Show that rotations and boosts lead to a combined boost

  • Thread starter ma18
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Homework Statement


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

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.
 

Answers and Replies

  • #2
34,654
10,796
You can expand the exponential of an infinitesimal rotation and then simplify.
 
  • #3
93
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You can expand the exponential of an infinitesimal rotation and then simplify.
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?
 

Attachments

  • #4
34,654
10,796
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.
 
  • #5
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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.
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?
 
  • #6
34,654
10,796
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.
 
  • #7
93
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Ah sorry, I used e in the equation instead of k, they are the same.
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.
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
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Should be right.
Great, thank you so much for your help
 

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