Environmental Engineering Reference
In-Depth Information
but that is because there is room for much confusion if these ideas are not properly
appreciated.
Let us return to the light-clock of Figure 6.1. In its rest frame, the time it takes
for light to do the roundtrip between the mirrors (one 'tick') is clearly
2
d
c
t
0
=
.
(6.1)
Now let us imagine what happens if the clock is moving relative to the observer.
To be specific let us put the clock in
S
and an observer in
S
where the two
frames are as usual defined by Figure 5.1. If the observer was in
S
then the
time for one tick of the clock would be just
t
0
.
Our task is to determine the
corresponding time when the observer is in
S
. According to this observer, the light
follows the path shown in Figure 6.2. We call
t
the time it takes for the light to
complete one roundtrip as measured in
S
. Accordingly the clock moves a distance
x
2
−
vt
over the course of the roundtrip. Using Pythagoras' Theorem, it
follows that the light travels a total distance 2
(d
2
x
1
=
v
2
t
2
/
4
)
1
/
2
. All of this is
as it would be in Galilean relativity. Now here comes the new idea. The light is
still travelling at speed
c
in
S
(in classical theory the speed would be
(c
2
+
v
2
)
1
/
2
by the simple addition of velocities). As a result, the time for the roundtrip in
S
satisfies
+
d
2
1
/
2
v
2
t
2
4
2
c
t
=
+
.
(6.2)
y
S
d
x
1
x
2
x
Figure 6.2
The path taken by the light in a moving light-clock.
Squaring both sides and re-arranging allows us to solve for
t
:
2
d
c
×
1
t
=
1
v
2
/c
2
.
(6.3)
−
The time measured in
S
is longer than the time measured in
S
and we are forced
to conclude that in Einstein's theory
moving clocks run slow
. This effect is also
known as 'time dilation', and it is negligibly small if
v/c
1 but when
v
∼
c
the effect is dramatic.
