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.