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In $n$ dimensional Euclidean space, we only need one radial coordinate, and $n-1$ angular coordinates, where one ranges from $[0, 2π)$ and the rest range from $[0, π]$

Spherical Minkowski coordinates are introduced as a cylyndrical like system, with a spherical spacial part and a cartesian time part.

Are there coordinates for Minkowski space that are only radial+3 angular coordinates?

I'm sure one of the angles would be strange, and r would lose its simple interpretation, but it doesn't seem impossible to do.

Craig
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These are roughly Rindler coordinates. The downside is you need to use different coordinate systems to describe future-timelike, spacelike, past-timelike and null 4-vectors.

Future-timelike ($c=1$ units): \begin{align*} t &= \tau\ \cosh\eta, \\ x &= \tau\ \sinh\eta\ \cos\theta, \\ y &= \tau\ \sinh\eta\ \sin\theta\ \cos\phi, \\ z &= \tau\ \sinh\eta\ \sin\theta\ \sin\phi. \end{align*} Here $\eta$ runs over all real numbers. But $\tau$ has the interpretation of proper time

For spacelike 4-vectors you can swap $\sinh$ and $\cosh$ (and maybe rename $(\tau,\eta)$ as $(\rho, \tau)$ or something). For past-timelike you need a minus sign in the expression for $t$. I either can't remember how to deal with null vectors or I never knew.

Subhaneil Lahiri
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