Lorentz transformation boosts as matrices

Question

I have seen Lorentz transformation boosts (say, along the x-direction) written as (in $c=1$ units):

$$ \left[\begin{array}{cccc}{\gamma} & {-v \gamma} & {0} & {0} \\ {-v \gamma} & {\gamma} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right] $$

as well as (where $\varphi$ is the boost parameter per Caroll, is this related to the rapidity?)

$$\left[\begin{array}{cccc}{\cosh \varphi} & {-\sinh \varphi} & {0} & {0} \\ {-\sinh \varphi} & {\cosh \varphi} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right]$$

What is the difference between these? If they are equivalent, how do they relate to one another? Should I think of one as a translation, one as a rotation, or something else? Under what circumstances would I use the top or the bottom? Any clarification would be much appreciated.

Answer 1

They are equivalent. The parameter $\varphi$ is the rapidity, just another way to describe how fast something is moving. The relationship between rapidity and speed is

$$\varphi=\tanh^{-1}\frac{v}{c}$$

so the rapidity goes from $0$, when $v=0$, to $\infty$ when $v=c$.

Don’t think of either matrix as any kind of translation.

You can think of both as rotations in a spacetime with a complex time coordinate $ict$, but don’t take that as anything more than a mathematical analogy to a true spatial rotation. Sometimes this transformation is called a “hyperbolic rotation” of Minkowski spacetime.

Rapidity is very nice to use when doing “relativistic addition” and “subtraction” of velocities in the same direction. The velocities don’t simply add and subtract, but the rapidities do, as you should be able to show by multiplying matrices with $\varphi_1$ and $\varphi_2$.