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Self-teaching General Relativity Mathematics for non initiates

  1. Oct 19, 2014 #1
    I am 72 and my background is in the humanities (I read classic authors in Latin and Greek, which I learnt in high school in Italy, I love history and I have a more recent M.A. in Philosophy from a Canadian University).

    Over the last few years, though, I developed a strong (and totally unexpected !) interest not so much in physics as in two of its most…esoteric branches, namely Relativity, in both its divisions, and the Quantum theory, an interest due in part, I guess, to my natural intellectual curiosity and also to their metaphysical underpinnings and the philosophical questions they raise...

    The problem is that my mathematical skills are inversely proportional to my relative proficiency in the humanities, which means almost null. Although I did maths in college, my relationship with this discipline has always been one of aversion and in the end this has compromised my mathematical literacy and the possibility of any further exploration in the sciences field.

    Mindful of my troubled relationship with maths, I thought at first, somewhat naïvely, that I could sail through the Special and General Relativity without maths, i.e. conceptually. I found this reasoning vindicated, at least in part, as far as Special Relativity is concerned, because the relativistic properties of space and time ( time dilation, space contraction, etc) , can be grasped, in spite of their apparent counter intuitiveness, with as little maths as some elementary algebra and the Pythagoras theorem.

    I think this is because time and space (at least the Euclidean space!) are entities which are after all apprehended by our senses in our common experience and we know what we are referring to when we speak of them, although imperfectly, because we are never exposed to velocities even remotely approaching c.

    When it comes to General Relativity, however, the counter intuitiveness of concepts like curved space ( and its interaction with matter) is compounded by the fact that we can’t even represent, for example, curved space in our mind and we have therefore to resort to maths to give it some kind of reality.

    If it were not for the fact that the postulates of both the General and Special Relativity have been experimentally verified over the last 100 years and have been successfully incorporated into many branches of technology because of their highly predictive power, one would be tempted to think ( as many, not necessarily uneducated, people do) that using maths to give an apodictic foundation to the “reality” of curved space and the like is as futile as trying to prove the existence of centaurs with some mathematical lucubration.

    This is not, of course, my view and I have therefore come to the conclusion that if I want to satisfy my intellectual curiosity in regard to Relativity and the like, I have to take the bull by the horns and learn maths.

    This brings me to my question.

    Can somebody lay out for me a tentative self-taught curriculum of the maths required to bring me up to the General Relativity level,( if this can be achieved, as I hope, within the rather limited and, alas, so rapidly shrinking time of my earthly existence )?.

    The C.O.W. ( Calculus on the Web) course provides, in my opinion, an excellent template suitable to my purpose. I took the pre-calculus and the 1st calculus level a couple of years ago just to test myself and to my surprise I was able to go through all the exercises, even though no keys are provided ( the correct solution is required to go to the next step, though.). If something similar were available through the Internet in the field of Relativity, I’d like to hear.

    Perhaps I should stress that my starting perspective is that maths, far from being a reality in itself, is but a language, whose goal is to explain an underlying reality, in this case the reality of the physical world, by conveying, through the shorthand of its symbols, concepts and logical links which cannot be otherwise conveyed by the verbal constructs of our common languages.

    The curriculum I am searching for should be therefore streamlined to what is strictly necessary to understand the basic concepts of the General Relativity Theory, ( which would be already quite enough of a hurdle!) without side-stepping into the more speculative and perhaps more unnecessary aspects of pure mathematics, in which, from what I read, even some physicists like to indulge…

    Any suggestions about reading material, and other learning tools will be appreciated.



    Thanks for your help


    Ittiandro
     
  2. jcsd
  3. Oct 19, 2014 #2
    As far as math, you will probably need more calculus, differential equations, and linear algebra. The rest you could probably learn along with physics (electromagnetism and classical mechanics). The one thing I would say to you about avoiding pure math, though, has to do with something you mentioned which is curvature. If you want to get a better feel for the meaning of curvature in higher dimensions, I would recommend studying the differential geometry of surfaces. So, it might be advantageous to actually delve into more math to get the kind of understanding you are looking for, at least in the case of that particular subject.

    For relavity, keep an eye on this
    http://www.worldscienceu.com/

    Right now, there's a course there in special relativity, and there should be one in general relativity in the future.

    Susskind has a lot of lectures you can watch posted online.

    I'm not sure if I can agree with that, exactly. Sometimes, the symbols are helpful for expressing things, but at other times, the meaning behind the math gets lost in the symbols, and would actually be better expressed in terms of everyday language, with the symbols being there mainly to make it precise and facilitate calculation. Also, I've never been fond of the idea that math is "just a language" because that seems to imply that it doesn't have content, which is far from the truth. It has just as much content as any other science. I am always having to explain to lay people that my PhD dissertation in math consisted mostly of words (definitions and proofs) and diagrams and only a few equations.
     
  4. Oct 19, 2014 #3
    I've never really understood math is a language saying, math is a tool imo, the more math you have the bigger your toolbox. Theres a reason the most famous physicists are also the most famous mathematicians

    Anyway

    For relativity you need differential geometry, there is no way around it, and to do differential geometry you need calculus and therefore algebra and trigonometry, and lastly geometry. Differential geometry is just applying calculus to geometry but its not as simple as it sounds.



    Don't get it twisted, you probably can learn the concepts of relativity without differential geometry but you'll still never have as good of a grasp as you would if you did know DiffGeometry. Its like taking highschool level electricity and magnetism (algebra based) you learn the concepts and the equations (which are all purely algebraic) but without knowing the multivariable calculus you can't truly derive the equations, and don't know the intuition behind the forming and creation of the concepts (which in my opinion makes problem solving 10 times easier).
     
  5. Oct 19, 2014 #4

    aleazk

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    Gold Member

    It's a rather hard goal the one you selected! For most of us, it took years of university studies! But I salute your determination.

    But, before you plunge into the deep waters of differential geometry, I recommend you to read the following two books:

    -https://www.amazon.com/General-Relativity-A-Robert-Geroch/dp/0226288641, by Robert Geroch.

    -https://www.amazon.com/Space-Time-G...2-4674337?ie=UTF8&refRID=1KBZ8NXDBEPCX0K7EWGS, By Robert M. Wald.

    These two books are popular accounts of the theory written by two of the world's leading experts on the field. But don't be fooled by the 'popular account' phrase here. Both Wald and Geroch are known for their conceptual and mathematical rigour, and certainly you will find that in these books. These are a must read even for students of the subject.

    I think that for people in your situation, these books will do the job pretty well. I recommend you to read them and take notes, make your diagrams, etc. You will learn a lot about both the fundamental, physical and mathematical, concepts of the theory.
     
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