Notes on Relativity and Cosmology - Physics Department, UCSB

90 CHAPTER 4. MINKOWSKIAN GEOMETRY
spacelike separation: Similarly, if the events are spacelike separated, there is
an inertial frame in which the two are simultaneous – that is, in which
∆t = 0. The distance between two events measured in such a reference
frame is called the proper distance d. Much as above,
d = √ ∆x 2 − c 2 ∆t 2 ≤ ∆x.
Note that this seems to “go the opposite way” from the length contraction
effect we derived in section 3.6. That is because here we consider the
proper distance between two particular events. In contrast, in measuring
the length of an object, different observers do NOT use the same pair of
events to determine length. Do you remember our previous discussion of
this issue?
lightlike separation: Two events that are along the same light ray satisfy
∆x = ±c∆t. It follows that they are separated by zero interval in all
reference frames. One can say that they are separated by both zero proper
time and zero proper distance.
4.1.2 Curved lines and accelerated objects
Thinking of things in terms of proper time and proper distance makes it easier to
deal with, say, accelerated objects. Suppose we want to compute, for example,
the amount of time experienced by a clock that is not in an inertial frame.
Perhaps it quickly changes from one inertial frame to another, shown in the
blue worldline (marked B) below. This blue worldline (B) is similar in nature
to the worldline of the muon in part (b) of problem 3-7.
d
t=4
R
a
c
B
x=-4 x=0 x=4
b
t=0
t=-4
Note that the time experienced by the blue clock between events (a) and (b)
is equal to the proper time between these events since, on that segment, the
clock could be in an inertial frame. Surely the time measured by an ideal clock
between (a) and (b) cannot depend on what it was doing before (a) or on what
it does after (b).
Similarly, the time experienced by the blue clock between events (b) and (c)
should be the same as that experienced by a truly inertial clock moving between
these events; i.e. the proper time between these events. Thus, we can find the
total proper time experienced by the clock by adding the proper time between
(a) and (b) to the proper time between (b) and (c) and between (c) and (d). We