Notes on Relativity and Cosmology - Physics Department, UCSB

116 CHAPTER 4. MINKOWSKIAN GEOMETRY
3
Left
Back
5
6
7
4
2
1
Front
8
Right
The new symmetry axis is shown above. Thus, with respect to the original
observer, this new observer is not moving along a line straight to the right.
Instead, the new observer is moving somewhat in the forward direction as well.
But wait.... something else interesting is going on here.... the light rays don’t
line up right. Note that if we copied the above symmetry axis onto the light
circle in the original frame, it would sit exactly on top of rays 4 and 8. However,
in the figure above the symmetry axis sits half-way between 1 and 8 and 4 and 5.
This is the equivalent of having first rotated the light circle in the original frame
by 1/16 of a revolution before performing a boost along the new symmetry axis!
The new observer differs from the original one not just by a boost, but by a
rotation as well!
In fact, by considering two further boost transformations as above (one acting
only backward, and then one acting to the right), one can obtain the following
circle of light rays, which are again evenly distributed around the circle. You
should work through this for yourself, pushing the dots around the circle with
care.
5
Left
4
3
Back
6
2
Front
7
8
Right
1
Thus, by a series of boosts, one can arrive at a frame of reference which, while it
is not moving with respect to the original fame, is in fact rotated with respect to
the original frame. By applying only boost transformations, we have managed
to turn our observer by 45 degrees in space. This just goes to show again
that time and space are completely mixed together in relativity, and that boost