Notes on Relativity and Cosmology - Physics Department, UCSB

96 CHAPTER 4. MINKOWSKIAN GEOMETRY
moves at more than one light-year per year in this frame. We see that Alphonse
is in fact moving more slowly than the light ray, which is good. However, we
also see that the speed of a light ray is not in general equal to c in an accelerated
reference frame! In fact, it is not even constant since the gold light ray appears
‘bent’ on Gaston’s diagram. Thus, it is only in inertial frames that light moves
at a constant speed of 3 × 10 8 meters per second. This is one reason to avoid
drawing diagrams in non-inertial frames whenever you can.
Actually, though, things are even worse than they may seem at first glance....
Suppose, for example, that Alphonse has a friend Zelda who is an inertial observer
at rest with respect to Alphonse, but located four light years on the other
side of Alpha Centauri. We can then draw the following diagram in Alphonse’s
frame of reference:
t = 6 yrs.
G
t = 3.1 yrs.
G
light ray
G
a
s
t
o
n
t =0 A
t = 15 yrs.
G
Alphonse
C
t =5
A
D
E
B
A
x =0
A
X
Alpha
Centauri
x A
=4
Z
Y
t = 0
G
W
V
U
T
t = 2.9 yrs.
G
t = 3 yrs.
G
t = -9 years
G
Once again, we simply can use Gaston’s lines of simultaneity to mark the events
(T,U,V,W,X,Y,Z) in Zelda’s life on Gaston’s diagram. In doing so, however, we
find that some of Zelda’s events appear on TWO of Gaston’s lines of simultaneity
– a (magenta) one from before the turnaround and a (green) one from after
the turnaround! In fact, many of them (like event W) appear on three lines
of simultaneity, as they are caught by a third ‘during’ the turnaround when
Gaston’s line of simultaneity sweeps downward from the magenta t = 2.9 to the
green t = 3.1 as indicated by the big blue arrow!
Marking all of these events on Gaston’s diagram (taking the time to first calculate
the corresponding positions) yields something like this: