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Thread: Minkowski’s Four-Dimensional Space

  1. #1 Minkowski’s Four-Dimensional Space 
    Time Lord
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    Actually I am reading from Einstein's book of which one of the chapters is titled "Minkowski’s Four-Dimensional Space".
    It is a long way ahead of my learning but a part of it intrigues me and and would appreciate perhaps a bit of help so that I cold perhaps understand exactly (or maybe just more precisely ) what he (Einstein ) is alluding to.

    This is the passage (and also the link - Chapter 17. Minkowski’s Four-Dimensional Space. Einstein, Albert. 1920. Relativity: The Special and General Theory)

    In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude

    ct proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same rôle as the three space co-ordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. It must be clear even to the non-mathematician that, as a consequence of this purely formal addition to our knowledge, the theory perforce gained clearness in no mean measure
    .




    So where could I find this result explained in detail (or the mathematics immediately relevant to it) ?

    Or do I have to start from scratch really and study it properly (perhaps beyond me)?

    Would I be right to think that for starters it seems to explain the negative sign for (ct)^2 in the the metric for flat Minkowski space-time (since ^2 =-1 )

    (I had naively assumed it would be positive ,the same as for the x, y, and z )

    To explain how I came to be (trying to ) reading that part of Einstein's Paper I googled "is spacetime real" (in quotes) which brought me to an old discussion on the Physics Forums (Spacetime doesn't really exist does it?) where this Paper of Einstein's was referred to.


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  3. #2  
    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by geordief View Post
    So where could I find this result explained in detail (or the mathematics immediately relevant to it) ?
    Any textbook on Special Relativity will do. It should be noted though that in modern texts the imaginary time notation is no longer used, instead one uses "ct" as time coordinate. The result is of course the same.

    Would I be right to think that for starters it seems to explain the negative sign for (ct)^2 in the the metric for flat Minkowski space-time (since ^2 =-1 )
    Well yes, you can understand the negative sign this way. Minkowski space-time isn't Euclidean, but hyperbolic geometry, so the metric signature becomes either (-,+,+,+) or (+,-,-,-), both of which are equivalent.
    The physical reason why the time coordinate has a different sign than the spatial coordinates is that the total line element ds must be an invariant, i.e. all observers in space-time must agree on the separation of the same two events in space-time, regardless of their state of relative motion. This is equivalent to saying that all observers experience the same laws of physics. On closer examination it turns out that this is possible and internally self-consistent only if the temporal coordinate has an opposite sign to the spatial coordinates.


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  4. #3  
    Time Lord
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    thanks

    without further research at this point I do find it quite surprising that , to quote "Formally, these four co-ordinates correspond exactly to the three space co-ordinates"

    Would this be an example of "mapping" so that
    the three space co-ordinates would not correspond exactly to the four co-ordinates (only vice versa)? Are we in Russian doll territory?
    If this is an example of a relationship between dimensions ?Does it follow a kind of "family tree" ,hierarchical structure?
    Last edited by geordief; March 26th, 2014 at 06:04 AM.
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  5. #4  
    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by geordief View Post
    thanks

    without further research at this point I do find it quite surprising that , to quote "Formally, these four co-ordinates correspond exactly to the three space co-ordinates"

    Would this be an example of "mapping" so that
    the three space co-ordinates would not correspond exactly to the four co-ordinates (only vice versa)? Are we in Russian doll territory?
    If this is an example of a relationship between dimensions ?Does it follow a kind of "family tree" ,hierarchical structure?
    To be honest I have no idea what the author means by this statement. Minkowski space-time is not the same as Euclidean 3-space.
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  6. #5  
    Time Lord
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    That is a quote from Einstein I thought (perhaps you knew that) . If it was poorly expressed or wrong wouldn't it have been dissected as such by now?

    It sounds like I am treating his every statement as the ultimate authority (which I realize is wrong and lazy ) but would I be wrong to anticipate an extremely high degree of understanding from him -even though this was written ~100 years ago and he was not considered to have maths as his strongest suit.

    If you did know it was from Einstein (and if I am not mistaken to think it was ) are you sure that he was saying that Minkowski space-time was the same as Euclidean 3-space? That is not what I thought he was saying even if I didn't myself know quite what he did actually mean.
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  7. #6  
    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by geordief View Post
    If you did know it was from Einstein
    I didn't, but I don't think it is relevant who wrote this.

    are you sure that he was saying that Minkowski space-time was the same as Euclidean 3-space
    No, which is why I said that I don't know what the author means by this statement. Perhaps you can reference the text, so that we can have a look what the context of this statement was.
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  9. #8  
    Moderator Moderator Markus Hanke's Avatar
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    I think what he is trying to say is simply that in SR, time becomes a geometric dimension, and is hence treated the same way as space in Euclidean geometry. Which is of course correct - time in relativity is a geometric aspect of the universe, just as space is.
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  10. #9  
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    Actually, Minkowski's original formulation did ransfer SR to the complex plane, as the link asserts.

    It is found in many texts with titles like "Mathematics for Physicists or similar, just as quoted.

    Let's see ......

    Let's assume spacetme with "labels" . But Minkowski was not so naive - he set , and using Einsteins basic postulate about the invariance of light velocity in SR writes

    as an invariant which implies that Notice that

    Assume the configuration of 2 observers moving with constant uniform velocity relative to each other in, say, the positive and shared direction

    Then the standard Lorentz transformation gives



    where

    Now any transformation (in this sense) has a matrix representation, so I write



    Now set where is Real.

    Since and therefore tend to a minimum of 1 when and tends to a maximum of when it follows that

    so we have the Lorentz matrix



    Which is simply a hyperbolic rotation in Minkowski space
    Last edited by Guitarist; March 29th, 2014 at 05:33 PM.
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  11. #10  
    Time Lord
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    thanks.That Maths doesn't look too outlandish.I might try and hunt down a textbook.
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