Notes on Relativity and Cosmology - Physics Department, UCSB

68 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
then I must find your stick to be shorter. As a result, consistency requires both
of us find the two meter sticks to be of the same length.
We conclude that the length of a meter stick is the same in two inertial
frames for the case where the stick points in the direction perpendicular to the
relative motion.
3.5.2 Light Clocks and Reference Frames
The property just derived makes it convenient to use such meter sticks to build
clocks. Recall that we have given up most of our beliefs about physics for the
moment, so that in particular we need to think about how to build a reliable
clock. The one thing that we have chosen to build our new framework upon is
the constancy of the speed of light. Therefore, it makes sense to use light to
build our clocks. We will do this by sending light signals out to the end of our
meter stick and back. For convenience, let us assume that the meter stick is one
light-second long. This means that it will take the light one second to travel
out to the end of the stick and then one second to come back. A simple model
of such a light clock would be a device in which we put mirrors on each end of
the meter stick and let a short pulse of light bounce back and forth. Each time
the light returns to the first mirror, the clock goes ‘tick’ and two seconds have
passed.
Now, suppose we look at our light clock from the side. Let’s say that the rod in
the clock is oriented in the vertical direction. The path taken by the light looks
like this:
However, what if we look at a light clock carried by our inertial friend who
is moving by at speed v? Suppose that the rod in her clock is also oriented
vertically, with the relative motion in the horizontal direction. Since the light
goes straight up and down in her reference frame, the light pulse moves up and
forward (and then down and forward) in our reference frame. In other words,
it follows the path shown below. This should be clear from thinking about the
path you see a basketball follow if someone lifts the basketball above their head
while they are walking past you.