Show that rotations and boosts lead to a combined boost

1. Oct 26, 2015

ma18

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. Oct 26, 2015

Staff: Mentor

You can expand the exponential of an infinitesimal rotation and then simplify.

3. Oct 26, 2015

ma18

Doing this I get to the equation:

Which after expanding leads me to

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

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

What am I missing?

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4. Oct 27, 2015

Staff: Mentor

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. Oct 27, 2015

ma18

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. Oct 27, 2015

Staff: Mentor

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. Oct 27, 2015

ma18

Ah I see, the RHS is supposed to be (where G_x represents the new boost around x_3)

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

and solving for G_x we get

Correct?

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8. Oct 27, 2015

Staff: Mentor

Should be right.

9. Oct 27, 2015

ma18

Great, thank you so much for your help