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 $uv=x^2-y^2$.

In the $(t,z)$ plane, $u=t-z$ is retarded time and $v=t+z$ is advanced time. The radial part of the Minkowski metric is $-d{t}^2+d{z}^2=-dudv$.

For $u=t-z$ and $v=t+z$, the four-dimensional Minkowski wave operator isEvery profile depending only on one null coordinate solves the wave equation when it is uniform in the transverse coordinates.