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Special relativity

  1. May 10, 2009 #1
    1. The problem statement, all variables and given/known data

    L1 and L2 are two lorentz trasformation.
    show that L3=L1 L2 is a lorentz trasformation too.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 10, 2009 #2
    I could be wrong but I think it would be enough to show the invariance of the dot product under L3.
     
  4. May 10, 2009 #3
    what is the dot product?
     
  5. May 10, 2009 #4
    The scalar product, or inner product or whatever it's called of 4-vectors. In Minkowski space it's [tex]r_1 \cdot r_2 = c^2 t_1 t_2 - x_1 x_2 - y_1 y_2 - z_1 z_1[/tex] or with the signs reversed depending on the convention. But if you haven't yet done the Minkowski notation then you could write L1 and L2 as Lorentz transformations with some velocities and show by applying them consecutively that L3 has the same form.
     
  6. May 11, 2009 #5
    I write better :

    I have: [tex]\Lambda [/tex] [tex]^{\mu}_{\nu}[/tex] and [tex]\Lambda\widetilde{} [/tex] [tex]^{\mu}_{\nu}[/tex] : lorentz trasformations.

    Show that [tex]\Lambda\overline{}[/tex] [tex]^{\sigma}_{\rho}[/tex] = [tex]\Lambda\widetilde{} [/tex] [tex]^{\sigma}_{\mu}[/tex] [tex]\Lambda [/tex] [tex]^{\mu}_{\rho}[/tex] is a lorentz trasformation
     
  7. May 11, 2009 #6

    George Jones

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    What is the definition of Lorentz transformation given in your notes and/or text?
     
  8. May 11, 2009 #7
    a trasfonmation that not change the space-time distance of a point to the origin.
     
  9. May 11, 2009 #8

    George Jones

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    Can you write a definition in terms of mathematics, i.e., [tex]\Lambda^\mu{}_\nu[/itex] is a Lorentz transformation iff ... ?
     
  10. May 11, 2009 #9
    ...if :

    x[tex]^{2}_{0}[/tex] - r [tex]^{2}[/tex]= ([tex]\Lambda[/tex] [tex]^{0}_{\nu}[/tex] x[tex]_{0}[/tex])[tex]^{2}[/tex] -( [tex]\Lambda[/tex] [tex]^{\mu}_{\nu}[/tex] r[tex]_{\mu}[/tex])[tex]^{2}[/tex]
     
  11. May 11, 2009 #10
    is it right?
     
  12. May 11, 2009 #11
    You don't have the indices right, your right hand side depends on on [tex]\nu[/tex] whereas the left hand side doesn't. It should be:

    [tex](x^0)^2 - r^2 = (\Lambda^0{}_\nu x^\nu)^2 - (\Lambda^j{}_\nu x^\nu)^2[/tex]

    Where j goes from 1 to 3. More compactly:

    [tex]g_\alpha_\beta x^\alpha x^\beta = g_\mu_\nu \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta x^\alpha x^\beta[/tex]

    Where g is the metric. Use the fact that L1 and L2 satisfy this to show that L3 satisfies it as well.
     
  13. May 11, 2009 #12
    L1 and L2 are two generical lorents tranformation. I must demostrate that their product is also a lorentz transormation.
    Can I demostrate :

    [tex]g_\alpha_\beta x^\alpha x^\beta = g_\mu_\nu \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta x^\alpha x^\beta[/tex]

    with \Lambda =L3= L1 L2 ?
     
  14. May 11, 2009 #13
    Yes you can.
     
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