How Do You Solve the Lorentz Transformation Condition Using Rapidity?

[ozone]

Hello,

I was working out of a text for my own knowledge and I ran into a slight snag which has become bothersome. First I was asked to find the condition on a linear transformation for which the dot product u \cdot u= u_0^2 - u_1^2{} is preserved. Easily I found that for u' = \Lambda u our condition is \Lambda^T \eta \Lambda = \eta where \eta is the metric tensor.

However the tricky part came in when I was asked "Solve this condition in terms of ra-
pidity.". It was quite clear that I was asked to derive the lorentz transformation matrix but I wasn't sure the best way to go about doing this from only this condition.

My best attempt was to populate \Lambda with 4 unknown functions f_1,f_2,.. of our rapidity \phi. This gave me 3 equations namely

f_1*f_2 = f_3*f_4
f_1^2 - f_3^2 = 1
f_2^2 - f_4^2 = -1

From which it seems clear that f_1 = cosh(\phi), f_2=sinh(\phi),...

However I have two questions/problems with my approach. (1) I do not know how to select a sign for my functions (it seems completely arbitrary what sign cosh or sinh take on, but most texts seem to have a definite convention and (2) my method does not seem completely rigorous and I was hoping to find one that was superior.

 

[Oxvillian]

Hi ozone!

(1) Yes there is a sign issue here. There are two possibilities for \det \Lambda: +1 and -1. This means that the Lorentz group splits into two. The transformations with \det \Lambda = +1 are the ones you're familiar with (called the "restricted Lorentz group") and the transformations with \det \Lambda = -1 involve parity flips, time reversals and things like that.

(2) I think your approach is fully rigorous

 

[ozone]

Alright fair enough, I have heard of the lorentz transformations having a + or - determinate, so this must be the manifestation of that, thank you.