Notes on Relativity and Cosmology - Physics Department, UCSB

70 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
t = us
1
1 - v 2/c2
x =0
us
A
x = 0
f
t = f
1
1 - v 2/c2
t = 1 sec
f
B
t
us
= 1 sec
Us
t =0 us
Note that I have marked two special events on the diagram. The dot labeled A
is the event where the moving (friend’s) clock ticks t = 1 second. It is an event
on the friend’s worldline. The dot labeled B is the event where our clock ticks
t = 1 second. It is an event on our worldline.
3.5.3 Proper Time
We have seen that 3 different observers in different inertial frames measure different
amounts of time to pass between two given events. We might ask if any
one of these is a “better” answer than another? Well, in some sense the answer
must be ‘no,’ since the principle of relativity tells us that all inertial frames
are equally valid. However, there can be a distinguished answer. Note that, if
one inertial observer actually experiences both events, then inertial observers in
other frames have different worldlines and so cannot pass through both of these
events. It is useful to use the term proper time between two events to refer to
the time measured by an inertial observer who actually moves between the two
events. Note that this concept exists only for timelike separated events.
Let’s work through at a few cases to make sure that we understand what is
going on. Consider two observers, red and blue. The worldlines of the two
observers intersect at an event, where both set their clocks to read t = 0.
1) Suppose that red sets a firecracker to go off on red’s worldline at t red = 1.
At what time does blue find it to go off? Our result (3.3) tells us that
t blue = 1/ √ 1 − (v/c) 2 .
2) Suppose now that blue sets a firecracker to go off on blue’s worldline at
t blue = 1. At what time does red find it to go off? From (3.3) we now have
t red = 1/ √ 1 − (v/c) 2 .
3 Still assuming that Einstein’s idea is right.