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?  

 A: Angles work very well in two dimensions (the Euclidean plane or 1+1 spacetime), where you only need one. In three or more dimensions, there are singularities in the representation of direction or rotation/orientation by angles, and other representations are more elegant.
Also, in 3 or more dimensions, the space of directions (velocities) and the space of rotations (Lorentz transformations) are different. You can't conflate them any more. This means that there is no analogue of "rapidity addition" in higher dimensions. Instead, you must either apply a rotation (Lorentz transformation) to a direction (velocity), or else compose two rotations, and these are different operations.
The most elegant way to represent a direction is by a unit vector pointing in that direction. In spacetime, this is the four-velocity. It isn't precisely analogous to the rapidity but it's the generalization that will give you the fewest headaches. In terms of the four-velocity $\mathbf v$ you have


*

*Lorentz factor: $\mathbf v \cdot \mathbf{\hat t}$

*Coordinate velocity: $\mathbf v / (\mathbf v \cdot \mathbf{\hat t}) - \mathbf{\hat t}$

*Proper velocity: $\mathbf v - (\mathbf v \cdot \mathbf{\hat t}) \mathbf{\hat t}$

*Total energy and momentum: $m\mathbf v$

*Proper acceleration: $\mathrm d\mathbf v/\mathrm dτ$
In 3D Euclidean space, the most elegant way to represent a rotation or an orientation is by a unit quaternion. The generalization of this to arbitrary dimension and signature is called the even Clifford algebra of the space. By multiplying elements of the Clifford algebra, you can see why vectors are inadequate for this purpose. For example, the composition of boosts in the $\mathbf v$ and $\mathbf u$ directions is $$(A + B \mathbf{\hat t u})(C + D \mathbf{\hat t v}) = AC + \mathbf{\hat t}(BC\mathbf u + AD\mathbf v) - BD\mathbf{uv}$$ where $A,B,C,D$ are scalars that I'm omitting for brevity. If $\mathbf u {\parallel} \mathbf v$ then $\mathbf{uv}$ is a scalar and this is equivalent to a boost in the same direction. Otherwise, it isn't a boost, because there is an additional rotation in the $\mathbf{uv}$ plane.
