special relativity - How to add together non-parallel rapidities?

How to add together non-parallel rapidities?

The Lorentz transformation is essentially a hyperbolic rotation, which rotation can be described by a hyperbolic angle, which is called the rapidity. I found that this hyperbolic angle nicely and simply describe many quantities in natural units:

• Lorentz factor: $\mathrm{cosh}\,\phi$


• Coordinate velocity: $\mathrm{tanh}\,\phi$


• Proper velocity: $\mathrm{sinh}\,\phi$



• Total energy: $m\,\mathrm{cosh}\,\phi$


• Momentum: $m\,\mathrm{sinh}\,\phi$


• Proper acceleration: $d\phi / d\tau$ (so local accelerometers measure the change of rapidity)

Also other nice features:

• Velocity addition formula simplifies to adding rapidities together (if they are parallel).


• For low speeds the rapidity is the classical velocity in natural units.

I think for more than one dimensions, the rapidity can be seen as a vector quantity.

In that case my questions are:

• What's the general rapidity addition formula?


• And optionally: Given these nice properties why don't rapidity used more often? Does it have some bad properties that make it less useful?