Environmental Engineering Reference
In-Depth Information
z
S
L
u∆
t
out
A
B
y
O
x
Figure 6.5
Measuring the length of a moving ruler.
S
. To measure the length of the ruler we shall mount a light-clock of equal length
next to it, as shown. The light-clock moves with the ruler. The light starts out from
one end of the ruler and reflects from a mirror located at the opposite end of the
ruler. Our strategy will be to determine the time taken for the roundtrip directly in
S
and equate this to the time dilation result. As a result of time dilation, the roundtrip
time in
S
is related to the roundtrip time in the rest frame of the ruler
t
0
by
γ
2
L
0
c
t
=
γt
0
=
.
(6.10)
We shall now endeavour to determine this time interval by considering the journey
of the light from the viewpoint of
S
. According to an observer in
S
, the total time is
t
=
t
out
+
t
in
,
(6.11)
where
t
out
is the time taken for the light to travel on its outward journey, i.e.
from A to B, and
t
in
is the time taken on the return journey. The figure shows
explicitly the two positions of the ruler when the light starts its journey (dashed
line) and when the light reaches the opposite end of the ruler (solid line). In
order not to clutter the picture we have not shown the third position of the ruler,
i.e. when the light finally returns back to its starting point. Since Einstein's 2nd
postulate tells us the speed of light according to
S
, we can write
ct
out
=
L
+
vt
out
L
⇒
t
out
=
v
.
(6.12)
−
c
Each side of the first of these equations is equal to the total distance travelled by
the light on its outward journey (according to
S
) and it takes into account the fact
that the light has to travel a little further than the length of the ruler
L
as a result
of the ruler's motion. Similarly for the return leg, the light has to travel a shorter
