Notes on Relativity and Cosmology - Physics Department, UCSB

106 CHAPTER 4. MINKOWSKIAN GEOMETRY
t =
d c sinh θ
t =0
d
x = d coshθ
x = 0
Again, we see that θ is really a measure of the separation of the two reference
frames. In this context, we also refer to θ as the boost parameter relating the
two frames. The boost parameter is another way to encode the information
present in the relative velocity, and in particular it is a very natural way to do
so from the viewpoint of Minkowskian geometry.
In what way is the relative velocity v of the reference frames related to the boost
parameter θ? Let us again consider the inertial observer passing from the origin
through event A on the hyperbola of constant proper time. This observer moves
at speed:
v = x cτ sinhθ
=
t τ coshθ = c sinhθ = c tanhθ, (4.13)
coshθ
and we have the desired relation. Here, we have introduced the hyperbolic
tangent function in direct analogy to the more familiar tangent function of
trigonometry. Note that we may also write this function as
tanhθ = eθ − e −θ
e θ + e −θ .
The hyperbolic tangent function may seem a little weird, but we can get a better
feel for it by drawing a graph like the one below. The vertical axis is tanhθ and
the horizontal axis is θ.
1
velocity
0
−1
−5 0 5
boost parameter
To go from velocity v to boost parameter θ, we just invert the relationship: