Show that rotations and boosts lead to a combined boost

[ma18]

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*J_x1*e^i*k*G_x2*e^i*k*J_x1*e^-i*k*J_x2 = e^-i*k2*G_x3

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.

 

[mfb]

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

 

[ma18]

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

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?

 

[mfb]

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.

 

[ma18]

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?

 

[mfb]

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.

 

[ma18]

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)

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

and solving for G_x we get

Correct?

 

[mfb]

Should be right.

 

[ma18]

Should be right.

Great, thank you so much for your help