Environmental Engineering Reference
In-Depth Information
z
S
z
′
S
′
u
y
y
′
x
O
′
x
′
0
x
′
O
Figure 6.6
A moving clock.
Subtracting these two equations gives
γt
,
=
−
=
t
t
2
t
1
which is the required result. Notice that to derive this result it was crucial to be
clear that the clock is at rest in S
.
Example 6.2.2
Use the Lorentz transformations to derive the formula for length
contraction.
Solution 6.2.2
We now consider the situation illustrated in Figure 6.7 where we
have placed a ruler in S
such that it lies along the x
-axis with one end located
at x
1
and the other at x
2
. The length of the ruler in its rest frame is therefore
x
=
x
1
. The question now is: 'what is the length of the ruler as determined
by an observer in S?'
x
2
−
S
′
z
S
z
′
u
y
y
′
x
′
O
O
′
x
x
′
1
x
′
2
Figure 6.7 A moving ruler.
Again there are two events to consider. Event 1 (measurement of one end of the
ruler) and event 2 (measurement of the other end of the ruler). The crucial point
now is that both events occur at the same time in S because that is what is meant
by a measurement of length. Let's call this time t
0
. Given that we know the location
of the two events in S
and the time of the events in S we should use Eq. (6.28a) to
give us the location of the events in S:
x
1
=
γ(x
1
−
vt
0
),
x
2
=
γ(x
2
−
vt
0
).
