Notes on Relativity and Cosmology - Physics Department, UCSB

104 CHAPTER 4. MINKOWSKIAN GEOMETRY
The point here is that the final result (4.6) bears a strong resemblance to
our formula for the addition of velocities. In units where c = 1, they differ only
by the minus sign in the deominator above. This suggests that the addition
of velocities can be simplified by using something similar to, but still different
than, the trigonometry above.
To get an idea of where to start, recall one of the basic facts associated with
the relation of sine and cosine to circles is the relation:
sin 2 θ + cos 2 θ = 1. (4.7)
It turns out that there are other natural mathematical functions called hyperbolic
sine (sinh) and hyperbolic cosine (cosh) that satisfy a similar (but different!)
so that they are related to hyperbolae.
cosh 2 θ − sinh 2 θ = 1, (4.8)
These functions can be defined in terms of the exponential function, e x :
sinhθ = eθ − e −θ
2
coshθ = eθ + e −θ
. (4.9)
2
You can do the algebra to check for yourself that these satisfy relation (4.8)
above. By the way, although you may not recognize this form, these functions
are actually very close to the usual sine and cosine functions. Introducing i =
√ −1, one can write sine and cosine as 6 .
sin θ = eiθ − e −iθ
2i
cosθ = eiθ + e −iθ
. (4.10)
2
Thus, the two sets of functions differ only by factors of i which, as you can
imagine, are related to the minus sign that appears in the formula for the squared
interval.
Now, consider any event (A) on the hyperbola that is a proper time τ to the
future of the origin. Due to the relation 4.8, we can write the coordinates t, x
of this event as:
t
= τ coshθ,
6 These representations of sine and cosine may be new to you. That they are correct may
be inferred from the following observations: 1) Both expressions are real. 2) If we square both
and add them together we get 1. Thus, they represent the path of an object moving around
the unit circle. 3) They satisfy d
d
sinθ = cos θ and cos θ = − sinθ. As a result, for θ = ωt
dθ dθ
they represent an object moving around the unit circle at angular velocity ω.