams with c*t axis being able to visualize "steepness" of curvature for escape velosity > speed of light

I was not able to find one via web search. Best I found was thread here mentioned above.
Visualization readily available is very good, any help with making one is appreciated too!

As a side note, the below visualizations cannot obviously show full spacetime (not 5D), but maybe they are not only useful tools and analogies but also correct visualizations of something?

P.S. I am not sure it is "basic", I am new here.

 

[A.T.]

In linked videos general idea of object going "straight"/geodesic line in spacetime is clear. However, I would like to find/see more informative visualization of spacetime curvature of spherical body with uniform density where it is clearly seen:
1) change from "gravitational force formula" outside a sphere to "falling" inside.
2) going though center and "raising" to opposite side of the sphere; extending it to several cycles
3) like on Minkowski diagrams with c*t axis being able to visualize "steepness" of curvature for escape velosity > speed of light

I think you are looking for something like this:

On the global scale spacetime is intrinsically curved (as shown below). It cannot be rolled out like the cone in the video above, which approximates just a small radial range:

The red path is the geodesic world-line of a free falling object, that oscillates through a tunnel through a spherical mass. Note that the geodesic always deviates towards the "more stretched" proper time, or towards greater gravitational time dilation. Gravitational time dilation has an extreme point at the center of the mass (gradient is zero), so there is no gravity there (but the maximal gravitational time dilation).

There is an interactive Flash version of that here (note that Flash is blocked by default by modern browsers):
http://www.adamtoons.de/physics/relativity.html

See also:
http://www.relativitet.se/Webtheses/tes.pdf
And other papers by Jonsson:
http://www.relativitet.se/articles.htmlBut note that this is not the usual space-coordiantetime diagram (Minkowski), but a space-propertime diagram as used in this book:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/mode/2up

Howerver, in the papers by Jonsson linked above, you will find similar embeddings for Minkowski diagrams.

As a side note, the below visualizations cannot obviously show full spacetime (not 5D), but maybe they are not only useful tools and analogies but also correct visualizations of something?

View attachment 273141

See here:
https://www.physicsforums.com/threa...-visualization-of-gravity.726837/post-4597121

View attachment 273142

See here:
https://www.physicsforums.com/threa...curvature-more-accurately.753672/post-4747390

 

[Martian2020]

Note that the geodesic always deviates towards the "more stretched" proper time, or towards greater gravitational time dilation.

http://www.relativitet.se/Webtheses/tes.pdf

Thank you!
It would take some time to go though resources you mentioned. In http://www.relativitet.se/Webtheses/tes.pdf I saw visualization like you included on page 9.
If you have a minute, could you help me clarify:
1) per quotation above, you mean "deviate" in our 3D space of cause (as geodesic is "straight")? Still it looks like you said only on going into sphere, on way back it does not look so...
2) If I make time scale more "dense" (fit more 8 minutes), looks to me I will be able to see several oscillations on one side w/out seeing through transparency. Is it correct way to do?
3) I could not quickly find visualization for space-coordianate time. Is it much different?
4) How can I see escape velocity on your visualization? Or is it possible only on space-coordianate time one?

 

[Keith_McClary]

I once saw a "coffee table" style picture book by Wheeler(?) that explained Einstein's equation visually, but I can't find it now.

 

[robphy]

I once saw a "coffee table" style picture book by Wheeler(?) that explained Einstein's equation visually, but I can't find it now.

I think you are referring to
A Journey into Gravity and Spacetime (Scientific American Library) by John A. Wheeler (1990)
https://www.amazon.com/dp/0716750163/?tag=pfamazon01-20
a review:
https://physicstoday.scitation.org/doi/10.1063/1.2810082

 

[Keith_McClary]

A Journey into Gravity and Spacetime (Scientific American Library

That looks like it!

 

[A.T.]

It would take some time to go though resources you mentioned. In http://www.relativitet.se/Webtheses/tes.pdf

Here are more papers:
http://www.relativitet.se/articles.html

1) per quotation above, you mean "deviate" in our 3D space of cause (as geodesic is "straight")? Still it looks like you said only on going into sphere, on way back it does not look so...

The first video you posted "How Gravity Makes Things Fall", around 3:30 shows what happens on the way up. You can approximate any small radial range of the curved surface in my post as such a cone section.

2) If I make time scale more "dense" (fit more 8 minutes), looks to me I will be able to see several oscillations on one side w/out seeing through transparency. Is it correct way to do?

The embedding-method has a free parameter, that determines how tight you roll the space time diagram. This will also affect the shape of the bulge. You cannot just change one axis scale on the same shape to describe the same situation.

3) I could not quickly find visualization for space-coordianate time. Is it much different?

Try this:
http://www.relativitet.se/Webarticles/2005AJP-Jonsson73p248.pdf
http://www.relativitet.se/Webarticles/2001GRG-Jonsson33p1207.pdf

Ths space-propertime embedding is described here in chapter 6:
http://www.relativitet.se/Webtheses/lic.pdf

4) How can I see escape velocity on your visualization? Or is it possible only on space-coordianate time one?

I'm not sure if you can directly see it in either of them. The geodesic can get quite complex on these surfaces, so it's not obvious what initial direction (in space-time) you need to escape.

 

[Martian2020]

But note that this is not the usual space-coordiantetime diagram (Minkowski), but a space-propertime diagram as used in this book:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/mode/2up

http://www.sciencechatforum.com/viewtopic.php?f=2&t=24943
"In his book "Relativity Visualized", Lewis Carroll Epstein introduced a kind of spacetime diagram called a Space-Proper-time diagram. It has a few issues and is hence not often used in the literature, but for a quick grasp of special relativity theory, there is none better."
These diagrams could be significantly different from "usual" ones. I want to understand them, in the meanwhile I also wait maybe my question maybe visualized in "square" coordinates.

 

[Martian2020]

I'm not sure if you can directly see it in either of them. The geodesic can get quite complex on these surfaces, so it's not obvious what initial direction (in space-time) you need to escape.

I wrote previous response not seeing these answers (not updated page in browser?). But I decided not to delete as it contains interesting link to discussion of Epstein diagrams. I'm going to study your links further now. Thank you.

 

[Martian2020]

Try this:
http://www.relativitet.se/Webarticles/2005AJP-Jonsson73p248.pdf
http://www.relativitet.se/Webarticles/2001GRG-Jonsson33p1207.pdf
Ths space-propertime embedding is described here in chapter 6:
http://www.relativitet.se/Webtheses/lic.pdf

You link lot of stuff of Rickard Jonsson. Could you explain why on link 1 to explain gravity, shape have larger radius closer to center, but on link 2 to explain time dilation shape have smaller radius closer to center?
http://www.relativitet.se/spacetime1.html
"Some properties of the full theory will be lost due to the trick, but many of the truly important features will remain."
http://www.relativitet.se/spacetime2.html
"The technique used here gives images that requires a bit more knowledge to interprete correctly than those of spacetime1. On the other hand, when interpreted correctly, all the features of the true spacetime of a line will be incorporated."
This seems contradictory to myself...re-visited it twice.
Maybe as noted in the accompanying text 2nd is more correct... But how then see objects fall on 2nd shape?

 

[A.T.]

You link lot of stuff of Rickard Jonsson. Could you explain why on link 1 to explain gravity, shape have larger radius closer to center, but on link 2 to explain time dilation shape have smaller radius closer to center?

It's two different types of space-coordinate time embedding, described in the two papers I linked.

The nice thing about the space-propertime embedding (3rd link, chapter 6) is that you can show both aspects in a single diagram. That's because space-propertime has an Euclidean metric, not the pseudo-Euclidean one.

 

[Martian2020]

It's two different types of space-coordinate time embedding, described in the two papers I linked.

The nice thing about the space-propertime embedding (3rd link, chapter 6) is that you can show both aspects in a single diagram. That's because space-propertime has an Euclidean metric, not the pseudo-Euclidean one.

I skimmed though chapter 6 (The Epstein-Berg way), looking at visualizations. I did not find one on which both fall and dilation was displayed. It could have been discussed... And also for Fig 6.3 "points does not correspond to events" (could not copy correctly from pdf, my wording). Not easy to understand that way...

 

[A.T.]

I skimmed though chapter 6 (The Epstein-Berg way), looking at visualizations. I did not find one on which both fall and dilation was displayed.

See the diagram in post #2, and make sure you understand space-propertime diagrams:

One simple to way to visualize this, is to see everything advancing in space-propertime at the same rate, because the path-length here is the coordiante time.

The time dilation is the ratio of the displacement along the propertime axis and the path-length (coordinate time). This includes both: kinetic time dilation (trough projection onto the propertime axis) and gravitational time dilation (trough stretching of the propertime dimension -> more path-length (coordinate time) for the same propertime difference)

You can download the full book where space-propertime diagrams are introduced here.
https://archive.org/download/L.Epst...isualized [elem txt] (1994, Insight)_text.pdf

"points does not correspond to events" (could not copy correctly from pdf, my wording). Not easy to understand that way...

Yes, if you are dealing with two objects, the crossing of their paths in space-propertime is not a meeting. They meet if they are at the same spatial-coordiante after the same path-length (coordinate time). Their offset along the propertime axis is then the differential aging.

 

[Frodo]

Summary:: I want to see correct visualization of curvature of spacetime (1d space+time) in 3d for spherical object of uniform density. Kind of Minkowski diagram.

Imagine throwing a ball to a person 5 feet away. The ball rises 4 feet and the total flight time is 1 second.

Viewed in space, the ball travels in a parabola, rising 4 feet and traveling 5 feet horizontally.

But viewed in spacetime you need to add in a time axis and the distance between the start and end points is 1 second. You need to convert 1 second into a distance - it is 186,000 miles. Remember that the speed of light, c, is the conversion factor between distance and time: 1 second = 186,000 miles.

So, you have a "box" which is 5 feet wide, 4 feet high and 186,000 miles long.

The ball travels from the front, lower left corner of the box to the rear, lower right corner of the box; and rises 4 feet during its journey. As you can see its path is virtually straight.

You can get a feeling for how much Earth bends space(time) - by about 4 feet in 186,000 miles.

 

[Martian2020]

As you can see its path is virtually straight.
You can get a feeling for how much Earth bends space(time) - by about 4 feet in 186,000 miles.

That is why I asked for sphere, not Earth specifically. I did not want to draw such long boxes ;-)
I've read time dilation in the center of Earth is some 0.000...%