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Nature of time

  1. Jun 28, 2007 #1
    Apparently, when one calculates a length in flat space-time, one must add the lengths in the three spatial dimensions, and subtract... square root(-1)ct, my first question is...
    is there some sort of physical significance to a dimension being imaginary or is this simply a mathematical trick so to speak?
    furthermore, why subtract? why is it that this weird equilibrium exists between space and time where motion through space takes away from motion through time (aging, time passage... you know what I mean)?
    Why do we have this default time motion? The other dimensions don't work like this, it's not like I move up at full speed, but when I move right or left, I move up slower... the components are separate, like in projectile problems
    on a somewhat separate note, why do photons have paths in space-time if they don't move through time?
     
  2. jcsd
  3. Jun 29, 2007 #2

    Mentz114

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    Probably the people who know about these things are not answering this because there are too many misconceptions in your questions. The time dimension is not usually taken to be imaginary in relativity.

    There are too many 'why's as well. Actually no-one knows 'why' anything.

    Get a straightforward book on relativity and don't ask 'why' too often.
     
  4. Jun 29, 2007 #3
    The time dimension can be taken as imaginary in special relativity (but not in general relativity).

    Some professors prefer that due to practical advantages. See for instance Gerard 't Hooft in http://www.phys.uu.nl/~thooft/lectures/genrel.pdf [Broken]

    My stance is that in special relativity one should be comfortable with both approaches instead of praising one and dooming the other.

    If you can use imaginary time in general relativity, where spacetime is curved, you deserve a price since no one has ever been successful in doing this. :smile:
     
    Last edited by a moderator: May 2, 2017
  5. Jun 29, 2007 #4

    Mentz114

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    SpitfireAce, you must be wondering if you can get a straight answer here without starting a squabble ( I refer to your other thread).

    Let me try to give a more technical answer than I did last time.

    The reason why we subtract the time extension is that it works. In special relativity it turns out that quantity that is preserved in transformations of co-ordinates is the difference between the spatial extent and the temporal extent ( or interval). So in flat space

    [tex]ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2[/tex]

    is invariant. Quantities that are invariant under transformations are important because they represent physical things rather than mathematical artefacts.

    But as to 'why' his particular structure works - who knows ?
     
  6. Jun 29, 2007 #5
    Get a straightforward book on relativity and you will obtain the perfectly rigorous physically and mathematically answer to your question: L.D. Landau and E.M. Lifsheetz, “Field Theory”, v.2; par. 1&2.

    The definition of the interval is 1-1 consequence of the statement that there exists the upper bound for the velocity of the signal propagation (A. Einstein) which is now generally accepted as the Principal Physical Postulate.

    i^2 =-1 is the notion introduced by stupid mathematicians (abstract algebra) since they were not able to understand that they should introduce the matrices and the matrix multiplication.

    Sorry, but your post #4 is completely wrong.

    Regards, Dany.
     
  7. Jun 29, 2007 #6

    Fra

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    > Get a straightforward book on relativity and don't ask 'why' too often.

    Or perhaps keep asking why is exactly they way to go, as perhaps you will actually start to understand WHY you keep asking why all the time. And that would be a substantial progress, and the result may be a refined version of the original why.

    I was sometimes told as a kid to not "think too much", because it could make you go nuts.

    /Fredrik
     
  8. Jun 29, 2007 #7
    When I was a kid, I came to Professor V.N. Gribov and said that I want him to be my teacher. He said that I should pass the Landau-min exams (L.D. Landau and E.M. Lifsheetz, “Field Theory”, v.2 to begin with). I ask what his requirements are. He said: very simple, when you will finish study, you close it and write the same by yourself. Later he taught me to ask questions.

    Regards, Dany.
     
    Last edited: Jun 29, 2007
  9. Jun 29, 2007 #8

    Fra

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    Yes, that reminds me of a funny teacher I had in a relativistic QM course which was a professor. He seemingly didn't like to explain calculations to students that asked if they did it right, and he always seemed very bothered and responded that when you understand the topic you will no longer ask these questions, you will know wether it's right on your own.

    /Fredrik
     
  10. Jun 30, 2007 #9

    Mentz114

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    Dany:
    I don't see anything in your post that justifies that !

    What is wrong with what I've said ?

    I bet you can't say why this so.

    I'm not trying to discourage people asking questions - but 'why' is not the right question. If you mean 'why do we believe...' then that can be answered.

    There have been other discussions on PF about this and the thing is to distinguish descriptions from reasons. Maxwells equations describe the EM phenomena exactly ( as far as experiments can tell) but tell us nothing about 'why' like charges repel.

    M.
     
    Last edited: Jun 30, 2007
  11. Jun 30, 2007 #10
    Sign.

    Regards, Dany.
     
  12. Jun 30, 2007 #11

    Mentz114

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    Dany: Do you object to the signature -+++ in the Minkowski metric ?

    M

    [edit] What I mean to say is that the interval is invariant whether one uses -+++ or
    +---. It's a matter of convention.
     
    Last edited: Jun 30, 2007
  13. Jun 30, 2007 #12
    Not. Physics is the empirical science. The notion of the invariance (covariance) is defined in physics only with respect to the inertial systems which represent the observers. One can’t justify the existence of the observer the communication with him is impossible. The existence of the invariant quantities together with the infinite set of the independent observers allows defining the notion of the objective reality.

    The best presentation of that I saw in the book referred above.

    Regards, Dany.
     
    Last edited: Jun 30, 2007
  14. Jun 30, 2007 #13

    Mentz114

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    I don't see the relevance of what you've posted to the sign convention of the metric. Nor does anything you say contradict any of my other statements about invariant quantities. Adds to it, maybe.

    Let's call it a day - I don't know what we're arguing about.

    M
     
  15. Jun 30, 2007 #14
    There is a simple way to see why the time term has to be negative. If it were positive, then the speed of light would have to be imaginary. Just set ds=0 ...
     
  16. Jun 30, 2007 #15
    When I learnt relativity at university the Mathematics department taught us to use the -+++ sign for the metric in Minkowski spacetime, whereas the Physics department taught us to use +--- (on the basis that spacetime intervals would be >0 for timelike separated events, which seems a more logical way of thinking about it to me)
     
  17. Jun 30, 2007 #16
    For the theory it really does not matter what signature you use.

    The best is to become comfortable with both signatures since they are both used in textbooks, papers and monologues.
     
  18. Jun 30, 2007 #17
    How can you write it (and make sense of how its presented) for both possibilites (and be variable for each/every sign) in the same/close amount of symbols? --or can it be?
     
    Last edited: Jun 30, 2007
  19. Jul 1, 2007 #18
    So Minkowski was a stupid mathematician?
     
  20. Jul 2, 2007 #19
    You should not take my words too literally. I used it to emphasize the difference between the mathematics and the physics. The matrices were introduced by W.R. Hamilton (the physicist indeed). H. Minkowski obviously was familiar with them. Also no doubt the abstract algebra is very useful development.

    My purpose was to point out that the physical requirements provide additional, very special restrictions imposed on the general mathematical consideration. For example, in my discussion above with Mentz114 clearly he is right mathematically (Noether “currents” and Casimir “charges”), but in the mathematics the notion of “observer” do not exist. That is the difference between the physics that deal with the “objective reality” and the mathematics that provide the infinite set of “subjective realities”.

    The special relativity as well as the general relativity is the physical theories that use only the relevant pieces of the mathematical constructs. A posteriori you may claim correctly that the physical theory is one of the particular realizations among all possible mathematical schemes, but it is impossible to find it without physical consideration. In QM the observable is the self-ajoint operator and it eigenvalues must be real. Therefore, you may forget about the imaginary time. In addition, it demonstrates the intrinsic mutual consistency between SR, GR and QM.

    Regards, Dany.

    P.S. Notice that the length in physics must be positive definite quantity. Mentz114 post #4 is not the first that I enjoy reading. I am sure that he understood what I said; only he did not agree.

    P.P.S. In order to appreciate how difficult is the problem of generalization, try to find the generalization of the matrix multiplication for the Cayley numbers.
     
  21. Jul 2, 2007 #20
    Well, when Minkowski introduced i (as sqrt(-1)) he used the term "mystic." Not stupid, but maybe a bit spiritual.
     
  22. Jul 3, 2007 #21
    ict & signature

    0. I think there was an excuse for using i=sqrt(-1) between 1905 & 1915. It helped to show a ‘rotational’ symmetry between x & ict, it saved using a metric, and only a few mathematicians/physicists would understand the subject anyway.
    1. But it introduced unnecessary abstraction or mystery (I agree with post #20). Even it you do use (x, ict), you have to translate it back to (x, ct) to apply to real measurements. And since GR, you need to use a metric anyway - so ict has saved nothing.
    2. After advanced QM came in, I gather there is an excuse to bring back ict. But not in introductory treatments. (I am out of my depth here - but I believe there may be a connection between statistical mechanics, time, inverse temperature & sqrt(-1).)
    3. Like post #15 I can really see why physicists like the +--- signature as it gives due emphasis on time.
    4. But I sympathise with mathematicians who start with +++ 3d and add in time as an after-thought. The signs involved with momenta, Hamiltonians & Lagrangians are more friendly. For 5d and above, these dimensions have a spatial rather then temporal signature.
    5. (I could be quite wrong here, but my own struggle in getting to grips with the Kaluza-Klein ‘miracle’ leads me to favour -++++ for 5d.)
    Regards, Mike.
     
  23. Jul 3, 2007 #22
    I agree. Abstraction and mystery can get in the way.

    Imposing mathematical formalism on physical theory is what makes it work. However, there is a tendency to regard the beauty in the mathematics as some kind of metaphysical truth, and this can mask the physical unknowns that remain. An example is the casting of relativity in terms of geometry, thereby eliminating any discussion of the ether. Although the early concepts of the ether were wrong, there does seem to be something there (gravity, the vaccum?). In solving physics problems we should strive to see past the mathematics.
     
  24. Jul 3, 2007 #23
    No. It is not right place. It was clearly demonstrated by C.N Yang and R.L. Mills, Phys. Rev., 96, 191 (1954) that imaginary units are connected with the phases (internal symmetries and fundamental interactions).

    Unfortunately, I have no idea what notion of “mystery” mean. On level of pure speculation I would say that it express the religion attitude which has much deeper roots in the human consciousness than the science.

    Regards, Dany.
     
  25. Jul 3, 2007 #24

    Mentz114

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    Can we agree that using i is always a mathematical trick and that any physical theory using i can be rewritten without i ( probably less elegantly) ?
     
  26. Jul 3, 2007 #25
    I don’t know yet. My intuition says that the answer is no. E.C.G. Stueckelberg demonstrated that the answer in Quantum Physics is no, but I guess that is not what you are asking. It is directly connected with the nature of time in the Schrödinger equation. Also I use for gravitation ei, i=1,2,…,7 with ei^2=-1. It can’t be written in terms of matrices with real matrix elements.

    Regards, Dany.
     
    Last edited: Jul 3, 2007
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