In the responses to my previous post, Cristi Stoica provided some very useful references that, for now, I am going to classify as “Rainich unification” since I needed something for the title of this post. What caught my eye was the following from the abstract of a paper by Misner and Wheeler from 1957:
Maxwell’s equations then reduce, as shown thirty years ago by Rainich, to a simple statement connecting the Ricci curvature and its rate of change. In contrast to unified field theories, one then secures from the standard theory of Maxwell and Einstein an already unified field theory. (Misner, C.W. and J.A. Wheeler (1957). “Classical physics as geometry”. Ann. Phy. 2: 525-603.)
Some related work was done by Penrose and Rindler in the 1980s but not much seems to have come of it. Whatever happened to this approach? Why was it dropped? What are its inherent flaws?
Quantum Moxie 
Dear Ian,
As far as I know, Wheeler’s Geometrodynamics program was concerned with obtaining as much as possible of the fundamental physics from spacetime geometry. It was based on the ADM formulation (http://arxiv.org/abs/gr-qc/0405109) and on the exploration of previously little considered aspects such as the topology, to obtain “X without X”, where “X” stand for various items. The “already unified theory” they rediscovered was only one part of this program. The ADM formulation led to another possibility of removing the singularity which appeared in classical physics when trying to describe the electron. The ADM Hamiltonian formulation of general relativity could be quantized and led Wheeler and Bryce DeWitt to their famous equation of Quantum Gravity. Wheeler was disappointed of the Geometrodynamics program when he started to believe that the fermionic spin cannot be obtained from the 4-dimensional spacetime, and that the spacetime should be replaced by a foam, due to quantum effects. Convinced that the spacetime geometry by itself is not enough, he abandoned Geometrodynamics, dedicating himself even more to the quantum world.
The research diminished very much, but it continued. A nice reference is this: http://arxiv.org/abs/0910.2574.
At the end of the eighties, the ADM’s Hamiltonian formalism started to be replaced in the preferences of the researchers in quantum gravity by another Hamiltonian formalism of General Relativity, the one discovered by Ashtekar – the “connection dynamics”. Ashtekar’s “new variables” provided low-degree polynomial dependence of the fields in term of basic variables, and the Poisson algebra of the constraints was closed, allowing a nicer quantization. It was more similar to Yang-Mills theory, and consequently well-suited for applying gauge theory techniques such as Wilson’s loops. Loop Quantum Gravity appeared.
On a personal note, I consider that spacetime geometry has still a lot of secrets to reveal. My approach is to see how much we can explain within the framework of the 4-dimensional spacetime, with some fiber bundles over it. We may have the surprise that Quantum Theory itself is much more compatible with General Relativity than we believed.
Best wishes,
Cristi
Thanks again for the references. I printed out Rainich’s paper just now and am about to sit down to read it. Regarding fiber bundles, I’m not that familiar with them, but I was active over on the nLab until this latest argument over spacetime broke out. I was trying to learn a bit about categories (and apparently fiber bundles have some sort of categorical representation). I’m still lurking and occasionally posting on the nForum, but the tongue-lashing I received was so harsh that it’s a bit awkward at the moment.
But I digress. I am very interested in some of these things so I will read up on them and get back to you. The biggest thing I’m curious about relating to fiber bundles is how they might be used to describe entanglement.
I put here a brief argument for using vector bundles in physics, and a short thought about their relation with the entanglement:
http://www.unitaryflow.com/2010/04/why-vector-bundles-physics.html
Which paper was that? Rainich was one of my undergrad mentors.
Hi Jim,
If you’re referring to the Rainich paper, it is here.
Thanks
Must’ve been interesting to have him as a professor.
It was marvelous! I owe to him my interest in associativity being an option. My last course with him was Theory of invariants – the class consisted of me and two physics students so it became Theory of invariants – of Maxwell’s Eqns. And then there was his unofficial introduction to reading Russian.
Hi Jim,
My apologies for not responding sooner (and maybe you will never see this since I waited too long!). Anyway, I absolutely love hearing stories like this. I have done a lot of work on the history of physics and tend to be a fan of history and genealogy and things like that. And I love anecdotes, especially things that make historical people more “real.”
Thanks so much for your remembrance!
Ian
If you have any other Rainich anecdotes, I would be happy to know them. History of math/physics intrigues me too.
I wish I did! I will be completely honest and say that I had never heard of him until I began looking into Rainich unification. (That, in itself, is a bit odd given that my PhD thesis examined Eddington’s Fundamental Theory and I’ve written papers on early unification theories.)
I don’t know if it has appeared yet but one volume of the Einstein correspondence includes an interchange with Rainich.
Misner clarifies:
Briefly, Hugh Everett III called to my attention a paragraph in Rainich’s textbook on GR explaining how, in the absence of other fields, some aspects of the Maxwell EM field could be extracted from the spacetime curvature. He (Everett) thought correctly that this would of interest to Wheeler. I then set to work with Wheeler’s encouragement to see whether the remaining details of the Maxwell EM field could also be extracted from the spacetime curvature.. After I solved this problem, Wheeler mentioned it to Bergmann, who directed us to Rainich’s paper containing these further results which he had not included in his text book.
This is cited in our paper “Geometrodynamics” published in Vol. 2 on the U.S. Annals of Physics 1957 ( pp.525-603), where footnotes cite
3. G. Y. RAINICH, Trans. Am. Math. Sot. 27, 106 (1925).
4. G. Y. RAINICH, “The Mathematics of Relativity.” Wiley, New York, 1950.
and mention his work prominently in the abstract.
Wheeler called this the “already unified theory”, but Roger Penrose later showed that it did not admit standard causality. The geometry of spacetime at one instant restricted the electromagnetic field severely, but not with enough precision to predict the future in a customary classical causal way. The entire past history might be needed to allow the future to be determined by the Rainich equations.
Hmmm. Interesting. Honestly, I’m a little shocked that Eddington didn’t talk about Rainich’s work, unless I missed it. I have studied Eddington’s Fundamental Theory for the better part of 12 years or so (my PhD thesis was actually on it) and I don’t recall ever coming across Rainich’s work.
At any rate, given Wheeler’s and Penrose’s comments, I need to take another look at it since I’m doing some related work right now.
What is an accessible reference for Eddington’s Fundamental Theory?
LOL, well, as far as I know there are only two publications in history that analyze it – a 1963 book by Clive Kilmister and O.J. Tupper and my PhD thesis. Notably, both only analyze the first part. However, I am on sabbatical this semester in an attempt to finish the rest (not to mention re-write some poorly written earlier chapters). Kilmister also wrote a book that deceptively has “Fundamental Theory” in its title, but it’s really about Eddington’s earlier work “Relativity Theory of Protons and Electrons.” There is a paper by Nikos Salingaros in Foundations of Physics back in 1985 that looks at Eddington’s E-number theory, which is the second half of FT (and also is developed a bit in Relativity Theory of Protons and Electrons), but it doesn’t really place the work in a larger context in conjunction with the statistical stuff. Most people find FT itself too dense to read.
Is http://en.wikipedia.org/wiki/Arthur_Eddington#Fundamental_theory
a fair summary? Can you say it better?
In my opinion, it is more complicated than that. It is true what the Wikipedia article says about the 1/136 v. 1/137 debate and he did place a great deal of emphasis on the fundamental constants of nature, but since it was published posthumously it lacks a bit of the philosophical context that I think was necessary to understand it. If you read his excellent book The Philosophy of Physical Science you get the impression (or at least I did) that he was willing to question everything, even his own theories. I don’t think he held them up as the answer, I think he merely saw those lines of reasoning as worth pursuing.
And it is worth pointing out that they are really two distinct lines of reasoning. The statistical theory and the E-number theory – the first and second halves, respectively, of Fundamental Theory – are really different approaches to the same problem, in a way. I think that point is often lost on people. When you realize that, you realize that what he was really doing was searching – as we all are – for the right path to unification and he didn’t assume that the path already existed. In other words, he didn’t assume that the popular theories of the day necessarily held the answer if you only combined them properly.
But he was so far ahead of his time in so many ways. He often saw things that others didn’t see, and that’s something I really admire about him. I wrote a paper about six years ago after discovering that he had proposed the concept of the solar wind in a footnote some 40+ years before it is generally recognized as having been first seriously proposed.
So, in short, I think a short summary such as the Wikipedia page (nor even Helge Kragh’s discussion in his book Quantum Generations) doesn’t do it justice.
I think you meant : does NOT do it justice.
When you have a chance, could you write something about how it should relate to Rainich?
LOL, yeah, I noticed that as soon as I posted it and just corrected it.
Anyway, I would be happy to. In fact I need an excuse to buckle down on some more work on that front anyway.
iWheeler & Misner were working on a paper, the story I heard was that Bergmann called to their attention a paper written by my father, Yuri Rainich, as “Rabinovich”. I understand another person may have been involved. Could this be clarified for me?
Interesting. I don’t know anything about this, but I would be interested in learning more. I’d have to look into it.