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Show that rotations and boosts lead to a combined boost

  1. Oct 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that applying an infinitesimal rotation of angle k<<1 around the axis x1, then a boost of speed -k along the axis x2 then the inverses of these is equal to a single boost of speed k^2 along the axis x3

    3. The attempt at a solution

    Putting this into mathematical terms I get

    e-i*k*Jx1*ei*k*Gx2*ei*k*Jx1*e-i*k*Jx2 = e-i*k2*Gx3

    However I don't really know where to go from here, the example we did in class only involved the rotations and we just used the matrix representations around the axis, but I don't know how exactly do it with the boosts.

    Any help would be much appreciated.
     
  2. jcsd
  3. Oct 26, 2015 #2

    mfb

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    You can expand the exponential of an infinitesimal rotation and then simplify.
     
  4. Oct 26, 2015 #3
    Doing this I get to the equation:

    upload_2015-10-27_0-24-32.png

    Which after expanding leads me to

    upload_2015-10-27_0-27-30.png

    Taking only the terms up to e^2 reduces this to

    upload_2015-10-27_0-29-57.png

    Which is not equal to the right hand side with only a single boost which expands to

    upload_2015-10-27_0-31-21.png

    What am I missing?
     

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  5. Oct 27, 2015 #4

    mfb

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    Where does e come from and where did k go?
    Your boost is along a different directions, the G on the different sides are not the same.
    Also, I think there is some error in the expansion of the right hand side.
     
  6. Oct 27, 2015 #5
    Ah sorry, I used e in the equation instead of k, they are the same. I see, as the G is different what would be my next step?
     
  7. Oct 27, 2015 #6

    mfb

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    Find a new G' such that the equation is true up to order k2.
    If you fix the expansion of the RHS, you are just one line away from this.
     
  8. Oct 27, 2015 #7
    Ah I see, the RHS is supposed to be (where G_x represents the new boost around x_3)

    upload_2015-10-27_18-22-1.png

    and then taking out the e^4 term this reduces to

    upload_2015-10-27_18-22-42.png

    and solving for G_x we get

    upload_2015-10-27_18-23-14.png

    Correct?
     

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  9. Oct 27, 2015 #8

    mfb

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    Should be right.
     
  10. Oct 27, 2015 #9
    Great, thank you so much for your help
     
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