Parametrizations of a four vector

Question

What are useful parametrizations of Minkowski 4-vectors for spherically symmetric problems?

I can only think of one, namely (for space-like intervals) $$ x^\mu = \rho(\cosh \alpha, \sinh \alpha \sin \theta \cos \phi, \sinh \alpha \sin \theta \sin \phi, \sinh \alpha \cos \theta) $$ Here $\rho = \pm \sqrt{x^\mu x_\mu}$ and $R=\rho \, \sinh \alpha$ is the radial distance which should be positive.

However, I'm not sure about the ranges for $\rho$ and $\alpha$. Should it be $(-\infty, \infty)$ or are we overcounting? Are there any other useful ways to parametrize 4-vectors?

Answer 1

Since $x^0\ne0$, $\rho$ matches its sign, so can be any nonzero real. To avoid overcounting, demand $\alpha\ge0$, so $\rho^{-1}x^i\operatorname{csch}\alpha$ is a unit vector in $\Bbb R^3$, each such vector counted once on the surface of fixed $\rho,\,\alpha$. (The case $\alpha=0$ yields one point per $\rho$, so at that point take e.g. $\theta=\phi=0$.)

As for other parameterizations, there's one of those for every general coordinate transformation; all you've done in particular is switched from the pedagogical starting point of Cartesian coordinates to a hyperboloid polar aalternative.