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Regarding notation for Lorentz transformation

  1. Aug 20, 2013 #1
    Difficulty regarding notation for Lorentz transformation

    Please can somebody explain to me the relation between Δ[itex]^{σ}[/itex][itex]_{μ}[/itex] and Δ[itex]_{σ}[/itex][itex]^{μ}[/itex] as symbols representing a Lorentz transformation?

    Thanks.
     
    Last edited: Aug 20, 2013
  2. jcsd
  3. Aug 20, 2013 #2

    Mentz114

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  4. Aug 20, 2013 #3

    DrGreg

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  5. Aug 21, 2013 #4

    Bill_K

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  6. Aug 21, 2013 #5
    It was silly of me to use Δ instead of [itex]\Lambda[/itex].

    Thanks everybody for the help given.
     
  7. Aug 21, 2013 #6

    vanhees71

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    No! Please use LaTeX. It's not only quicker to type but also better to read. You write for the reader and not for yourself. A bit effort to make your text as readable as possible increases to probability to be read tremendously!
     
  8. Aug 21, 2013 #7

    Bill_K

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    That's why I use UTF.
     
  9. Aug 21, 2013 #8
    So one can write

    x[itex]^{μ^{'}}[/itex]x[itex]_{μ^{'}}[/itex]= λ[itex]^{μ^{'}}[/itex][itex]_{α}[/itex]x[itex]^{α}[/itex] λ[itex]_{μ^{'}}[/itex][itex]^{β}[/itex] x[itex]_{β}[/itex] = [itex]\delta[/itex][itex]^{β}_{α}[/itex] x[itex]^{α}[/itex]x[itex]_{β}[/itex] = x[itex]^{α}[/itex]x[itex]_{α}[/itex]

    showing the invariance of x[itex]^{α}[/itex]x[itex]_{α}[/itex] under the Lorentz transformation.

    Hence the notation of the indices itself suggests the position of the various indices.
     
  10. Aug 21, 2013 #9

    vanhees71

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    That's also a very misleading notation. Don't put the primes at the indices but on the symbol, i.e., the above equation you should write
    [tex]x'^{\mu} x'_{\mu}={\Lambda^{\mu}}_{\alpha} {\Lambda_{\mu}}^{\beta} x^{\alpha} x_{\beta}=\delta_{\alpha}^{\beta} x^{\beta} x_{\alpha}=x^{\alpha} x_{\alpha}.[/tex]
    It can be pretty misleading to put the primes on the indices rather than the symbol, because if you rename simply dummy indices in a Einstein-convention sum it's a trivial identity:
    [tex]x^{\alpha} x_{\alpha}=x^{\mu'} x_{\mu'}.[/tex]
    Here the dummy index on the rhs of the equation simply is called [itex]\mu'[/itex] and that's it.
     
  11. Aug 22, 2013 #10

    Bill_K

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    What part of Λμν = ημσ Λστ ητν do you find difficult to read?
     
  12. Aug 22, 2013 #11

    dextercioby

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    Not all Greek letters look that Greek: tau, nu.
     
  13. Aug 23, 2013 #12
    Yes if one renames simply dummy indices in a Einstein-convention sum then it's a trivial identity and such use of indices IS misleading.

    But denoting a Lorentz transformation by [itex]\Lambda[/itex][itex]^{μ'}[/itex][itex]_{β}[/itex] will show that the Lorentz transformation is between the frame of reference labelled by the primed indices and the frame of reference with the unprimed indices.

    Hence if one is talking about three Lorentz transformations involving frames of reference labelled by xμ, xμ' and xμ'' one can distinguish which transformation one is using.

    Thanks everyone for your help. I am a starter in all this!
     
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