Environmental Engineering Reference
In-Depth Information
relationship must be of the linear form
ax
+
=
x
bt
(5.3)
where
a
and
b
are to be determined. We'll not dwell on this, but if the relationship
were not linear then it would violate the relativity principle. We also know that the
point
x
=
=
0 travels along the
x
-axis with speed
v
, i.e. along the line
x
vt
.From
=
this it follows that
b
v
. The relativity principle can be used again to figure out
a
since one can equally well think of
S
as being at rest and
S
as moving along
the negative
x
-axis with speed
v
. This implies that
x
=
ax
−
vt .
(5.4)
Substituting for
x
1 and we have proved the result.
Notice that we did not need to invoke the idea of absolute space to derive this
result: all that was needed was the relativity principle and the assumption that time
is absolute. The equations (5.1) and (5.2) tell us how to relate the co-ordinates
of an event in two different inertial frames and they are often referred to as the
Galilean transformations.
In what follows we shall often speak of 'observers'. These are the real or fictitious
people who we suppose are interested in recording the co-ordinates of events using
a specified system of co-ordinates. For example, we might say that 'if an observer
at rest in
S
measures an event to occur at the point
(x,y,z)
then an observer at rest
in
S
will measure the same event to occur at
(x
,y
,z
)
where the co-ordinates in
the two frames are related to each other by the Galilean transformations.'
into Eq. (5.3) implies that
a
=
Example 5.1.1
A rigid rod of length 1m is at rest and lies along the x-axis in an
inertial frame S. Show that if space and time are universal, the rod is also 1m long
as determined by an observer at rest in an inertial frame S
which moves at a speed
v relative to S in the positive x direction.
Solution 5.1.1
It is tempting to think that this result is so self-evident that it needs
no proof but as we shall see, it is not true in Einstein's theory so it is a good idea
for us to work through the proof here assuming that Eq. (5.2) holds. We shall also
go very slowly and spell out explicity exactly how the length is measured in each
inertial frame. For this question this level of analysis may be a little over the top
but it will prepare us well for later, trickier, problems.
We can refer to Figure 5.1 and imagine two observers, one at rest in S and the
other at rest in S
. Suppose that the observer at rest in S measures the positions of
each end of the rigid rod. In doing so, she records the space and time co-ordinates
of two events. The first event is the measurement of one end of the rod and the
second event is the measurement of the other end of the rod. To specify an event
we need to specify four numbers: the three spatial co-ordinates and the time at
which the event took place. The first event has co-ordinates (x
1
,
0
,
0
) and occurs
at time t whilst the second event has co-ordinates (x
2
,
0
,
0
) and also occurs at time
t . Obviously these two events take place at the same time since that is what we
mean by making a measurement of length: we measure the positions of the ends of
