Notes on Relativity and Cosmology - Physics Department, UCSB

4.3. MORE ON MINKOWSKIAN GEOMETRY 105
x = cτ sinhθ. (4.11)
t =
τcosh
θ
τ
A
t =0
x = c τ sinhθ
x = 0
On the diagram above I have drawn the worldline of an inertial observer that
passes through both the origin and event A. Note that the parameter θ gives
some notion of how different the two inertial frames (that of the moving observer
and that of the stationary observer) actually are. For θ = 0, event A is at x = 0
and the two frames are the same, while for large θ event A is far up the hyperbola
and the two frames are very different.
We can parameterize the points that are a proper distance d from the origin in
a similar way, though we need to ‘flip x and t.’
t = d/c sinhθ,
x = d cosh θ. (4.12)
If we choose the same value of θ, then we do in fact just interchange x and t,
“flipping things about the light cone.” Note that this will take the worldline of
the above inertial observer into the corresponding line of simultaneity. In other
words, a given worldline and the corresponding line of simultaneity have the
same ‘hyperbolic angle,’ though we measure this angle from different reference
lines (x = 0 vs. t = 0) in each case.