Environmental Engineering Reference
In-Depth Information
where
g
is the acceleration felt at the origin (recall it is not possible to build
a rigidly accelerating frame such that all points within it accelerate at the same
rate). Using the Equivalence Principle this must also be the invariant distance in
a particular static gravitational field
1
. Armed only with this information we can
go ahead and deduce that clocks run faster higher up in a gravitational field. As
illustrated in Figure 14.2, let us consider a clock at rest in the gravitational field
at a height
h
above the observer. Then for an observer
A
adjacent to the clock the
space-time interval between two ticks of the clock is given by
(s)
2
(ct
A
)
2
.
=
(14.45)
A
g
h
B
Figure 14.2 A clock in a static gravitational field. The double arrow indicates the direction
of the acceleration due to gravity.
=
Now consider a second observer
B
for whom the clock is located at
x
h
.For
them the same space-time interval is
(ct
B
)
2
1
2
gh
c
2
(s)
2
=
+
,
(14.46)
where
g
is the acceleration at
B
. Equating these two intervals gives
1
c
2
t
B
gh
t
A
=
+
(14.47)
which means that according to the observer at
B
the clock runs faster than it
does according to the observer at
A
. One might worry that this is not a very
realistic situation because the metric presented in Eq. (14.44) corresponds to a
rather artificial field in which the acceleration varies with height according to
Eq. (14.28). There is in fact no cause for concern since, provided we assume that
gh/c
2
1, it is sufficient to take
g
as a constant in Eq. (14.47). In any case, we
now aim to improve things and describe a truly uniform gravitational field.
1
Although not one in which the field is uniform.
