Parametrizations of a four vector

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?

• Since $(x^0)^2>\sum_i(x^i)^2$, surely this is time-like.
– J.G.
Aug 20, 2021 at 16:20

1 Answer

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.