Notes on Relativity and Cosmology - Physics Department, UCSB

3.4. RELATIONS BETWEEN EVENTS IN SPACETIME 61
in what is to come.
Before moving on, let us get just a bit more practice and ask what set of events
our friend (the moving observer) finds to be simultaneous with the origin (the
event where the her worldline crosses ours)? We can use light signals to find
this line as well. Let’s label that line t f = 0 under the assumption that our
friend chooses to set her watch to zero at the event where the worldlines cross.
Drawing in a carefully chosen box of light rays, we arrive at the diagram below.
x = 0
f
Us
x =0 us
t = const
f
A
B
t
us
=+1sec
t = 0
f
friend’s line of
C
simultaneity
α α
t =0
us
Note that we could also have used the rule noticed above: that the worldline
and any line of simultaneity make equal angles with the light cone.
As a final comment, note that while we know that the line of simultaneity drawn
above (t f = const) represents some constant time in the moving frame, we do
not yet know which time that is! In particular, we do not yet know whether it
represents a time greater than one second or a time less than on second. We
were able to label the t f = 0 line with an actual value only because we explicitly
assumed that our friend would measure time from the event (on that line) where
our worldlines crossed. We will explore the question of how to assign actual time
values to other lines of simultaneity shortly.
Summary: We have learned that events which are simultaneous in one inertial
reference frame are not in fact simultaneous in a different inertial frame. We
used light signals and postulate II to determine which events were simultaneous
in which frame of reference.
3.4 Relations between events in Spacetime
It will take some time to absorb the implications of the last section, but let
us begin with an interesting observation. Looking back at the diagrams above,
note that a pair of events which is separated by “pure space” in one inertial