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Minkowski space: basics

  1. Nov 28, 2006 #1
    hey,

    i'm just trying to learn about special and general relativity and i figure a good place to start is with minkowski space since that is the basis of special relativity. I have a few questions though, i hope you forgive me because these questions will sound rather ignorant and silly i guess but please understand that i've never seen tensor calculus or anything like this before.

    The first thing i'm confused about is this concept of a Minkowski metric. You can see on page 3 of this pdf what i'm refering to exactly in the rest of this post:
    http://preposterousuniverse.com/grnotes/grtinypdf.pdf [Broken]

    I don't understand what this metric is... what is a metric? Or why is the signature of this metric (-1,1,1,1). So as you can see i have a total lack of basic understanding.

    Also, the dot product of this metric confuses me. The two vectors are A and B, which i gather are two fixed vectors in this minkowski space right... and the dot product is given as
    A . B = n(uv)A^uB^v = -A^0B^0 + A^1B^1 + A^2B^2+A^3B^3+A^4B^4

    Well that's messy and i'm sorry, i hope you understand what i am trying to write there. I can see that signature in there with the (-,+,+,+,+) pattern, and if i'm not mistaken the negative term is the time coordinate right? I really need some help to understand the basics of this! It doesn't make sense why A.B equals NuvA^uB^v what does A^u and B^v mean anyway? Thankyou.
     
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  3. Nov 28, 2006 #2

    robphy

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    You might need to start off with something on Euclidean vector-algebra and its tensorial form... just to get the notation down and basic concepts. Without knowledge of your background, I can't make a definite suggestion.

    There are ways to learn the important concepts of relativity without having to study tensors first. This is probably the best place to start: https://www.amazon.com/General-Relativity-B-Robert-Geroch/dp/0226288641" would be the next stop... although the maroon 1966 version is better.

    If you want to get more into the tensorial approach, you might start here with Kip Thorne's course: http://www.pma.caltech.edu/Courses/ph136/yr2004/
     
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  4. Nov 28, 2006 #3
    Basically i just have highschool maths and physics... I'm reading odd articles on tensors and just touching on Lorentz transformations for the first time. If anyone knows any good resources online (or books) for a beginner then that would be fantastic.

    The Lorentz transformation is defined on one site as 'transformation that connects space-time in two inertial frames'

    Could someone maybe elaborate on what that means exactly in simple simple language? Thx.

    robphy, thx for the link to the caltech courses, they're helping a great deal.
     
    Last edited: Nov 28, 2006
  5. Nov 28, 2006 #4
    See http://www.math.ucr.edu/home/baez/relativity.html [Broken] for some links to web tutorials.
     
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  6. Nov 28, 2006 #5

    robphy

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    While the Lorentz Transformations are at heart of many relativity presentations and textbooks, one might think that it is NECESSARY to understand them (or at least regard them as primary) in order to understand special relativity. To a mathematician, it is primary... in the Felix-Klein viewpoint. However, to a physicist, it is not primary... and it is arguably likely to cloud the physics. It's rarely appreciated that you could get quite far first focusing on the physics and the operational interpretation of [radar] measurements, then formulating the Lorentz Transformation. In fact, if one writes things vectorially and tensorially, and works with vectorial operations (like dot products) as opposed to components, one rarely needs to explicitly write down the Lorentz Transformations.

    Since you are a beginner, I'm curious how well such an approach might work for you... that is, a treatment that delays the Lorentz Transformation. Of course, one needs a good treatment that follows this plan. The Geroch book is one such treatment. Some others [in increasing difficulty] are https://www.amazon.com/Relativity-Common-Sense-Herman-Bondi/dp/0486240215".
     
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  7. Nov 28, 2006 #6
    robphy,

    well basically i'm thinking about transfering into a theoretical physics degree at Imperial College London next year. I'm fascinated by physics... so until that time i want to study up as much as i can on physics... especially relativity. What do you think of these two books as a starting point?

    Geometrical Methods for Mathematical Physics

    and

    A First Course in General Relativity

    both by Bernard F. Schutz.

    I get the impression that they both complement eachother... although i am as green behind the ears as someone can be with physics, i am a fast learner and i would like a book which isn't all words but can challenge me a bit. Flat and Curved Space-Times looks like it might be similiar to 'A First Course in General Relativity'... do you know which might be better?

    Also im curious what do you think about my approach to learning theoretical physics? I'm thinking about focusing on classical physics, specifically SR and GR first, and then branching off into the other areas that i'll need to know for an undergraduate degree course... would that be a logical enough way to self study physics?
     
  8. Nov 28, 2006 #7
    While tensors may well cloud the physics of SR, I certainly wouldn't discourage anybody employing the Minkowski metric (over basic gedunkan) as early as possible. Personally, I disregarded the tensorial approach preferred by a course I studied on SR :uhh: but I regretted that after the following GR course.
     
  9. Nov 28, 2006 #8
    I haven't seen that math methods book. I'd see how the intro to tensors in Schutz works for you first before getting it. [Er, bad grammar, the antecedant is ambiguous. I mean, try Schutz first.]
     
    Last edited: Nov 28, 2006
  10. Nov 28, 2006 #9
    I didn't really say so above, but, yes, Schutz is a good book. Another popular and well regarded intro is Gravity: An Introduction to Einstein's General Relativity by James B. Hartle.

    Here are some more books. Don't forget to do a search on this topic, there have been lots of threads on this.

    Space, Time, and Gravity: The Theory of the Big Bang and Black Holes by Robert M. Wald. A semi-popular book that does a good job on spacetime diagrams.

    It's About Time: Understanding Einstein's Relativity by N. David Mermin. Pedagogically careful book on SR.

    Spacetime Physics by Taylor & Wheeler. As discussed here recently, many of us prefer the older edition that can be found in libraries.

    Some "pre-tensor" books on GR:

    Exploring Black Holes: Introduction to General Relativity by Taylor & Wheeler

    Flat and Curved Space-Times by George F. R. Ellis

    Spacetime, Geometry, Cosmology by William L. Burke. Actually, he does an excellent job introducing one-forms, tensors, and manifolds, but he doesn't "go all the way" and develop the full machinery of Riemannian curvature. Unfortunately out of print, so look for it at the library.
     
  11. Nov 28, 2006 #10

    robphy

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    I didn't say that tensors cloud the physics of SR.
    ...Quite the contrary, when used correctly.
    It's the "premature overemphasis of the Lorentz Transformations" that cloud the physics of SR.

    Let me be clear on this point:
    I encourage the use of geometrical objects and their operations [4vectors, tensors, dot products, projection-tensors, etc...], and I discourage and de-emphasize component-based descriptions and transformations of coordinates.

    I did say above "In fact, if one writes things vectorially and tensorially, and works with vectorial operations (like dot products) as opposed to components, one rarely needs to explicitly write down the Lorentz Transformations."
     
  12. Nov 28, 2006 #11

    robphy

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    Imperial seems like a nice place. (I just visited for a week-long conference back in September.... although I did find the use of the swipe cards for entry and exit rather annoying.)


    These Schutz books are good, and they do complement each other. These books were helpful to me for understanding tensors, geometrically and component-wise. I recall a useful discussion of tensors in polar coordinates and of the use of differential forms in the Maxwell [thermodynamic] relations.

    Flat and Curved Space-Times is at a lower mathematical level than the "First Course" book... but it deals well with some conceptual issues in SR and GR that one doesn't find in other books.

    By the way, Schutz has a new book out http://www.gravityfromthegroundup.org/" [Broken].


    Your approach sounds okay.... the Thorne course might be a good reference for you.... although I would suggest that try to work in Quantum Mechanics as soon as possible. Another suggestion: work out lots of problems... in detail.
     
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  13. Nov 28, 2006 #12

    robphy

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    http://www.ucolick.org/~burke/home.html" [Broken] ].
     
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  14. Nov 29, 2006 #13

    Hurkyl

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    aeroboyo: have you learned matrices yet?
     
  15. Nov 29, 2006 #14
    I'm learning about matricies for the 1st time just now, the only other time i've ever come across any kind of vector analysis was in a course on statics... so it's all new to me.

    I have a couple of basic questions after reading last night about special relativity.

    A vectors is a straight line between two events right... and a tensor is a linear function of vectors. What purpose is there for dealing with tensors in SR?

    I've learnt that intervals between events are invarient in Minkowski space-time... what else is invarient in Minkowski space?
     
    Last edited: Nov 29, 2006
  16. Nov 29, 2006 #15

    dextercioby

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    Nope, the Lorentz transformations affect in a direct way objects (spinors, tensors) in Minkowski space, which is the flat 4-dimensional space of Special Relativity. They are linked with how these objects behave when subject to change of inertial observers.


    This is thorny subject that doesn't have a unique answer. Directly put, there's no unique, by all accepted, receipt of passing from Minkowski space to curved space.

    Daniel.
     
  17. Nov 29, 2006 #16
    So if i were to buy those two Schutz books, can anyone recommend any good maths books that will help me to reach the level at which those two books start? I'm refering to 'Geometrical Methods for Mathematical Physics' and 'A First Course in General Relativity'. I get the impression that those books do require a half decent grip on maths, and all i know right now is basic calculus and some very basic methods of solving first order differential equations. So basically i guess i'm wondering what kind of maths should i learn to help me to understand and get the most out of these two Schutz books? An introductory text to vector calculus, tensors, matricies, differential geometry etc is what i need i think.
     
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  18. Nov 29, 2006 #17

    robphy

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  19. Nov 29, 2006 #18
    Oh, sorry, I thought you were referring to a different Geometrical Methods of Mathematical Physics :uhh:

    Yeah, Schutz's Geometrical Methods is great. However, his GR book is fairly self-contained, and you shouldn't really need his math book until you try to tackle books like Wald's General Relativity.

    Another excellent math methods book is Frankel, Geometry of Physics.
     
  20. Nov 29, 2006 #19
    http://www.bookfinder4u.com/IsbnSearch.aspx?isbn=038797606X&mode=direct. Uses differential forms throughout.

    Schutz's GR book does a good job on tensors.

    Also the aforementioned book by Burke, Spacetime, Geometry, Cosmology.

    And don't neglect basic physics at the level of, e.g., the Feynman Lectures volumes 1 & 2.
     
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  21. Nov 29, 2006 #20
    Daverz do you think that 'Geometry of Physics' is suitable for self study by someone like me with a very limited understanding of maths?

    Also would Geometry of Physics complement Schutzs 'A First Course in GR'? It's just that i dont have that much money so i'd like to only invest in two books for now... so i want to make an informed choice. Thanks to everyone for showing interest in my learning!
     
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  22. Nov 29, 2006 #21

    robphy

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  23. Nov 29, 2006 #22
    So Frankel would be more of a graduate level book?

    If 'A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry' is simpler then i might just go for it. It's a shame there are no reviews of it on amazon by im guessing you've read it robphy... I have read the table of contents and it looks like it does start off with the basics which is what i'd need.

    Also i have a question about Mathematical Physics... is it a good foundation for an aspiring theoretical physicist? From what i can tell its the hardcore maths branch of physics...
     
    Last edited: Nov 29, 2006
  24. Nov 29, 2006 #23

    Chris Hillman

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    Learning gtr, again

    Hi, aeroboyo,

    Sounds like you want to teach yourself gtr, fast, but currently are at the pre-linear algebra level in terms of mathematical background. In view of this lack, you might be particularly interested in my advice since (as outlined in a recent post to this forum) I taught myself gtr, fast, from the classic textbook Gravitation, by Misner, Thorne, & Wheeler, at a time when I knew only high school algebra and trig, with two crucial supplements:

    1. I had some knowledge of differentiation (and made free use of a table of integrals when I needed to integrate anything),

    2. Most important of all, from the original edition of Spacetime Physics by Taylor & Wheeler, I already had a good intuition for the geometry of Minkowski spacetime.

    My experience immediately suggests three comments:

    1. The most important background you need to get STARTED on gtr is hyperbolic trig and good intuition for Minkowski geometry. You should think of a boost in the Minkowski plane as a direct analog of a rotation in the euclidean plane. How direct? Well, you should write out a table comparing the circular and hyperbolic trig definitions. (In some Wikipedia article I once introduced such a table, but I forget which one, and of course anyone could have mucked with the signs since then, which would be disastrous for any student trying to verify the table.)

    2. Nonetheless, since you are a registered university student, you should not fail to take advantage of this to follow a more systematic and standard route than I did, by planning to take lots of math courses in the standard sequence. If you wish to study gtr, you will need to at least concurrently study a large variety of mathematical topics, which fortunately are all valuable in many other areas (should your interests change). These topics include the theory of vector spaces, linear algebra, matrix theory, differential and integral calculus of one real variable, the usual theory of odes and pdes, the theory of perturbations, some real analysis including special functions, multivariable Taylor expansions, and asymptotic expansions, as well as complex variables, vector calculus, differential forms, manifold theory and some topology, modern algebra (groups, rings, linear associative algebras), Lie algebras and Lie groups, and symmetries of (systems of) odes and pdes. (Some of these are often considered level graduate topics, but they are all taught to undergraduates at the best universities, and of course an sufficiently capable undergraduate can hold his own in a first year graduate course.) A course in mathematical modeling and exposure to tools like Mathematica and Maple will also be invaluable.

    3. It sounds like you prefer to plunge right in, rather than being extremely systematic, so regarding the choice between the two books by Schutz (both of which I recommend!), although the geometrical methods textbook should by rights be a prerequisite for the gtr textbook, I'd encourage you to try reading the gtr book first, but only after reading the fine popular book by Robert Geroch which someone else already mentioned, since you certainly won't be able to get started with Schutz or another gtr textbook until you have mastered both Minkowski geometry and its physical interpretation. (However, to reiterate what I said above, facility in drawing and interpreting spacetime diagrams is highly critical, but prior knowledge of the topics mentioned above is not essential, although mastery of these will be essential for mastery of the gtr).

    But be systematic in your attempt to jump over all the usual prerequisites--- e.g. by following my advice to set your first goal as constructing the above mentioned table comparing in detail circular and hyperbolic trig (diagrams, power series, geometric interpretation, the lot). You could follow that up, as I did, by learning just enough differential geometry (fortunately, I found an introductory calculus textbook which covered just enough) to figure out how to compute path curvature in euclidean and Minkowski planes, and then make a table comparing these in detail. You should see the pattern: analytically, everything is the same except for a systematic change of signs here and there.

    The notion of making such tables is one of the most valuble things I picked up from Wheeler, incidently! You can infer his preference for teaching himself by ignoring prerequistes but being very systematic in every other way, in all of his textbooks--- bearing in mind that authors tend to write for an imaginary student based upon an internal vision of their youthful self.

    Fortunately for autodidacts, students of gtr are blessed with an unusually large and diverse collection of truly excellent textbooks, which is not the case at all for many other subjects of equally compelling interest. Should you decide the Schutz is not working for you (although I do think this is one of the very best textbooks around!), you can find a long list of suggested reading at http://www.math.ucr.edu/home/baez/RelWWW/reading.html [Broken]

    Enjoy!

    Chris Hillman
     
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  25. Nov 29, 2006 #24
    He assumes a knowledge of multi-variable calculus, vector calculus, and basic linear algebra, e.g. that you know what a Jacobian is. That's why I recommended the Bressoud book.

    The bulk of it is not absolutely necessary for GR on a first pass. I'd concentrate on physics (introductory mechanics and E&M), multi-variable calculus, vector calculus, and basic linear algebra. If you have one of those monster calculus books it might have some of the multi-variable and vector calc material.

    Well, check these out at the library if you can. Sorry, I know what book lust on a student's budget is like :tongue2:
     
  26. Nov 29, 2006 #25
    Perhaps you're thinking of Frankel's Gravitational Curvature?

    Frankel's Geometry of Physics has a more intuitive approach than Szekeres. For example, Frankel first introduces the covariant derivitive for an embedded surface, where it has a simple and logical geometrical expression. Szekeres just starts with the usual abstract definition as a derivation on a manifold.

    The value of Szekeres's book is his attention to algebra and topology. He based his book partly on Choquet-Bruhat, but Szekeres is much easier to read.

    All this may not be very relevant for our OP for a while, since all of these math methods books require the second year math courses he will probably be occupied with soon enough.
     
    Last edited: Nov 29, 2006
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