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General Relativity Problem

  1. Jun 14, 2013 #1
    Hello,

    I was working out of a text for my own knowledge and I ran into a slight snag which has become bothersome. First I was asked to find the condition on a linear transformation for which the dot product [itex]u \cdot u= u_0^2 - u_1^2{}[/itex] is preserved. Easily I found that for [itex] u' = \Lambda u[/itex] our condition is [itex] \Lambda^T \eta \Lambda = \eta [/itex] where [itex] \eta [/itex] is the metric tensor.

    However the tricky part came in when I was asked "Solve this condition in terms of ra-
    pidity.". It was quite clear that I was asked to derive the lorentz transformation matrix but I wasn't sure the best way to go about doing this from only this condition.

    My best attempt was to populate [itex] \Lambda [/itex] with 4 unknown functions [itex] f_1,f_2,..[/itex] of our rapidity [itex] \phi [/itex]. This gave me 3 equations namely

    [itex] f_1*f_2 = f_3*f_4 [/itex]
    [itex] f_1^2 - f_3^2 = 1 [/itex]
    [itex] f_2^2 - f_4^2 = -1 [/itex]

    From which it seems clear that [itex] f_1 = cosh(\phi), f_2=sinh(\phi),... [/itex]

    However I have two questions/problems with my approach. (1) I do not know how to select a sign for my functions (it seems completely arbitrary what sign cosh or sinh take on, but most texts seem to have a definite convention and (2) my method does not seem completely rigorous and I was hoping to find one that was superior.
     
    Last edited: Jun 14, 2013
  2. jcsd
  3. Jun 14, 2013 #2
    Hi ozone!

    (1) Yes there is a sign issue here. There are two possibilities for [itex]\det \Lambda[/itex]: +1 and -1. This means that the Lorentz group splits into two. The transformations with [itex]\det \Lambda[/itex] = +1 are the ones you're familiar with (called the "restricted Lorentz group") and the transformations with [itex]\det \Lambda[/itex] = -1 involve parity flips, time reversals and things like that.

    (2) I think your approach is fully rigorous :smile:
     
  4. Jun 17, 2013 #3
    Alright fair enough, I have heard of the lorentz transformations having a + or - determinate, so this must be the manifestation of that, thank you.
     
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