Notes on Relativity and Cosmology - Physics Department, UCSB

3.3. SIMULTANEITY 59
one about the speed of light. Note that there is no problem in determining
whether or not two things happen (like a door closing and a firecracker going
off) at the same event. The question is merely whether two things that occur
at different events take place simultaneously.
Suppose that we have a friend in an inertial frame and that she emit a flash
of light from her worldline. The light will travel outward both to the left and
the right, always moving at speed c. Suppose that some of this light is reflected
back to her from event A on the left and from event B on the right. The
diagram below makes it clear that the two reflected pulses of light reach her at
the same time if and only if A and B are simultaneous. So, if event C (where
the reflected pulses cross) lies on her worldline, she knows that A and B are in
fact simultaneous in our frame of reference.
Note: Although the light does not reach our friend until event C (at t = 2 sec.,
where she ‘sees’ the light), she knows that the light has taken some time to
travel and he measurements place the reflections at t = 1 sec.
x f =0
C
Signals return at the same time
if and only if the
events are simultaneous
C
x =0 f
A
B
t =+1sec
f
A
B
t
f
=+1sec
Friend
t =0
f
Friend
In fact, even if we are in a different reference frame, we can tell that A and B
are simultaneous in our friend’s frame if event C lies on her worldline.
Suppose that we are also inertial observers who meet our friend at the origin
event and then move on. What does the above experiment look like in our
frame? We’ll take the case of the figure on the left above.
Let’s start by drawing our friend’s worldline and marking event C. We don’t
really know where event C should appear, but it doesn’t make much difference
since I have drawn no scale on the diagram below. All that matters is that event
C is on our friend’s worldline (x f = 0).
x =0 us
x = 0
f
t =0
f
C
Us
t =0 us