One-dimensional Lorentz group ($SO(1,1)$)

The determinant-one Lorentz group in one space and one time dimension has two connected components. Its identity component consists of boostsand the other component is its negative.

A boost of rapidity $\phi$ acts on null coordinates by $u\mapsto e^{\phi }u$ and $v\mapsto e^{-\phi }v$, preserving $uv=x^2-y^2$.