I've never seen a precise definition of a Penrose diagram for a $(1+1)d$ Lorentzian metric, merely the statements that it's conformally equivalent to the true metric and that null trajectories are oriented at 45 degrees. These requirements do not uniquely specify the quantitative form of the diagram, so I wasn't clear whether general Penrose diagrams are merely qualitative depictions of the causal structure, or actually quantitatively precise. I asked a GR postdoc, who told me that there is a unique and general algorithm for drawing a quantitatively precise Penrose diagram for a completely general $(1+1)d$ Lorentzian manifold (subject only to some weak smoothness requirements). But I haven't encountered any textbooks on general relativity that actually do so. Carroll only explains how to draw the Penrose diagrams for the Minkowski, Einstein static universe, and certain FRW metrics. Misner, Thorne, and Wheeler only explain the Penrose diagrams of the Minkowski and Schwarzchild metrics, and neither Wald nor Thorne and Blandford have any systematic discussions of Penrose diagrams at all that I could find. What is the general (quantitatively precise) algorithm for mapping an arbitrary $(1+1)d$ metric to a unique Penrose diagram?
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3$\begingroup$ Here's a lecture with a good description of how to construct a Penrose diagram: youtube.com/watch?v=nAT1PDkufso $\endgroup$– md2perpeCommented May 26, 2017 at 17:45
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$\begingroup$ I am not sure if there exists any explicit algorithm. They way Penrose diagrams are drawn for the usual metrics (like the ones you mentioned in your question), it seems like we keep on applying Weyl Transformations until we can bring all the events within a finite patch of coordinate space. Not sure though. $\endgroup$– user87745Commented Jun 19, 2017 at 13:28
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$\begingroup$ @Dvij My postdoc friend claims that there is, and points me to section IB of these notes: phys.ufl.edu/~det/6607/public_html/grNotesMetrics.pdf $\endgroup$– tparkerCommented Jun 19, 2017 at 20:24
1 Answer
There's a recent paper addressing exactly this question: https://arxiv.org/abs/1802.02263.
(Disclaimer: I'm a friend of the authors and happy to plug their work.... but I promise you'll find it right on topic.)
There is currently no general procedure, but they give the procedure for a large class of metrics.
To elaborate...
As you've noted, the most important part of the definition is that null trajectories are at 45 degrees and the diagram coordinate patch is finite. You can give a precise definition encapsulating that (above paper does), but don't really need to.
There's no known general algorithm for all spacetimes. It's true that every 2D spacetime is locally conformally flat, and there's a known formula for coordinates where this is explicit (as postdoc must be referring to). That formula gets you a Penrose diagram for any individual coordinate patch, but in general not for the whole spacetime, since many spacetimes can't easily be covered in a single patch. If you want a diagram of the whole thing, you have to cover it in local patches and match them up properly.
Diagrams can be either qualitative or quantitative, depending how they are generated. Most of the time they are qualitative, and that's usually good enough. Most of the quantitative variety are special cases (Minkowski, Schwarzschild, R-N, dS, AdS,...), but these are not often plotted quantitatively even when they could be. A systematic method for generating qualitative diagrams for metrics $ds^2 = - f(r) \, dt^2 + f(r)^{-1} \, dr^2+ r^2 \, d\Omega^2 $ is given by Walker's 1970 paper (https://doi.org/10.1063/1.1665393). The paper I cited at the top gives a quantitative version of the same diagrams.
When they exist, Penrose diagrams are not unique. They can always be stretched by arbitrary functions along null directions. It may be true that they are unique under some suitable equivalence relation, but I'm not sure.
