Notes on Relativity and Cosmology - Physics Department, UCSB

100 CHAPTER 4. MINKOWSKIAN GEOMETRY
hyperbolae.
Us
1 sec. proper time
to the future
of origin
1 ls proper distance
to the left
of origin
1 ls proper distance
to the right
of origin
t =0
us
1 sec. proper time
to the past
of origin
We see that changing the inertial reference frame simply slides events along
a given hyperbola of constant time or constant distance, but does not move
events from one hyperbola to another.
Remember our Euclidean geometry analogue from last time? The above observation
is exactly analogous to what happens when we rotate an object 4 . The
points of the object move along circles of constant radius from the axis, but do
not hop from circle to circle.
4 Actually, it is analogous to what happens when we rotate but the object stays in place.
This is known as the different between an ‘active’ and a ‘passive’ rotation. It seemed to me,
however, that the main idea would be easier to digest if I did not make a big deal out of this
subtlety.