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How to start with Tensors

  1. Mar 16, 2004 #1
    What began as a highly motivated inquiry into understanding Special Relativity has come to a grinding halt on Tensors. I hadn't heard the term before yesterday; now I've spiraled so far away from the topic of SR that I'm wholly determined to learn Tensors FIRST because I believe they're very necessary to learning and understanding SR.

    However my last math course was Multivariate Calculus and that was over 3 years ago. I'm rusty in calc but not disabled. I've never had a class on Differential Equations or Linear Algebra. I've never had a class on Differential Geometry.

    I have a couple leads from another thread in this discussion forum and I'm progressing through Tensors as best I can (beginning with subscript notation and summation/free indexes), but could someone let me know if I'm in WAY over my head by not starting with a more fundamental concept? Is jumping from Calculus to Tensors completely out of step? I have a feeling it is but I'm baffled with which concept is the more fundamental one.

    Calculus -> ? -> ? -> Tensors

    Please and thank you very much...

  2. jcsd
  3. Mar 16, 2004 #2
    In order to learn SR you don't need to study tensors in detail. Some algebra and knowledge of 4-vectors (which are ofcourse tensors) will suffice. You WILL need tensors when you're studying general relativity though.
  4. Mar 16, 2004 #3
    Goooo! Thank god. I had intended for GR to follow up plenty of research into SR first, so thank you very much for the reassuring reply.

    Reading Chapter 1 from Kip Thorne's online textbook here:
    section 1.3 introduces tensors and tensor calculus without coordinates. I assumed this meant I would need to learn about tensors first, but I can pretty easily comprehend 4-vectors. In my brief romance with tensors I learned that 4-vectors are tensors, but I'll happily put the topic aside knowing they're not necessary.

    Actually, now that you mention it something just jogged...are tensors not necessary for SR because SR deals with Minkowski spacetime? This spacetime is not curved, is that why tensors are not needed?
  5. Mar 16, 2004 #4
    You definitely do not have to learn tensors before you learn special relativity. You don't even have to know what a 4-vector is. You can relly on simple algebra and basic calculus. In the end you'll have learned what a 4-vector is and not even know it.

    But in reality tensors are required for SR. But you don't have to go learn tensors before this. You can learn SR and in doing so you'll end up learning tensors - and not even know it if you do it right! :-)

    I created this web site for a similar purpose


    There is almost everything you need to know about SR in that site.

    First step to realize is that tensors are not that difficult to understand - but the stigma of it makes a person believe that it is hard before they go into it. You've started with the Kip Thorne from scratch and I can see why you're having a problem. I meant that as supplemental reading rather than to learn if from there.

    Try going through my web site above from the top link down. Follow each derivation with a fine tooth comb. Work out all the details yourself (although I tried to do the itty bitty details that most people leave out). Let me know when you run into trouble. This should help you learn it and help me prefect that web site.

  6. Mar 16, 2004 #5
    multivariable calculus to tensors is probably OK. it would be helpful know some linear algebra, and for a much deeper understanding of tensors, some understanding of manifolds, but for now, just calculus is plenty. give it a shot.
  7. Mar 16, 2004 #6
    To pmb_phy:

    Your reference to Thorne was challenging and engaging, but ultimately I became confused because the discussion became the removal of coordinates from a type of mathematics which I'd never learned! Haha, you're right, that's exactly why I flipped a bit.

    I'll begin your website in earnest and let you know my thoughts from my perspective in this thread.

    To lethe:

    I'll attempt SR first and become comfortable, then hopefully I'll figure out why Tensors are required for GR. I have faith that with enough sweating and studying I could pound out the mechanics of Tensors, but I want to make sure I know the 'why' and 'what it all means' of these strange new beings before I try to learn the ins and outs of their behavior. Thanks for the confidence tho!
  8. Mar 17, 2004 #7
    Hey pmb_phy, I printed out the first 4 excersizes of your website. I was very pleased because the math was predominant and all the analogies to spaceships and NLS travel mumbo-jumbo was left out...I thought bingo! This is what I've been looking for.

    There were a few typos and I can tell you those if you like.

    Everthing through the Lorentz Contraction was great. I followed a long nicely and felt I was doing well.

    Once I progressed to the "Lorentz Contraction - Version 2" everything went to heck. First off you construct the diagrams, then throw in the Velocity Transformation Rules. I was using printed page without the use of hyperlinks so you can imagine my dismay upon finding this beast of a definition with no aparent derivations. Now that I've logged on I can give that another go, but you've also seamlessly introduced the terms alpha (lower case) and Beta (upper case) into those derivaitons. I've never seen alpha or Beta.

    Am I supposed to have some other reading with these pages? I was under the impression that they were stand-alone, but if I had the wrong idea it's no problem.

    So far it's good, the illustrations are good, the few typos made it a bit confusing in cases, and now I'm just curious if I'm supposed to have some supplementary material. Thanks!

    PS Sorry if I sounded a bit frustrated during this post. Your site is VERY nice and almost ideal for me. I wanted to get down to the details and you to just that! Plus you're making me drudge up all my past math classes a little at a time
    Last edited: Mar 17, 2004
  9. Mar 17, 2004 #8
    Great! I'm very glad to hear that. Mind you this web site was only created for the express purpose for me communicating to others over the internet - It was never intended to be a text of any kind.

    re - There were a few typos and I can tell you those if you like.

    Yes! That'd be great!

    Ingore that page. It was created to describe the physics to someone who had an incorrect notion of what a beam of light is. They thought that since the beam is perpendicular to the motion of the source that the photons must also be moving parallel to the beam.

    Please ignore that page.

    I had intentioned to have links to things in there but it was never designed to be a text so there are lots of holes in it. With your help perhaps I can fill these holes.

    Thanks for pointing this out. I defined alpha right after I used it, i.e. right above Eq. (10). It's basically alpha = velocity/c where "velocity" refers to the velocity of a particle rather than the relative velocity of the frames of reference. Simply follow what it says - i.e. divide Eq. (3)-(5) [I see I left "5" out and out "4"].
  10. Mar 17, 2004 #9
    Here are the typos I found, optimized for an electronic "text find." I'm a tech writer by profession so I'm a bit oversensitive to this type of thing :wink:

    Typos in "Light Clock"
    "two events as measured on ideal clock", replace with
    "two events as measured on an ideal clock"

    "equal the time time", replace with
    "equal the time"

    Typos in "Lorentz Contraction"
    "the same direction in originated", replace with
    "the same direction it originated"

    Is the v(delta)t2 arrow supposed to have "prongs" at both ends?

    Typos in "Lorentz Contraction - Version 2"
    "In frame S the components of the velocity of the photons of beam A in frame S are", replace with
    "In frame S the components of the velocity of the photons of beam B in frame S are"

    As per your instructions I'll skip over "Version 2" and hit "Lorentz Transformation" next, although I may return to your derivation of the Velocity Transformation Rules to brush up on my calc some more. Up to this point it has been smooth sailing with some elementary geometry and algebra.
  11. Mar 17, 2004 #10
    To pmb_phy:

    I'm deriving through the Lorentz Transformation. In equation (7) (the entire derivation process) can I ask why

    [tex]v \gamma^2 t - (\gamma^2 - 1) x = v \gamma^2 t - \gamma^2 \beta^2 x[/tex]

    Obviously this implies that

    [tex]\gamma^2 - 1=\gamma^2 \beta^2[/tex]

    Is there a derivation for this somewhere?

    ::: UPDATE :::

    Well a partial nevermind to that. I just spent 70 seconds squaring the Time Dilation equation (assuming [tex]\tau=1[/tex]) and achieved the equality after substituting [tex]\beta=v^2/c^2[/tex], but I would never have tried this normally. Why throw Beta into the middle of the derivation? Is it in an attempt to work with fewer terms?
    Last edited: Mar 17, 2004
  12. Mar 17, 2004 #11
    I wrote that a while back, a few months ago if memory serves. I don't recall why I put things in or left things out. But substitutions like including beta are done to make the equations easier to deal with.
  13. Mar 17, 2004 #12

    Thanks! I appreciate all of your help, this topic is very interesting and I feel good about my understanding of it so far. Though I admit toward the middle of deriving the two transformation equations I lost sight of the goal. As it ends up I assume we were shooting to derive x' and t' in terms of the S reference frame.

    You may want to check out http://www.geocities.com/physics_world/sr/acceleration_trans.htm . It's ugly...literally.
  14. Mar 18, 2004 #13
  15. Apr 17, 2004 #14

    I am a beginner to SR and am in the process of studying your web site. After working through your discussion of light clocks, I ran into a few questions.

    You argue:

    "Therefore the time, t, between the same two events as measured in the frame S' will be larger. E.g. if the clock is moving at 99% of the speed of light and if Tau = 1 second then t = 7.09 seconds. Hence moving clocks run slower than the same clock at rest."

    Given what you are saying here, wouldn't the moving clock run faster? Between the same two events, the moving clock would tick ~ 7 times while the clock at rest would only tick once. Where am I going wrong?

    One thing that confuses me is the precise definition and measurement of time in each reference frame, S and S'. I would think that in S, the time, Tau, refers to the number of clock ticks that occur as the light pulse moves to the mirror and back. But what does the time, t, refer to in S'? What is the reference or basis for its measurement?

    Thanks for any help, and also for the great website,
    Last edited: Apr 17, 2004
  16. Apr 19, 2004 #15
    Hi brianparks, and welcome to the world of SR. I was in your shoes about a month ago.

    First off...reference frames. It's common notation to refer to frames of reference as S and S'. If we assume that S' is moving at 99% the speed of light with respect to S, how long will one second in S' take with respect to S? This is where T=1 and t=7.09 come from. It takes 7.09 seconds in frame S for one second to pass in frame S'.

    Notice the miracle of symmetry between S and S'. If we were to consider S' as the stationary frame, S' would measure frame S seconds at 7.09 of its own seconds. So both observers will report that the other clock is running slower.

    Finally Proper Time (Tau) is the term used for the measurement of time between two events with respect to the events' reference frame. Or in other words, the measure of change in time between two events in the frame of reference in which the events happen at the same place in space. This is essentially the measure of time that suffers no dilation (with respect to the stationary observer) due to motion.

    I hope my comments help. To sum up:

    No. Moving clocks run slower. Between the same two events, the moving clock would tick once while the clock at rest would tick ~7 times.

    You're right, Tau is the number of clock ticks that take place as the light beam travels to the mirror and back (event A and event B). Time t refers to the amount of time an observer moving at speed v with respect to S would measure between the same two events. The time t always reflects the amount of time that passes for an observer in motion, in units of Tau. Remember to keep your frames of reference straight. One of the top causes for errors in SR is forgetting/confusing which frames of reference you're referring to. If you switch to the moving frame you must also switch your equations accordingly; the "moving frame" S' (even though neither frame is preferred, as SR asserts) will measure the "stationary frame" S as running slower by an equal factor as S will measure about S'.

    Last edited: Apr 19, 2004
  17. Apr 21, 2004 #16
  18. Sep 9, 2004 #17


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    In Michael Spivak's book Caklculus on Manifolds, chapter 2 is differentisation, chapter 3 is integration and chapter 4 begins with tensors ("algebraic preliminaries"), and then moves on the general stokes theorem, so tensors are now considered part and parcel of several variable calculus.
  19. Sep 14, 2004 #18
    these days iam getting mad...because i started learning tensor notation of vectors.i am very much new to this .can anybody tell me where to start and how to proceed.actually i want to study general theory of relativity.
    Last edited: Sep 14, 2004
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