Notes on Relativity and Cosmology - Physics Department, UCSB

30 CHAPTER 1. SPACE, TIME, AND NEWTONIAN PHYSICS
1. Does this quantity actually exist? (Can we perform the above operations
and find the position and time of an event?)
2. Is this quantity unique or, equivalently, is the quantity “well-defined?”
(Might there be some ambiguity in our definition? Is there a possibility
that two people applying the above definitions could come up with two
different positions or two different times?)
Well, it seems pretty clear that we can in fact perform these measurements, so
the quantities exist. This is one reason why physicists like operational definitions
so much.
Now, how well-defined are our definitions are for position and time? [Stop
reading for a moment and think about this.]
⋆ One thing you might worry about is that clocks and measuring rods are
not completely accurate. Maybe there was some error that caused it to give
the wrong reading. We will not concern ourselves with this problem. We will
assume that there is some real notion of the time experienced by a clock and
some real notion of the length of a rod. Furthermore, we will assume that we
have at hand ‘ideal’ clocks and measuring rods which measure these accurately
without mistakes. Our real clocks and rods are to be viewed as approximations
to ideal clocks and rods.
OK, what else might we worry about? Well, let’s take the question of measuring
the time. Can we give our friend just any old ideal clock? No..... it is very
important that her clock be synchronized with our clock so that the two clocks
agree 3 .
And what about the measurement of position? Well, let’s take an example.
Suppose that our friend waits five minutes after the event and then reads the
position off of the meter stick. Is that OK? What if, for example, she is moving
relative to us so that the distance between us is changing? Ah, we see that it
is very important for her to read the meter stick at the time of the event. It is
also important that the meter stick be properly ‘zeroed’ at that same time.
So, perhaps a better definition would be:
time: If our friend has a clock synchronized with ours and is present at an
event, then the time of that event in our reference frame is the reading of
her clock at that event.
position: Suppose that we have a measuring rod and that, at the time that
some event occurs, we are located at zero. Then if our friend is present
at that event, the value she reads from the measuring rod at the time the
event occurs is the location 4 of the event in our reference frame.
3 Alternatively, if we knew that her clock was, say, exactly five minutes ahead of ours then
we could work with that and correct for it. But the point is that we have to know the
relationship of her clock to ours.
4 Together with a + or - sign which tells us if the event is to the right or to the left. Note
that if we considered more than one dimension of space we would need more complicated
directional information (vectors, for the experts!).