Necessary conditions for a Penrose diagram?

[bcrowell]

What conditions are necessary if it's to be possible to make a Penrose diagram for a 3+1-dimensional spacetime?

It seems that rotational symmetry is not necessary, since people draw Penrose diagrams for Kerr black holes. If you don't have rotational symmetry, how do you know what 2-surface is represented by a given point on the 2-dimensional diagram? Would this have to be defined on a case-by-case basis?

Conformal flatness doesn't seem to be necessary, since the Schwarzschild spacetime isn't conformally flat.

Asymptotic flatness isn't necessary, because we can make Penrose diagrams for FRW spacetimes.

IIRC Penrose distinguishes strict conformal diagrams from diagrams that aren't strict. (I may be misremembering the word.) How does this factor in?

 

[PeterDonis]

people draw Penrose diagrams for Kerr black holes.

But IIRC they are different for different "cuts" through the spacetime; for example, the diagram for the equatorial plane is different from the diagram for a cut through the "poles" of the hole.

 

[bcrowell]

bump

 

[tom.stoer]

Interesting question; I do not have an answer, but a couple of questions.

1) What is the defining property of a Penrose diagram?
2) What is the mathematical definition of its construction?

Unfortunately I was not able to find Penrose's original paper Conformal treatment of infinity. I would try to answer (1) and (2) as follows: First one "projects" 4-dim spacetime to 2-dim spacetime; then one applies a conformal rescaling. Both mappings must preserve the conformal structure, i.e. map light rays to light rays, preserve the causal relation between any two points i.e. preserve the causal past an future of any two points etc. In addition the conformal rescaling shall bring the 2-dim. spacetime to the standard "diamond".

I guess that the existence of the conformal transformation can be proven quite easily. The main problem for me seems to be the "projection" b/c one must map topological 2-spheres to points. First question which I cannot answer is whether this is valid for arbitrary spacetimes w/o any special symmetry.

 

[bcrowell]

Unfortunately I was not able to find Penrose's original paper Conformal treatment of infinity.

Here are a couple of online sources:
Winitzki, https://sites.google.com/site/winitzki/index/topics-in-general-relativity
http://backreaction.blogspot.com/2009/11/causal-diagrams.html

There is also a discussion in Hartle and Hawking, and also a very extensive but nontechnical one in Penrose's popular-level book Cycles of Time.

I don't have the two books handy, just my notes on them.

First one "projects" 4-dim spacetime to 2-dim spacetime; then one applies a conformal rescaling. Both mappings must preserve the conformal structure, i.e. map light rays to light rays, preserve the causal relation between any two points i.e. preserve the causal past an future of any two points etc. In addition the conformal rescaling shall bring the 2-dim. spacetime to the standard "diamond"

I think there is also an initial step here where you adjoin idealized surfaces at infinity such as $\mathscr{I}^+$. I guess this would typically, but not always, go as far as constructing the maximal extension of the original spacetime.

The diamond isn't a requirement, and it isn't how most spacetimes come out. It's just how Minkowski space comes out.

From Peter Donis's #2, it also sounds like there is the option of taking a 3-slice and then drawing that as 2 dimensions, and that can be used in cases where there is a lower symmetry. In fact, it's not really clear to me whether this is better thought of in general as a projection or a slice. People certainly do customarily describe it as a projection -- they talk about each point as representing a 2-sphere. But that actually violates the condition you suggested that the projection should preserve the light-cone structure. For example, let event A be me, here, right now, and let event B be me at a time that's one second later. Let A* and B* be the corresponding points on a Penrose diagram of Minkowski space. Then B* represents a huge 3-sphere stretching around the universe, most of whose points are outside A's light cone.

 

[bcrowell]

I think I may understand this better now. There is some relevant discussion in section 3.2.2 of the Winitzki book, https://sites.google.com/site/winitzki/index/topics-in-general-relativity .

For example, the Schwarzschild spacetime isn't conformally flat, but we can draw a Penrose diagram for it. What we do is to project out two dimensions, and do so in such a way that lightlike geodesics in the full spacetime still look like lightlike geodesics in the 2-d version. Every 2-dimensional manifold is conformally flat, so we're guaranteed to be able to make a Penrose diagram after that.