Easiest possible way to derive the Lorentz transformation

[m4r35n357]

One more thing about rapidity (sorry!).

Well, there is also the fact that you can use it to appease people who are unhappy with a finite/maximum speed of light ;)

 

[robphy]

One more thing about rapidity (sorry!).

It's well-known that rapidities represent hyperbolic angles. What often goes unstated, though—perhaps because it's thought to follow trivially from the previous sentence—is that when Bob measures Alice's rapidity, he's measuring the hyperbolic angle between her four-velocity and his own.

The big deal with rapidities is that they are additive (since Minkowski-arc-length along the "Minkowski-circle" [the hyperbola] is additive).
This is contrasted with velocities.
Note:
$$v_{AC} =\tanh\theta_{AC} =\tanh\left( \theta_{AB}+\theta_{BC}\right) =\frac{\tanh\theta_{AB}+\tanh\theta_{BC}}{1 + \tanh\theta_{AB}\tanh\theta_{BC}} =\frac{v_{AB}+v_{BC}}{1 + v_{AB}v_{BC}}$$

In addition, $\exp(\theta_{AC})=\sqrt{\frac{1+V_{ac}}{1-V_{ac}}}$, the Doppler factor [which is an eigenvalue of the Lorentz boost].
It's worth noting that there are Galilean-rapidities, as well.
It's the Galilean-spacelike-arc-length along the "Galilean-circle" [the vertical line on a position-vs-time graph].
The analogue of the "tangent function of the Galilean-rapidity" in Galilean spacetime geometry is the Galilean-rapidity itself.
Because of this, Galilean-velocities are also additive (since Galilean-rapidity is additive).
This situation of the additivity-of-Galilean-velocities is one of those prejudices making up our common sense that makes relativity hard to understand.

 

[SiennaTheGr8]

The big deal with rapidities is that they are additive (since Minkowski-arc-length along the "Minkowski-circle" [the hyperbola] is additive).
This is contrasted with velocities.

Yes, and what I'm trying to emphasize is that in addition to the mathematical elegance this gives you (additivity is nice!), there's a conceptual elegance. Even if you aren't drawing a diagram, rapidity has a simple geometric interpretation (angle between an object's four-velocity and your own).

 

[SiennaTheGr8]

The other "big deal with rapidities" is that differences/changes in them are invariant under a collinear boost (or in the 1+1D case), though you need to allow for signed rapidities for this to work in all cases.

More generally, under a boost in the $x$-direction, it's differences/changes in the quantity $\tanh^{-1} \beta_x$ that are invariant. This is sometimes called the "longitudinal rapidity" (or just "rapidity" in the context of particle physics). Note that this quantity is not the $x$-component of the rapidity vector $\vec \phi = \phi \hat \beta$.