Notes on Relativity and Cosmology - Physics Department, UCSB

1.2. REFERENCE FRAMES 29
The worldline of
a car door
+t
t=0 t=+1s
The door
slams
(an event)
+x
t=-1s
x=-1m x=0 x=+1m
That definition was a simple thing, now let’s think deeply about it. Given an
event (say, the opening of a door), how do we know where to draw it on a
spacetime diagram (say, in your reference frame)? Suppose it happens in our
1-D world.
• How can we find out what time it really happens?: One way is to give
someone a clock and somehow arrange for them be present at the event.
They can tell you at what time it happened.
• How can we find out where (at what position) it really happens?: We could
hold out a meter stick (or imagine holding one out). Our friend at the
event in question can then read off how far away she is.
Note that what we have done here is to really define what we mean by the
position and time of an event. This type of definition, where we define something
by telling how to measure it (or by stating what a thing does) is called an
operational definition. They are very common in physics. (Food for thought:
Are there other kinds of precise definitions? How do they compare?)
Now, the speed of light thing is really weird. So, we want to be very careful in
our thinking. You see, something is going to go terribly wrong, and we want to
be able to see exactly where it is.
Let’s take a moment to think deeply about this and to act like mathematicians.
When mathematicians define a quantity they always stop and ask two questions: