Notes on Relativity and Cosmology - Physics Department, UCSB

3.5. TIME DILATION 71
3) Suppose that (when they meet) blue plants a time bomb in red’s luggage
and sets it to go off after 1sec. What times does blue find it to go off? The
time bomb will go off after it experiences 1sec of time. In other words, it
will go off at the point along its worldline which is 1sec of proper time later.
Since red is traveling along the same worldline, this is 1sec later according
to red and on red’s worldline. As a result, (3.3) tells us that this happens at
t blue = 1/ √ 1 − (v/c) 2 .
4) Suppose that (when they meet) red plants a time bomb in blue’s luggage
and he wants it to go off at t red=1 . How much time delay should the bomb be
given? This requires figuring out how much proper time will pass on blue’s
worldline between red’s lines of simultaneity t red=0 and t red=1 . Since the
events are on blue’s worldline, blue plays the role of the moving friend in
(3.3). As a result, the time until the explosion as measured by blue should
be t blue = √ 1 − (v/c) 2 , and this is the delay to set.
3.5.4 Why should you believe all of this?
So far, we have just been working out consequences of Einstein’s idea. We have
said little about whether you should actually believe that this represents reality.
In particular, the idea that clocks in different reference frames measure different
amounts of time to pass blatantly contradicts your experience, doesn’t it? Just
because you go and fly around in an airplane does not mean that your watch
becomes unsynchronized with the Cartoon Network’s broadcast schedule, does
it?
Well, let’s start thinking about this by figuring out how big the time dilation
effect would be in everyday life. Commercial airplanes move at about 4 300m/s.
So, v/c ≈ 10 −6 for an airplane. Now 5 , √ 1 − (v/c) 2 ≈ 1 − 1 2 (v/c)2 + ... ≈
1 − 5 × 10 −13 for the airplane. This is less than 1 part in a trillion.
Tiny, eh? You’d never notice this by checking your watch against the Cartoon
Network. However, physics is a very precise science. It turns out that it is
in fact possible to measure time to better than one part in a trillion. A nice
form of this experiment was first done in the 1960’s. Some physicists got two
identical atomic clocks, brought them together, and checked that they agreed to
much better than 1 part in a trillion. Then, they left one in the lab and put the
other on an airplane (such clocks were big, they bought a seat for the clock on
a commercial airplane flight) and flew around for awhile. When they brought
the clocks back together at the end of the experiment, the moving clock had in
fact ‘ticked’ less times, measuring less time to pass in precise accord with our
calculations above and Einstein’s prediction.
⋆⋆ You should not underestimate the significance of what I have just said. Just
a moment ago, we were merrily exploring Einstein’s crazy idea. While Einstein’s
4 Those of you with physics background may recognize this as roughly the speed of sound
in air. Travel faster than the speed of sound is difficult and therefore expensive, so most
commercial planes lurk as just below sound speed.
5 Note that I am using the expansion from problem (2-2). It really is handy.