# Show that rotations and boosts lead to a combined boost

## 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

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

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?

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mfb
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.

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
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.

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?

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mfb
Mentor
Should be right.

ma18
Should be right.
Great, thank you so much for your help