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:
- Make an $n$-dimensional section or projection, where usually, but not always, $n=2$.
- Do a conformal transformation to reduce the resulting manifold to a flat one of finite size.
- 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