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Minkowski space: basics

  1. Nov 28, 2006 #1

    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:

    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.
  2. jcsd
  3. Nov 28, 2006 #2


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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: Geroch - General Relativity from A to B (Don't let the lack of equations fool you... There is a lot of conceptual and operational content here that you won't find in your typical relativity text.) Taylor/Wheeler's Spacetime Physics 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/
    Last edited: Nov 28, 2006
  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
  6. Nov 28, 2006 #5


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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 Bondi - Relativity and Common Sense, Moore - A Traveler's Guide to Spacetime, and Ellis/Williams - Flat and Curved Space-Times.
  7. Nov 28, 2006 #6

    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


    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


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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


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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 Gravity from the Ground Up.

    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.
  13. Nov 28, 2006 #12


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    Burke continued in "Applied Differential Geometry" [which introduced me to Schouten's visualizations of differential forms, as well as a geometrical formulation of thermodynamics]. You might also find these notes interesting: http://www.ucolick.org/~burke/class/grclass.ps [from the "Plain text" page: http://www.ucolick.org/~burke/ ].
  14. Nov 29, 2006 #13


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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


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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.

  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.
    Last edited: Nov 29, 2006
  18. Nov 29, 2006 #17


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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
    Bressoud, Second Year Calculus: From Celestial Mechanics to Special Relativity. 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.
    Last edited: Nov 29, 2006
  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!
    Last edited: Nov 29, 2006
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