Notes on Relativity and Cosmology - Physics Department, UCSB

4.5. HOMEWORK PROBLEMS 117
transformations are even more closely related to rotations than you might have
thought.
A boost transformation can often be thought of as a “rotation of time into
space.” In this sense the above effect may be more familiar: Consider three
perpendicular axes, x, y, and z. By performing only rotations about the x and
y axes, one can achieve the same result as any rotation about the z axes. In the
above discussion, the boosts are analogous to rotations about the x and y axes,
while the rotation is indeed a rotation ‘about the t axis.’
4.4.4 Other effects
Boosts in 3+1 dimensions and higher works pretty much like it they do in
2+1 dimensions, which as we have seen has only a few new effects beyond the
1+1 case on which we spent most of our time. There is really only one other
interesting effect in 2+1 or higher dimensions that we have not discussed. This
has to do with how rapidly moving objects actually look; that is, they have to do
with how light rays actually reach your eyes to be processed by your brain. This
is discussed reasonably well in section 4.9 of Inside Relativity Mook and Vargish,
which used to be a required textbook for this class. This section is probably
the greatest loss due to no longer using this text as it is quite well done. It
used to be that I would save time by not talking about this effect directly and
simply asking students to read that section. At the time of writing these notes,
I had not decided for sure whether I would discuss this material this time, but
probably time will simply not allow it. For anyone who is interested, I suggest
you find a copy of Mook and Vargish (I think there are several in the Physics
Library) and read 4.9. If you are interested in reading even more, the references
in their footnote 13 on page 117 are a good place to start. One of those papers
refers to another paper by Penrose, which is probably the standard reference
on the subject. By the way, this can be an excellent topic for a course project,
especially if you are artistically inclined or if you like to do computer graphics.
4.5 Homework Problems
4-1. The diagram below is drawn in some inertial reference frame using units
of seconds for time and light-seconds for distance. Calculate the total time
experienced by a clock carried along each of the four worldlines (A,B,C,D)
shown below. Each of the four worldlines starts at (t = −4, x = 0) and ends
at (t = +4, x = 0). Path B runs straight up the t-axis. Remember that
the proper time for a path that is not straight can be found by breaking
it up into straight pieces and adding up the proper times for each piece.