Notes on Relativity and Cosmology - Physics Department, UCSB

118 CHAPTER 4. MINKOWSKIAN GEOMETRY
A
C
D
t=4
B
t=0
x=-4 x=0 x=4
t=-4
4-2. This problem is to give you some practice putting everything together.
Once again, we and our friends, Alice and Bob, are inertial observers who
all meet at a single event. At this event, our clock, Alice’s clock, and Bob’s
clock all read zero and a firecracker explodes. Alice moves to our right at
c/2 and Bob moves to our left at c/2.
Draw a single spacetime diagram in our reference frame showing all of the
following:
(a) Alice, Bob, and the outgoing light from the explosion.
(b) The curve representing all events that are a proper time of one second
to the future of the explosion. Also draw in the curves representing
the events that are: i) one second of proper time to the past of the
explosion, ii) one light second of proper distance to the left of the
explosion, and iii) one light second of proper distance to the right of
the explosion.
(c) The events A, U, and B where Alice’s clock reads one second, where
our clock reads one second, and where Bob’s clock reads one second.
(d) Finally, suppose that we (but not Alice or Bob) are holding the middle
of a stick that is two light seconds long (and which is at rest relative
to us). Draw in the worldlines of both ends of that stick. Also mark
the events X and Y occupied by the ends of that stick on the line
t us = 0.
4-3. Redraw everything in problem (4-2) using Alice’s frame of reference.
4-4. Let’s get one more perspective on the twin paradox. It is always interesting
to ask what each twin sees during the trip. Now, note that what you
actually see has to do with light rays, and with when a bit of light happens
to reach your eye. So, to study this question, we should study light rays
sent from one twin to the other. We will again have Gaston go off to
Alpha Centauri (4 light-years away) and back at .8c while Alphonse stays
at home.
(a) Let’s first think about what Gaston (the traveling twin) sees. Start
by drawing a spacetime diagram for the trip in any inertial frame (it