Learning General Relavity and Differential Geometry.

Since my ski accident 2 month ago and the operation in the middle, I have been immobile for the past 2 mo and have another month to go.  In the downtime and instead of waving my arms, I have decided to learn General Relativity.

The motivation
I have been talking about this 'elastic world' for some time (and the past blog entry) but realized it was time to move from 'intuition' to 'mathematics'. What truly motivated me was that in the first week of downtime I came upon the realization that the anti-symmetric part of the strain tensor of space would give a trivial account of electro-magnetism.  This is fancy way of saying that Electro magnetism is the result of the torsion of space (rotation) and I don't mean at the hand waving level but that I could rebuild Maxwell equations with a couple of lines of calculation by identifying the ElectroMagnetic potential with the displacement and running some math.   In digging further, this book comes up.  It is highly underground in the sense that only it references itself, and the folks behind it are rather enigmatic, but in there is the full development of this mathematics of 'space as an elastic ball'.  Of course the language is not as pedestrian as what I like to use (swimming pools and elastics) and one will be met by talks of metrics, covariant derivatives representing an elastic space-time and the anti-symmetric part of the connection representing such an elastic space time mapping to the Faraday tensor. For the little story, the book was supposedly written by a kid of 16-18 years old.  I find it hard to believe that such a genius would exist but have done enough research on the kid to believe he MAY exist and that if he is real, he has serious health issues (which he alludes to in the book). I would not be surprised that such a genius would suffer mentally and physically.

In a way this is mean because it obscures the rather silly nature of what is behind it (that the universe is elastic) and that I can explain to any 12 year old.  But in another way, this is the only way forward.  So I want to understand that language and what the paper says in detail. In the book, you have the mathematical justification of how EM and Gravity unify and interact in a visual and mathematical framework.

The irony
But back to serious science. In theory, this should be a 'relearning' for me since, again in theory, I majored in theoretical physics and had to do a 'end of year thesis" on Einstein field equations. What a joke that was in retrospect.  I did not understand a single thing I was doing back then.  Both in the math and the physics department I learned more in 2 months of self study, than in 2 years of wasting time. I realize I was a virgin, a complete virgin and that the high grade education I received was purely a selection mechanism having little to do with the subject at hand but rather my capacity to regurgitate it.

A short criticism of the french educational system
For those that know me, you know I attended the most blue blood schools in France, the Ecole Polytechnique and the Ecole Normale.  For my american friends the Polytechnique is a cross between Harvard and MIT with a lot more social prestige.  And therein lies the rub. The social prestige is such that no one really works while there, or after really. And if one works, one works to become a civil servant. What a waste of talent and fine teaching.  There is also the question that most of us slack off in said schools once we have passed the very selective entrance exams (I know I certainly did) and cram a lot of study in 2 years paced by end of term exams.  As mentioned I made more progress in 2 month than in 2 years. I also took a lot more pleasure in learning the topic than I ever did back then.  Back then all I had was frustration, late night cramming, fear, and an immense sense of inadequacy, even though I ended up with a PhD. It was a waste of my time.  Those topics, at least for me, were not meant to be studied in the hurried and distracted setting of university but rather in the calm immobility of a broken leg.  I am a lot slower and at the same time a lot faster. It is hard to explain.

3 levels of understanding, visual, symbolic, components
The mathematical apparatus behind General Relativity is formidable. Very simply put it is downright intimidating (try the link above if you need any proof) but on many levels is actually rather trivial, once understood. The question is how to get to that understanding. Everything, at least to me, is extremely hard and takes sometimes a full day on a single page or an exercise. I use several books to get various points of view.  The outstanding one is the classic "Misner, Thorne, Wheeler" (MTW).  I love MTW because it is so complete and makes perfectly clear that the 3 levels of understanding of the topic are necessary.   1/ A pictorial view.  This is where we draw little picture of manifolds and tangent spaces and Lie derivatives and bla bla bla.  With an emphasis on bla bla bla, because words are silly at this level as most of the objects of study are rather trivial visually.  2/ A abstract, symbolic view.  This is the "object" layer of the understanding. It talks about vectors and forms and tensors and all the objects that inhabit our space. Sometimes the knowledge is trivially derived in this layer (Jacobi identity being one).  3/ A  component/index view.  This is what you see in the book above and in most literature.  It is the most rigorous treatment one can give but also the most obscure as the equations with index components look like 'index salad' and sometimes hide the trivial geometric meaning they are trying to convey and sometimes make perfectly TRIVIAL the relationships.  Most books shine in one or two layers.  Some touch a little bit of everything without going deep but are essential to my progress by gradually introducing me to the topic. I need all layers and many books in order to achieve any measure of understanding.

The art of coding: an approach to learning
In a way, even though I was trained in science and the deductive approach, the encounter with computers prepared me better for the topic.  I struggled with learning computer programming when I was younger with the fact I was not able to get the topic from one presentation and I kept wanting to get the ONE BOOK that would unlock the key for me. No one thing did it. In retrospect, it was a combination of approaches, practice and insights that hit me bit by bit.  At first I was looking for that linear introduction but ended up picking bit and pieces in different places until it slowly coalesced in 'knowledge'.  In retrospect there is no singular moment of divine inspiration but rather a long series of little gratifying insights that make all the pieces fall in place. That has been my experience with Differential Geometry as well. Every day I am afraid to go fight the mountain again (yet, totally excited at the same time) and every day I come away with a little bit of the mountain, somedays just a chip.  But over time, I know it will amount to the whole thing and that is what gives me hope. And if I fail, I am not afraid to, this will have been one of the hardest endeavors that I have undertaken.

Alchemy and the art of patience
The first difficulty, for me, then is a symbolic one.  You are met with a impenetrable wall of symbols.   I need to get to a level of fluency in the mathematical language spoken where I can understand what is being said in links like the one above. For "normal folks" like me, learning GR and DG is already a luxury and I hope to one day really claim I belong to the tiny club of people who truly 'get' GR but for others, the knowledge just appears by divine inspiration.  I am always grateful and in awe of such talent even if slightly jealous.  In my case, the knowledge doesn't fall from the sky, but rather requires a long and arduous path of self-study in obscure (if classic) books.  This study is probably the most rewarding thing I have done, both in its difficulty and the pleasure I derive from it.  It teaches patience in a way I have never learned in my academic or professional life where speed is valued above all (speed of learning, speed of execution).  It is also one of the biggest sources of physical pleasure in those moments when I realize I have made progress, the little release of endorphins when a chip comes away and a small insight achieved never gets old.  Each day feels like an infinite stretch of time where the progress feels infinitesimal and frankly almost null, the mountain is still there and feels almost untouched. Yet over time I realize the sum of it all gives me definite jumps.


Anonymous said…
Marcf said…
thank you very much for this link

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