Notes on Relativity and Cosmology - Physics Department, UCSB

64 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
B
simultaneous frame
A
frame in which A happens first
In this case we say that they are timelike related. Note that the following
things are true in this case:
a) There is an inertial observer who moves through both events and
whose speed in the original frame is less than that of light.
b) All inertial observers agree on which event (A or B) happened first.
c) As a result, we can meaningfully speak of, say, event A being to the
past of event B.
case 3) A and B are on each other’s light cones. In this case we say that they
are lightlike related. Again, all inertial observers agree on which event
happened first and we can meaningfully speak of one of them being to
the past of the other.
Now, why did we consider only inertial frames with relative speeds less than
c? Suppose for the moment that our busy friend (the inertial observer) could
in fact travel at v > c (i.e., faster than light) as shown below at left. I have
marked two events, A and B that occur on her worldline. In our frame event
A occurs first. However, the two events are spacelike related. Thus, there is
another inertial frame (t other , x other ) in which B occurs before A as shown below
at right. This means that there is some inertial observer (the one whose frame
is drawn at right) who would see her traveling backwards in time.