# Penrose diagrams, reference request.

• martinbn
In summary, books on Penrose diagrams that I have looked at either provide brief general comments about the diagrams and a few specific examples, or present a nonmathematical overview of the subject with the intention of directing readers to other more detailed books.
martinbn
Where can one find a general and detailed exposition of Penrose diagrams? What I have seen so far, in the books, are relatively brief general comments and a couple of specific examples. Usually Minkowski and Schwarzschild metrics, sometimes one or two more.

I like the discussion in Penrose's popularization Cycles of Time. The good thing about it is that although it's nonmathematical, he explains lots and lots of examples.

For a more detailed mathematical treatment, you could try Hawking and Ellis, or Winitzki's online book http://sites.google.com/site/winitzki/index/topics-in-general-relativity . Winitzki has nice discussions of things that I haven't seen elsewhere, like the properties that are required for the mapping in the Minkowski case. (E.g., Hawking and Ellis just give a mapping, but they don't say anything about why they choose that particular one.) Wald has a careful, detailed mathematical treatment of the definition of asymptotic flatness.

What I find hard about the subject is that it seems to have undergone a process of gradual generalization, but Penrose's is the only attempt I've seen to go back and give a synoptic view, and Penrose's treatment is nonmathematical. I haven't seen a clear, mathematically detailed description of what the fundamental principles are that apply in general. Everybody just presents certain special cases.

Last edited:

bcrowell said:
What I find hard about the subject is that it seems to have undergone a process of gradual generalization, but Penrose's is the only attempt I've seen to go back and give a synoptic view, and Penrose's treatment is nonmathematical. I haven't seen a clear, mathematically detailed description of what the fundamental principles are that apply in general. Everybody just presents certain special cases.

Yes, I have looked at only a few books, but this was my impression too, and I would like to see a mathematically detailed description.

The book of Penrose is on my "to read" list, now I will bump it up. I have looked at Hawking and Ellis, but that part wasn't to my liking. Probably I wasn't ready for it yet. I'll give it another try. Winitzki's books, on first glance, seems exactly what I was looking for, and I have immediate access to it. It'll certainly help me understand better.

Thanks a lot.

## 1. What is a Penrose diagram?

A Penrose diagram is a graphical representation of the spacetime structure of a particular region in a curved spacetime. It was developed by physicist Roger Penrose in order to visualize the causal relationships between different events in a spacetime.

## 2. How are Penrose diagrams constructed?

To construct a Penrose diagram, one first chooses a coordinate system for the spacetime and then performs a conformal transformation on the coordinates. This transformation maps the entire spacetime onto a finite region, with the boundary of the region representing infinity. The resulting diagram is a two-dimensional representation of the four-dimensional spacetime.

## 3. What is the purpose of using Penrose diagrams?

Penrose diagrams are useful for understanding the global structure of a spacetime, particularly in cases where the spacetime is highly curved or contains black holes. They also allow for easier visualization of the paths of light rays and other particles through the spacetime.

## 4. Are there any limitations to using Penrose diagrams?

Penrose diagrams are only applicable to spacetimes that are stationary, meaning that they do not change over time. They also do not accurately represent the distances between different points in the spacetime, as the conformal transformation distorts the distances.

## 5. Can you recommend any references for learning more about Penrose diagrams?

Yes, there are several books and articles that provide more in-depth explanations and applications of Penrose diagrams. Some recommended references include "The Road to Reality" by Roger Penrose, "The Geometry of Spacetime" by James J. Callahan, and "Black Holes: The Membrane Paradigm" by Kip S. Thorne.

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