Environmental Engineering Reference
In-Depth Information
S
is using clocks at rest in
S
. Thus it is the case that each concludes that the
other is aging more slowly. Reflecting upon Einstein's 1st postulate we can see
that this symmetrical situation must be correct for otherwise one could distinguish
between the two inertial frames. Of course if the two observers were to meet up
and compare notes then at least one of them must have undergone an acceleration.
This would break the symmetry between the two and leads to the fascinating
possibility that one of the observers would be genuinely older than the other upon
meeting (see Section 14.1.1).
We have been very careful to explain what we mean by measurements of time
and have stressed that they have nothing to do with seeing events with our eyes.
Nevertheless, people do see things and it is interesting to ask how our perception
of things changes in Special Relativity. Referring to Figure 5.1 we could imagine
an observer situated at the origin
O
who is watching a clock speed away from
them. We suppose that the clock is at rest at the origin
O
in
S
. If one tick of the
clock takes a time
t
in
S
what is the corresponding interval of time seen by
the observer in
S
? The key word here is 'see'. Observations of events as we have
hitherto been discussing them have referred explicitly to a process which does not
depend upon the observer actually watching the event nor on where the observer
is located when the event takes place. In contrast, the act of seeing does depend
upon things like how far the observer is away from the things they are watching
and the quality of the eyesight of the person doing the seeing. That distance is
important when watching a moving clock becomes apparent once one appreciates
that the clock is becoming ever further away and as a result light takes longer and
longer to reach the observer. With this in mind, we can tackle the question in hand
and attempt to work out the time interval
t
see
perceived by our observer at the
origin
O
. According to all observers in
S
, including our observer standing at the
origin, the time of one tick of the clock is given by the time dilation formula, i.e.
t
γt
. However this is not what we want. The time interval
t
see
is longer
than
t
by an amount equal to the time it takes for light to travel the extra distance
the clock has moved over the course of the tick, i.e. light from the end of the
clock's tick has to travel further before it reaches the observer by an amount equal
to
vt
. Therefore the perceived time interval between the start and the end of the
tick is
=
t
1
1
/
2
γt
1
v
c
v
c
+
v/c
γt
+
γt
t
see
=
=
+
=
.
(6.8)
1
−
v/c
It is very important to be clear that this extra slowing down of the clock
is an 'optical illusion', in contrast to the time dilation effect which is a real
slowing down of time. To emphasise this point, if light travels at a finite speed
then moving clocks will appear to run slow even in classical theory such that
t
see
=
t
(
1
v/c)
.
Eq. (6.8) leads us on nicely to the Doppler effect for light. Let us consider
the situation illustrated in Figure 6.4. A light source is at rest in
S
and is being
watched by someone at rest in
S
. The time interval
t
could just as well be the
time between the emission of successive peaks in a light wave, i.e. the frequency
ofthewaveis
f
=
+
1
/t
. The person watching the light source will instead see
