Notes on Relativity and Cosmology - Physics Department, UCSB

4.3. MORE ON MINKOWSKIAN GEOMETRY 103
Thus, it is still moving at a good fraction of “the speed of light.” Anyway, if
the water is also flowing (say, toward us) at a fast rate, then the speed of the
light toward us is given by the above expression in which the velocities do not
just add together. This is just what Fizeau found 5 , though he had no idea why
it should be true!
Now, the above formula looks like a mess. Why in the world should the composition
of two velocities be such an awful thing? As with many questions, the
answer is that the awfulness is not in the composition rule itself, but in the filter
(the notion of velocity) through which we view it. We will now see that, when
this filter is removed and we view it in terms native to Minkowskian geometry,
the result is quite simple indeed.
Recall the analogy between boosts and rotations. How do we describe rotations?
We use an angle θ. Recall that rotations mix x and y through the sine and cosine
functions.
x 1= 0
x = r sin θ 2
r
y = r cosθ
2
y 1= 0
x 2 = r sin θ,
y 2 = r cosθ. (4.5)
Note what happens when rotations combine. Well, they add of course. Combining
rotations by θ 1 and θ 2 yields a rotation by an angle θ = θ 1 + θ 2 . But
we often measure things in terms of the slope m = x y
(note the similarity to
v = x/t). Now, each rotation θ 1 , θ 2 is associated with a slope m 1 = tanθ 1 ,
m 2 = tanθ 2 . But the full rotation by θ is associated with a slope:
m = tanθ = tan(θ 1 + θ 2 )
= tan θ 1 + tanθ 2
1 − tan θ 1 tan θ 2
= m 1 + m 2
1 − m 1 m 2
. (4.6)
So, by expressing things in terms of the slope we have turned a simple addition
rule into something much more complicated.
5 Recall that Fizeau’s experiments were one of the motivations for Michelson and Morely.
Thus, we now understand the results not only of Michelson and Morely’s experiment, but also
of the experiments that prompted their work.