Exploring a Conformal Transformation Between 2-D Space-Times

[honeytrap]

I have two 2-dimensional space-times. One of them is flat the other one has not-vanishing curvature (Riemann tensor). But they seem to have a similar global and causal structure.
Of course, because of the 2-dimensional case they are local conformally flat.

I am looking for a relation between them that could explain the similar causal structure and I think that a conformal transformation would be nice.

1) How do I know (prove) whether there exists a (global) conformal transformation between them?
Is there a way to prove that there exists one (I do not need the transformation mapping itself, only the proof of existence)?


2) Are there other global properties of space-times that are worth discussing? What are the
typical global properties (my guess: Horizons, causal light cone structure...what else)?

 

[bcrowell]

By 2-dimensional, you mean 1+1?

I guess conical singularities are not possible in 1+1 dimensions, so there's probably no way to get any singularities of any kind.

In n+1 dimensions with n>1, Penrose diagrams only work if there's symmetry. In 1+1 dimensions, I guess you should be able to represent the entire conformal structure using a Penrose diagram, regardless of symmetry. Have you tried drawing Penrose diagrams?

Are they asymptotically flat?

 

[honeytrap]

By 2-dimensions I mean a 2-dim manifold with Lorentz metric (-,+).

As a mathematician, I am not familiar with the Penrose diagrams, so I didn't draw them.
I thought there might be a necessary requirement/ premise or assumption so that we can conclude a "conformal relation".

But, what kind of information would a Penrose diagram reveal?
-> Provided the Penrose diagrams are identical, would that be a "proof" for a conformal relation (or causal and global similarity) between those two spacetimes?

If so, it would probably make sense to learn about the Penrose diagrams.Here are the two spacetimes/ metrics that seem to have the same global and causal properties:
http://tbf.me/a/OikJJ

 

[honeytrap]

They are not asymptotically flat because of the identification of one coordinate; I call it asymptotically flat under identification. But without the identification they would be both asymptotically flat.

By 2-dimensional, you mean 1+1?

I guess conical singularities are not possible in 1+1 dimensions, so there's probably no way to get any singularities of any kind.

In n+1 dimensions with n>1, Penrose diagrams only work if there's symmetry. In 1+1 dimensions, I guess you should be able to represent the entire conformal structure using a Penrose diagram, regardless of symmetry. Have you tried drawing Penrose diagrams?

Are they asymptotically flat?

 

[honeytrap]

i) Sorry, I posted the wrong metrics. Here are the correct two metrics:
http://tbf.me/a/Bu4jVZii) Here are the two penrose diagrams (I believe they are correct):
http://tbf.me/a/BntYLL

In both diagrams the red line is a horizon (chronology/ cauchy).
Could someone help me read/ compare the diagrams?
Do the diagrams show that both spacetimes have the same causal and global structure (which should be the case)? Why?

THANK YOU!

 

[zonde]

I have two 2-dimensional space-times. One of them is flat the other one has not-vanishing curvature (Riemann tensor). But they seem to have a similar global and causal structure.
Of course, because of the 2-dimensional case they are local conformally flat.

I am looking for a relation between them that could explain the similar causal structure and I think that a conformal transformation would be nice.

1) How do I know (prove) whether there exists a (global) conformal transformation between them?
Is there a way to prove that there exists one (I do not need the transformation mapping itself, only the proof of existence)?

Isn't curvature of 2-dimensional manifold a Gaussian curvature?
And if so then I believe you can't have global transformation between flat and curved space that preserves distances and angles. Or maybe conformal transformation requires only preservation of angles but not distances?

 

[bcrowell]

Isn't curvature of 2-dimensional manifold a Gaussian curvature?

But isn't that only in a Riemannian space?

 

[honeytrap]

Isn't curvature of 2-dimensional manifold a Gaussian curvature?
And if so then I believe you can't have global transformation between flat and curved space that preserves distances and angles. Or maybe conformal transformation requires only preservation of angles but not distances?

Yes, in 2 dimensions the Gaussian curvature (K) can be directly derived from the Riemann curvature (R) by K=-R.
Thus, there is no isometry between those two spaces. But conformal transformations are only angle preserving and do not preserve distances. So I believe that there must be a conformal transformation.
(?)

 

[zonde]

Yes, in 2 dimensions the Gaussian curvature (K) can be directly derived from the Riemann curvature (R) by K=-R.
Thus, there is no isometry between those two spaces. But conformal transformations are only angle preserving and do not preserve distances. So I believe that there must be a conformal transformation.
(?)

Hmm, obviously you can't have conformal transformation between surface of sphere and flat plane so I believe there should be additional conditions to allow for conformal transformation.

Another thought. Before you look for a proof that (under certain conditions) there is conformal transformation between the two it would be reasonable to look if it works for some simple case, right?

 

[zonde]

But isn't that only in a Riemannian space?

I am not sure what is Riemannian space.
I asked about this because I believe I more or less understand what is Gaussian curvature so I tried to tie the question with the thing that I know.

 

[TrickyDicky]

Hmm, obviously you can't have conformal transformation between surface of sphere and flat plane so I believe there should be additional conditions to allow for conformal transformation.

This is false. The surface of a sphere is conformally flat, so a conformal transform. can turn it to a flat plane.

 

[zonde]

This is false. The surface of a sphere is conformally flat, so a conformal transform. can turn it to a flat plane.

Indeed. Turns out this can be done using stereographic projection.