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I have heard a phrase somewhere, which can be reduced to the following two points: 1) There exists a handy and underused mathematical apparatus applicable to GR, comparing to which tensor calculus is obsolete. 2) The apparatus is related to graphs.

Does anybody have an idea, what could this apparatus be?

Alexey Bobrick
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  • I am really really unsure but I would say something related to twistor theory maybe? – toot Jul 08 '12 at 20:31
  • I haven't really worked with it, but what I remember is that: 1) It is vaguely similiar to spinor theory in curved spacetime, 2) It was introduced with a hope to produce quantum gravity, and it didn't give much. So, you might be right, but it is not related to graphs (as far as I know). – Alexey Bobrick Jul 08 '12 at 20:47
  • Cartan formalism. . – Nick Sep 11 '12 at 21:45

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Maybe people talked about Penrose graphical notation or related inventions.

(The question is a little vague and I'm not use if "obsolete" is really the word. Watch out, the last time someone complained about tensors here, Ron Maimon gave a raging answer, which resembed an anvanced lecture in theoretical linguistics.)

Nikolaj-K
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  • :) Thank you for both the answer and the warning! I cautiously remark again that it was what I heard: "at least in some respects the formalism is superior to tensors". I think it might be what I've been looking for. Thank you for the interesting link! – Alexey Bobrick Jul 08 '12 at 20:56
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    @AlexeyBobrick: It was of course not a real warning. Rons "answer" in that thread was pretty awesome. – Nikolaj-K Jul 08 '12 at 21:06
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    +1, but Penrose graphs do make the traditional tensor expressions a little obsolete, so I have nothing to rant about. It's just a picture for each tensor expression, nothing more, but it's useful because certain manipulations (like raising and lowering contracted indices, or finding symmetry) become null operations. The formalism is most useful in twistors, where the number of indices really gets out of hand (the Riemann tensor has 8 spinor indices), and it makes the index manipulations really pretty. It's a wonderful way of showing how pretty indices really are underneath, linguistically. – Ron Maimon Jul 09 '12 at 03:53
  • @RonMaimon: Do you have an introductionary books or something of that sort in mind, which deals with the question what can and what can not be represented in graphs this way. I mean abstract definite mathematical considerations about representation of structures with 2D (or maybe 3D) puictures, constructed just to store information. – Nikolaj-K Jul 09 '12 at 07:10
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    @NickKidman: For Penrose specifically, Spinors and Space Time tells you what you want. But in general, it's an interesting question. I thought about it. Generally, you can represent anything in any formalism, of course, but there's a question of redundancy--- if you want (more or less) unique expressions, there are interesting constraints. I don't think this question has been investigated in any depth, but there are some graphical formalism (one due to David Harel called "higraphs", and I did a generalization that I like) that are efficient in representing computational structures in nature. – Ron Maimon Jul 09 '12 at 07:52
  • This article Graphical representation of invariants and covariants in general relativity by S. Ichinose seems to invoke similar ideas. – Qmechanic Jul 09 '12 at 21:01
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    @Qmechanic: I think Nick Kidman's concern was about how to demonstrate when a graphical formalism is provably less redundant than an algebraic formalism with parentheses. This is more a linguistics question than a physics question, but it is understudied because linguistics doesn't often consider graphical formalisms. Harel considers this question in 1988 with Higraphs, and gives a formalism for finite state automata which is relatively optimal. This formalism can be extended to arbitrarily large and growing automata, and then it describes biological protein interaction economically precisely. – Ron Maimon Jul 10 '12 at 20:27
  • @RonMaimon Even if Nick Kidman's interesting concerns are valid, the diagrams seem definitely useful. I'm practicing the notation at the moment as I rehabilitate my 20 years in suspended animation GR: the way I see it, index free notation is for communication with others, indices are for private calculations and not for public places unless absolutely necessary! It's very like cutting something like C++ code: fairly easy to get your head around if you're the writer, terribly thorny if not: in general a dreadful communication medium, which is exactly why Penrose conceived his graphs. – Selene Routley Nov 15 '13 at 05:22