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Thread: Lorentz transformation derivation.

  1. #1 Lorentz transformation derivation. 
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    I had been looking at the derivation of the lorentz transformation for a while and i came across the first statement which stated something as this



    where and implies the frame and the unprimed the frame.

    My question is how do i understand such equation. thnks in advance.


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    Quote Originally Posted by newspaper View Post
    I had been looking at the derivation of the lorentz transformation for a while and i came across the first statement which stated something as this



    where and implies the frame and the unprimed the frame.

    My question is how do i understand such equation. thnks in advance.
    The quantity varies linearly from frame to frame, hence the


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  4. #3  
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    Quote Originally Posted by xyzt View Post
    Quote Originally Posted by newspaper View Post
    I had been looking at the derivation of the lorentz transformation for a while and i came across the first statement which stated something as this



    where and implies the frame and the unprimed the frame.

    My question is how do i understand such equation. thnks in advance.
    The quantity varies linearly from frame to frame, hence the
    I thought of the same thing but my mind just does not digest it...If you could elaborate a little more on this it would be helpful..
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    Quote Originally Posted by newspaper View Post
    Quote Originally Posted by xyzt View Post
    Quote Originally Posted by newspaper View Post
    I had been looking at the derivation of the lorentz transformation for a while and i came across the first statement which stated something as this



    where and implies the frame and the unprimed the frame.

    My question is how do i understand such equation. thnks in advance.
    The quantity varies linearly from frame to frame, hence the
    I thought of the same thing but my mind just does not digest it...If you could elaborate a little more on this it would be helpful..
    The spatial isotropy and homogeneity dictates that the relationship should be linear.
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  6. #5  
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    Quote Originally Posted by xyzt View Post
    The spatial isotropy and homogeneity dictates that the relationship should be linear.
    why should spatial isotropy and homogeneity dictates that the relationship should be linear? isnt there any simple understandable reason..
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    Quote Originally Posted by newspaper View Post
    Quote Originally Posted by xyzt View Post
    The spatial isotropy and homogeneity dictates that the relationship should be linear.
    why should spatial isotropy and homogeneity dictates that the relationship should be linear? isnt there any simple understandable reason..
    Because only linear dependencies preserve the homogeneity and isotropy, all non-linear dependencies, do not.
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  8. #7  
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    Quote Originally Posted by xyzt View Post
    Quote Originally Posted by newspaper View Post
    Quote Originally Posted by xyzt View Post
    The spatial isotropy and homogeneity dictates that the relationship should be linear.
    why should spatial isotropy and homogeneity dictates that the relationship should be linear? isnt there any simple understandable reason..
    Because only linear dependencies preserve the homogeneity and isotropy, all non-linear dependencies, do not.
    how do we prove it?
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    I looked into this a few months back and spent a good 2 weeks scratching my head (and paper) about it.It is actually Einstein's maths from his own book I think.

    I also wondered how he adopted a linear formulation so readily and I think the answer I came to was "because it was the simplest and because when you go through the maths it all works out in the end".

    Perhaps in hindsight it may be that "it had to be linear" I can't say one way or the other.

    I approached the subject with the assumption that it was going to be something complicated and it didn't occur to me it could be linear .

    You are welcome to look at the thread I started about this derivation back in Feb (nobody answered ,probably for a good reason).

    Einstein's mathematics in his derivation of the Lorentz transformation.
    It is unlikely to be illuminating but at least you might sympathise (or gloat at ) with my struggling with the same maths that you are now looking at.
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    Quote Originally Posted by newspaper View Post
    how do we prove it?
    The Lorentz transformations are those operations that preserve the metric, and only those operations. As such they have a very specific form, which ( thankfully ) happens to be linear, in the sense mentioned earlier.
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    Quote Originally Posted by Markus Hanke View Post

    The Lorentz transformations are those operations that preserve the metric, and only those operations. As such they have a very specific form, which ( thankfully ) happens to be linear, in the sense mentioned earlier.
    I understand the invariance of the metric due to lorentz transformation, but my question is more to do with the derivation of the lorentz transformation itself..
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    Quote Originally Posted by geordief View Post
    I looked into this a few months back and spent a good 2 weeks scratching my head (and paper) about it.It is actually Einstein's maths from his own book I think.

    I also wondered how he adopted a linear formulation so readily and I think the answer I came to was "because it was the simplest and because when you go through the maths it all works out in the end".

    Perhaps in hindsight it may be that "it had to be linear" I can't say one way or the other.

    I approached the subject with the assumption that it was going to be something complicated and it didn't occur to me it could be linear .

    You are welcome to look at the thread I started about this derivation back in Feb (nobody answered ,probably for a good reason).

    Einstein's mathematics in his derivation of the Lorentz transformation.
    It is unlikely to be illuminating but at least you might sympathise (or gloat at ) with my struggling with the same maths that you are now looking at.
    well, i believe you had a same curiosity, which have very little thing to do with understanding algebra but to be able to visualize....
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    Well I did have to relearn my rusty algebra in the derivation .It took me about 20 or 30 hours of actual work , I would say to go through the workings of this page:

    Appendix 1. Simple Derivation of the Lorentz Transformation. Einstein, Albert. 1920. Relativity: The Special and General Theory

    I was interested also in visualisation but I am not very good at geometry.
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    Quote Originally Posted by newspaper View Post
    I understand the invariance of the metric due to lorentz transformation, but my question is more to do with the derivation of the lorentz transformation itself..
    What is was trying to point out was that the situation is rather the other way around - the invariance of the metric is the cause, rather than the effect of the Lorentz transformations. The Minkowski metric can more generally be viewed as a quadratic form, and one then asks what kind of operations leave that form unchanged ( reason being that all observers should see the same laws of physics ); closer inspection reveals that the set of all such operations forms a group, the generalised orthogonal group O(1,3). This is a matrix Lie group, with the Lorentz transformation matrices being its elements, and from the group axioms it is then possible to explicitly derive the coordinate representation of the Lorentz transformations :

    Derivations of the Lorentz transformations - Wikipedia, the free encyclopedia

    To put it simply - the Lorentz transformations are derived from the very simple requirement that, in Minkowski space-time, all inertial observers should see the same laws of physics, regardless of their states of relative motion. This is possible only if the measurements these observers perform are related in such a way that they all agree as to the separation of two fixed events in space-time, meaning they all see the same metric. That is the geometric justification behind Lorentz transformations.
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