Rapidity - Aetherwisp

The rapidity $y$ is an alternative quantity to represent the speed of a Particle, typically in the context of relativity. Given the relativistic speed coefficient $\beta=v/c$ , where $c$ is the speed of light, the rapidity is defined as its hyperbolic arctangent:

y=\text{arctanh } \beta

Rapidity, by construction, is a relativistic invariant.

The benefit of rapidity is that, since it is defined from a hyperbolic arctangent, it goes from $-\infty$ to $+\infty$ as $v\to \pm c$ . Moreover, rapidities add like in classical physics ( $y_{1}+y_{2}=y_\text{total}$ ) instead of having to follow the Einstein velocity addition rule. These facts combined make rapidity quite similar to how velocity works in classical physics, which can make it a more intuitive way of describing movement.

Derivation#

The rapidity is the value for which the following relations hold:

\begin{cases} e^{y}=\gamma(1+\beta)=\gamma\left(1+ \frac{v}{c}\right)=\sqrt{\frac{1+ \frac{v}{c}}{1- \frac{v}{c}}}=\sqrt{\frac{1+\beta}{1-\beta}} \\ e^{-y}=\ldots=\sqrt{\frac{1-\beta}{1+\beta}} \end{cases}

Then $y=\ln(\gamma(1+\beta))$ and

\begin{cases} ct-x=e^{-y}(ct'-x') \\ ct+x=e^{y}(ct'+x') \end{cases}

\frac{e^{y}+e^{-y}}{2}=\gamma\quad;\quad \frac{e^{y}-e^{-y}}{2}=\beta\gamma

from which we find

\begin{cases} \gamma=\cosh(y) \\ \beta\gamma=\sinh(y) \end{cases}

and putting them together

\beta=\tanh(y)

As a hyperbolic rotation#

Then we can interpret a Lorentz transformation as a hyperbolic rotation in spacetime, with rapidity representing the angle of rotation:

\pmatrix{ct' \\ x' \\ y' \\ z'}=\pmatrix{\cosh y & -\sinh y & 0 & 0 \\ -\sinh y & \cosh y & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1}\pmatrix{ct \\ x \\ y \\ z}