Lorentz Transformation: R_2=R_1^{-1}?

[facenian]

Let L_w be a Lorentz transformation between to systems that coincide at t=0(paralell axes assumed) and with relative velocity w along x_1. If L_u is the Lorentz transformation when the relative velocity u is in any direcction then we have that L_u=R_2 L_w R_1 where R_2 and R_1 are sapce rotations, R_1 is such that the direction of u and x_1 are the same.
My question is: is it correct that under this circuntances R_2=R_1^{-1}?

 

[bcrowell]

I think the answer is yes. It seems straightforward to verify that it holds for the case where u lies along x_2, i.e., the case where the angle theta between the w and u vectors is pi/2. Also, if it holds for rotations by theta in the 1-2 plane, then it also holds for 2theta. The combination of these two facts makes me think that it holds for any rotation in the 1-2 plane, and since the choice of the 1-2 plane is arbitrary, I think it has to hold for any rotation.

[When I previewed the post above, I saw some of the math rendered incorrectly. The first reference to theta is rendered by some other, unrelated math, (1-epsilon)c. I remember that this is a known bug in the software used by PF, but I don't remember if there is any way to fix it.]

[Later edit: it now seems to be rendered correctly.]

[Gah, now it's rendering incorrectly again, after I made another edit. I'll just remove all the math.]

 

[facenian]

yes there seems to problems when you write in latex that's why I don't use it anymore. Thank you for your answer