Environmental Engineering Reference
In-Depth Information
and since
u(t)
2
is always positive, the integrand is always less than unity and the
time registered on the clock of the twin who underwent an acceleration is less
than the time interval registered on the clock of the twin who did not. Indeed we
see that inertial observers present at any pair of events necessarily experience the
maximum possible time interval between those events.
Example 14.1.1
Show that for small enough speeds the accelerating spaceship in
our discussion of the twins paradox follows the trajectory x
1
2
gt
2
.
≈
Solution 14.1.1
Eq. (14.10) tells us that, for gt/c
1
d
x
d
t
≈
gt
(14.14)
1
2
gt
2
for x
and hence x
≈
=
0
at t
=
0
.
14.1.2
Accelerating frames of reference
In the last subsection we considered an accelerating observer but from the point
of view of an infinite set of inertial frames and we did not invoke the idea of an
accelerating frame of reference. Sometimes we may wish to investigate a piece of
physics directly in an accelerating frame of reference, just as we did in Chapter 8
when we discussed non-inertial frames in classical physics. Of course we expect
that in so doing we should encounter non-inertial forces.
However, the construction of an accelerating reference frame is not as straight-
forward in Einstein's theory as it is in classical physics. The most natural way
to think would be to suppose we erect a rigid system of rulers and clocks and
use these to locate the position of events in the accelerating frame. However, as
we shall shortly see, the rods will tend to be bent or buckled and it will not be
possible to synchronize the clocks so that they always read the same time. The
lack of synchronicity of the clocks need not be a problem; we could accept that
time might tick at different rates throughout an accelerating frame. However the
buckling of the rulers would make life difficult since we'd need to know all about
the physical properties of the rulers in order to compare theoretical predictions for
the relationships between events in the accelerating frame and the corresponding
observations.
To illustrate these points we'll focus our attention on a very special accelerating
frame of reference. Namely one in which the acceleration is time independent and
the distance between points in the frame do not vary with time. Clearly this is an
appealing frame of reference however, as we shall soon see, it is not a very practical
frame. Our goal will be to figure out the equivalent of the Lorentz transformation
formulae that relate the co-ordinates of an event in the accelerating frame,
S
,to
the co-ordinates of the same event in our typical inertial frame
S
. To visualize the
accelerating frame let us consider Figure 14.1, which shows a fleet of tiny rocket
ships located at different points in
S
. The little rockets define the locations of
events, i.e. the rocket ship located closest to an event can be used to label the
position of that event. The rockets are arranged so that they all accelerate in the
