Can Every Spherically Symmetric Spacetime Be Represented by a Penrose Diagram?

[mersecske]

Thanks.

If Penrose-Carter diagram means:

1) compactificated space-time diagram and
2) null-like world lines are 45 degree lines

than, every spherically symmetric space-time has Carer-Penrose diagram?
If Yes, can you cite a paper about proving the existence?

Else:
on the 2D Carter-Penrose diagram only radial motion can be studied,
what about 3D=2+1 Penrose-Carter diagrams? Can you draw it?
The 2D diagram is simply rotated?

 

[JesseM]

Thanks.

If Penrose-Carter diagram means:

1) compactificated space-time diagram and
2) null-like world lines are 45 degree lines

than, every spherically symmetric space-time has Carer-Penrose diagram?

Yes, I found a reference confirming this--see p. 48 of 'Gravitational Collapse and Spacetime Singularities on google books, in the paragraph right before section 2.7.4 the author says:

The structure of infinity for any spherically symmetric spacetime can be depicted by a similar Penrose diagram.

Likewise, p. 20 of http://www.mittag-leffler.se/preprints/0809f/files/IML-0809f-14.pdf talks about how to define the black hole interior region as a complement of the exterior region described in terms of a Penrose diagram, and says:

For spherically symmetric spacetimes arising as solutions of the Cauchy problem for (2), one can show that there always exists a Penrose diagram, and thus, a definition can be formalised along precisely these lines (see [60]).

Here reference [60] is M. Dafermos http://iopscience.iop.org/0264-9381/22/11/019/

If Yes, can you cite a paper about proving the existence?

I think the Dafermos paper above probably proves it, although understanding it would probably require more knowledge of topological reasoning in GR than I have...

on the 2D Carter-Penrose diagram only radial motion can be studied,
what about 3D=2+1 Penrose-Carter diagrams? Can you draw it?
The 2D diagram is simply rotated?

No, it can't be a simple rotation, that would imply that in the maximally extended Schwarzschild solution one could travel around in a circle from the exterior region I on the right to the other exterior region III on the left of the diagram, which is supposed to represent a "different universe"! In fact I would guess (though I don't know this for a fact) that if Penrose diagrams are specific to spherically symmetric spacetimes, then they would always be two dimensional, with the radial dimension shown and the angular coordinates suppressed. It may help to think in terms of the fact that any if you take any spacelike surface through a Penrose diagram (or a Kruskal-Szkeres diagram, which looks basically identical except that it doesn't compress the spacetime to a finite size), which will just be any line closer to the horizontal than 45 degrees, there will be an http://www.bun.kyoto-u.ac.jp/~suchii/embed.diag.html showing the curvature of space in that spacelike surface. This embedding diagram will depict one of the two angular dimensions in addition to the radial dimension depicted on the Penrose/Kruskal-Szekeres diagram. For example, p. 528 of Gravitation by Misner/Thorne/Wheeler shows embedding diagrams for various spacelike slices through the Kruskal-Szekeres diagram:

Each point on a Penrose/Kruskal-Szekeres diagram actually represents a spherical region of space at constant radius (with both angular coordinates allowed to vary), so in the embedding diagrams each point on the spacelike surface becomes a circle (since the embedding diagram only shows 2 of the 3 spatial dimensions)--you can see that a cross-section of any of the above embedding diagrams would be a circle (several of these embedding diagrams illustrate a Schwarzschild wormhole or 'Einstein-Rosen bridge', see also the animations in the section of http://casa.colorado.edu/~ajsh/schww.html titled 'Instability of the Schwarzschild Wormhole'). Here, on the other hand, you can see on the right a Kruskal-Szekeres diagram for a more realistic black hole that forms from a collapsing star, with no white hole interior region or second exterior region on the left:

And here was my guess about what embedding diagrams for spacelike slices through this diagram would look like: