Imagining spacetime curvature more accurately

[HowardHughes]

I am intrigued to see what spacetime curvature is like in reality. Most images or ways to imagine it tend to look at spacetime as a fabric which it is not precisely. So how would be best to imagine it... Do any of the picture demonstrate this? What is the best way to imagine it?

 

[Dale]

Our forum member A.T. has some good visualizations. The important thing is to include time in the visualization, which none of the pictures you linked do.

 

[pervect]

To add just a few things to Dale's remarks:

1) It's a good idea to label your diagrams , so you know what's they are representing. Diagrams without any explanation are ambiguous at best.

2) Before you try to represent curved space-time, you probably want to know how to represent non-curved space-time. The usual technique is a 2 dimensional diagram called a space-time diagram, which contains 1 space dimension and one time dimension. It's worth doing some research if you aren't familiar with space-time diagrams.

3) If space-time diagrams are too confusing, you might start out with timelines as a base as to how to represent time. The general idea between any of the diagrams that we've been talking about is the notion that there is a 1:1 correspondence between points on the diagram, and something in reality.

4) Without going deeply into the mathematical details of curvature, we can say that the surface of a sphere is "curved", and that a plane is "not curved". The popular notion of curvature is rather vague and ambiguous, the particular notion of curvature that general relativity is concerned with is known as "intrinsic curvature". In particular, the spheres are intrinsically curved, and cylinders are not intrinsically curved, which may be confusing if one is not using the applicable definition of "curved".

 

[PeterDonis]

In particular, the spheres are extrinsically curved, and cylinders are not extrinsically curved, which may be confusing if one is not using the applicable definition of "curved".

I think you mean "intrinsically" here.

 

[pervect]

I think you mean "intrinsically" here.

Ooops- yes, fixed.

 

[A.T.]

I am intrigued to see what spacetime curvature is like in reality. Most images or ways to imagine it tend to look at spacetime as a fabric which it is not precisely. So how would be best to imagine it... Do any of the picture demonstrate this? What is the best way to imagine it?

The pictures you posted try to show curved space, not space-time. As you see there no time axis in them, just 3 spatial dimensions.

They are also incorrect: 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.

And this visualization problem gets even worse if you include the 4th dimension (time), which is crucial to understand gravity in General Relativity. One way around this is to reduce the number of dimensions you show to just 2. A curved 2D surface can be embedded in non-curved 3D space, while preserving its correct geometry (distances within the surface). This way you have a more correct but limited picture of the distorted geometry. You can choose between 2 spatial, or 1 spatial & time dimension in one diagram. Here some examples:

https://www.youtube.com/watch?v=DdC0QN6f3G4

http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

http://www.relativitet.se/spacetime1.html

http://www.adamtoons.de/physics/gravitation.swf

 

[xox]

The pictures you posted try to show curved space, not space-time. As you see there no time axis in them, just 3 spatial dimensions.

They are also incorrect: 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.

And this visualization problem gets even worse if you include the 4th dimension (time), which is crucial to understand gravity in General Relativity. One way around this is to reduce the number of dimensions you show to just 2. A curved 2D surface can be embedded in non-curved 3D space, while preserving its correct geometry (distances within the surface). This way you have a more correct but limited picture of the distorted geometry. You can choose between 2 spatial, or 1 spatial & time dimension in one diagram. Here some examples:

https://www.youtube.com/watch?v=DdC0QN6f3G4

http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

http://www.relativitet.se/spacetime1.html

http://www.adamtoons.de/physics/gravitation.swf

Very nice illustration, congratulations.

 

[MikeGomez]

https://www.youtube.com/watch?v=DdC0QN6f3G4

A.T. This video confuses me. Before :28 the horizontal axis is marked ‘time’ and the vertical axis is marked ‘space’.

At :28 when the grid bends up to make the fall worldline straight, the labels all disappear.

Then the labels reappear again at :32, and the ‘time’ axis and ‘space’ axis have transformed along with the grid. But what did they transform into? What are the new horizontal and vertical axes representing?

Is time still horizontal and space still vertical? From the graphics it looks like maybe something like a ‘local’ coordinate system has been transformed inside a more ‘global’ coordinate system, but I can only guess and guessing isn’t good.

Is there any way you can label the ‘global’ axes during and after the straightening period from :28 onward?

 

[A.T.]

Is time still horizontal and space still vertical?

On the cone like diagram time is circumferential and space is radial, as indicated by the labels. Space here means one spatial dimension, away form the the mass.

Is there any way you can label the ‘global’ axes during and after the straightening period from :28 onward?

The meaning of the grid lines doesn't change in that transition, just the distances between coordinates. Look at 1:00 where both diagrams are shown side by side with labels. It should be pretty obvious what corresponds to what.

From the graphics it looks like maybe something like a ‘local’ coordinate system has been transformed inside a more ‘global’ coordinate system,

It is not so much about local vs. global, but rather inertial vs. non-inertial. The distorted grid is one way to model a non-inertial reference frame, like the rest frame of the tree branch.

Here a nice diagram by DrGreg:

More info in his post:
https://www.physicsforums.com/showthread.php?p=4281670&postcount=20

 

[MikeGomez]

Thanks A.T. I will take a little time and try to grok it, but have patience as I'll probably be back with more questions.

 

[homeomorphic]

Here's one way to think of curvature, which is Riemann's original idea. We can make sense of curvature for surfaces (technicallly, Gaussian curvature is what we want). For a surface, there isn't really a directional quality to curvature. It's just a function of what point you are at. Some number at each point. But for a higher-dimensional space, there will be different curvature in all directions (sectional curvature). To measure this, you can just pick two directions, form a surface tangent to them and measure its curvature. This surface should be a geodesic surface (it has curvature, but sort of "lies flat" in the space it lives in). So, in 3 dimensions, you'd need 3 of these surfaces to captures all this information. In 4 dimensions, you need 4 choose 2 or 6 surfaces (one for each coordinate plane--you pick whatever tangent plane you want, but the coordinate planes suffice to capture everything). The Riemann curvature tensor sort of puts all these sectional curvatures together in one package. Einstein's equation uses some gadget that is derived from this curvature tensor, so that mass/energy gives partial information about it.

 

[homeomorphic]

Except, come to think of it, I think you need more than just the coordinate plane directions. But the idea is that it's just curvature of surfaces, except there are more directions.