# Is there an analog to polar coordinates for Minkowski 4-vectors?

Is there a way to represent a Minkowski 4-vector using a 4D polar coordinate system, i.e. a single radial coordinate and 3 angular coordinates?

I know this can be done in Euclidean 4-space with spherical coordinates. And I know that 3D spherical coordinates can be used for the spacelike part of a Minkowski 4-vector. But I'm not sure how to combine the spacelike and timelike parts into one set of polar coordinates.

• At the most basic level a four-vector is just a list of four real numbers, so you can use the Euclidean spherical coordinates if you want. Is there some additional special property you want the coordinates to have? May 3, 2017 at 12:06
• Well, I'm simulating elastic collisions, which I'm presently doing by Lorentz-transforming 4-momentums in/out of the center-of-momentum frame. So, presumably I need 4-vectors that can be Lorentz-transformed, and that allow me to find the CoM frame easily. I'm wondering if polar coordinates might help with numerical stability. May 3, 2017 at 12:25
• web.mit.edu/edbert/Alexandria/notes2.pdf May 3, 2017 at 13:16

You can imagine starting with a four-vector in its rest frame, $p=(m,0,0,0)$, where $p^2 = m^2$. Then you can boost in the $z$ direction with rapidity parameter $\eta \in [0,\infty)$, giving you $p = m(\cosh\eta,0,0,\sinh\eta)$. You can check that this left the Minkowski norm invariant. Finally, you can do a spatial rotation to make the three-vector part point any direction you want. $p = m(\cosh\eta,\sinh\eta \sin\theta\cos\phi,\sinh\eta\sin\theta\sin\phi,\sinh\eta\cos\theta)$.
For physical interpretation, note that $E = m\cosh\eta$, so $\cosh\eta$ is the same as the Lorentz factor $\gamma$, and $\sinh\eta = \beta\gamma$.
Note this form is only for time-like vectors. For lightlike vectors or spacelike vectors you have to start with different initial four-vectors, like $m(1,0,0,1)$ or $m(0,0,0,1)$.