Conditions for existence of Penrose diagrams

Question

A Penrose diagram (also known as a conformal diagram or Carter-Penrose diagram) is a technique for visualizing the causal (light-cone) structure of a 3+1-dimensional manifold. Usually the diagram is two-dimensional, but it is also possible to have more dimensions. My question is whether there are definite conditions for the existence of a Penrose diagram for a given spacetime, and if so, what they are.

It's not clear to me that this question has a definite answer, partly because the construction of these diagrams seems to be more like a body of techniques than a systematic theory. There does not seem to be any widely recognized definition of exactly what is meant by a Penrose diagram. Penrose,[Penrose 2010] decades after originating the idea, says he wants to distinguish "strict" from "schematic" diagrams. To explain the distinction he gives a reference to Carter 1966, which, unfortunately, does not use those terms and does not seem to make any such distinction.

My understanding of the process is that it's something like this. We start with some given spacetime, then:

1. Make an $n$-dimensional section or projection, where usually, but not always, $n=2$.
2. Do a conformal transformation to reduce the resulting manifold to a flat one of finite size.
3. Adjoin idealized surfaces and points at infinity.

At step 1, we want to take advantage of any symmetries, such as rotational symmetry, so that the final result will be informative, be representative of the whole spacetime, and accurately depict causal relationships in the original spacetime. At this step we also need to make sure that lightlike geodesics in the original space correspond properly to lightlike geodesics in the submanifold.

Step 2 is only possible if our $n$-dimensional space is conformally flat, but this is automatic in the usual case where $n=2$. In many cases of interest, the original 3+1-dimensional manifold is also conformally flat (which in four dimensions is equivalent to vanishing of the Weyl tensor).

Can anyone shed any light on the distinction between strict and conformal Penrose diagrams, and the conditions for their existence? From the sketch above, it seems to me that $n=2$ diagrams would always exist, but they might not be useful and informative. Penrose 2010 says that exact spherical symmetry is needed for strict conformal diagrams but not for schematic ones, but it's not clear to me why this is a requirement, or what would be so special about spherical symmetry as opposed to any other symmetry.

Carter 1966 - "Complete Analytic Extension of the Symmetry Axis of Kerr's Solution of Einstein's Equations," Phys. Rev. 141, 1242, http://luth2.obspm.fr/~luthier/carter/trav/Carter66.pdf

Penrose 2010 - Cycles of time: an extraordinary new view of the universe, Bodley Head, p. 106

Answer 1

There is no definitive answer to your specific question, so I'm going to talk around the topic and hope that it's informative.

As you've noted the classical examples all basically look like special cases. This isn't helped by, pretty much all, books and articles using the same examples. There are a few common threads (which you've noted) and those threads are exemplified in Carter's paper.

My guess, and this really is a guess, is that Penrose is pointing to those threads. My experience of Penrose's writing, particularly on these kinds of "meta-techniques", is that he does not provide strict definitions and is quite happy to point towards something while giving details in specific cases. If I was him (which I'm not) I'd do this because I thought something was interesting but had other things I wanted to study.

The "strict" vs "schematic" differentiation might very well be an example of this. I haven't found that terminology anywhere else in the literature on this kind of thing... and I did my PhD on it.

My best advice is to shrug your shoulders, note that it is interesting that complicated global conformal structure can be accurately represented in two-dimensional drawings and then to move on.

Why? Because this type of question is what birthed a subfield of General Relativity called "Boundary Constructions". This is a subfield that divides the community. Some think it's great, others not so much. I once had a paper recommended for rejection on the sole basis that (para-phrasing slightly) "the field should be allowed to die in this generation as it is an exercise in academic abstraction." (At the time I thought this reviewer probably also objects to category theory for the same reason.)

It is no overreach to say that most, if not all modern studies in GR have been influenced by Penrose diagrams and the even more important Penrose embeddings. For example, AdS/CFT is based on the conformal boundary. The conformal boundary is an example of the boundary of a Penrose embedding. Using Penrose embeddings Penrose gave, I think, the first geometric interpretation of the BMS group and super-translations. Even now one of the fundamental areas of study in numerical relativity is to do with propagation "at infinity" defined, you guessed it, by Penrose embeddings. If you look closely enough at a research area you'll see Penrose diagrams and embeddings influencing it.

But Penrose embeddings (which have a definition unlike Penrose diagrams) are only defined for weakly asymptotically flat space-times. Moreover, if you want certain fields to be nicely behaved on the boundary you'll also need the space-time to be weakly asymptotically empty. These are strict conditions that apply only to isolated gravitating systems.

Researchers wanted a tool that replicated the success of Penrose's embeddings that could be applied to a wider class of space-times... and thus was born the field of "Boundary Constructions".

The field has had some success, most notably in the importance attached the causal relations (e.g. the causal sets quantum gravity thing), Krolak's weak cosmic censorship results and in application to the study of singularities. And there is a perennial interest in it (e.g. most recently, by people in condensed matter physics looking to replicate the AdS/CFT correspondence for aniostropic fields). But these are few and far between, particularly for the level of sophistication that has been reached in the most recent constructions.

So Boundary Constructions are a dangerous road to tread. You'll immediately get offside with a large segment of the community, but there is something there. Something that a lot of people have chased and not quite got a handle on. This is very very hard stuff with little to no idea of what the pay off will be. You'd be much better served by working in a more active area.

Now, what if you want to completely ignore my advice? Take a look at my own most recent (and probably last) contribution http://arxiv.org/abs/1401.1287. And have a look at the references. Ashley's PhD thesis has a good review of the state of the art from around the turn of the millennium (https://digitalcollections.anu.edu.au/handle/1885/46055). You should also take a look at Senovilla's work on the iso-causal boundary, http://iopscience.iop.org/article/10.1088/0264-9381/22/9/R01, as well as the "controversy" around it, http://arxiv.org/abs/1103.2083. The guys working on the causal boundary, e.g. http://arxiv.org/abs/1001.3270 and the papers that cite it, have also staked a claim to generalising Penrose's constructions. These are all good places to start.

To summarise: It certainly feels like there is something deep about Penrose diagrams, that feeling has been enough to produce its own subfield and absorb substantial amounts of research. But, that subfield hasn't done anything truly astounding and that "deep" feeling might just be the same kind of thing as discovering that taking taylor expansions about infinity can be really useful.