Environmental Engineering Reference
In-Depth Information
and
τ
is the proper time measured by a clock located at
x
. Hence, keeping
x
fixed
so d
x
/
d
τ
=
0, gives
c
2
g
x
c
2
d
t
2
2
g
2
c
2
,
+
=
d
τ
i.e.
d
t
d
τ
1
1
gx
/c
2
=
.
(14.24)
+
This equation relates the co-ordinate time in
S
, which is the proper time recorded
on a clock at the origin in
S
, to the proper time on a clock at any other
x
.Since
they are not equal we see that it is impossible to synchronize the clocks in
S
for
all time. The acceleration of the little rocket at
x
can now be determined once
we appreciate that the four-acceleration of a particle moving through Minkowski
space satisfies Eq. (14.6), i.e.
α
2
,
A
·
A
=−
(14.25)
where
α
is the acceleration as determined in the rest frame of the particle. For the
rocket at
x
, its four-acceleration is
d
t
d
τ
g
c
c
2
g
+
x
cosh
gt
c
,
c
2
x
sinh
gt
c
,
0
,
0
d
d
τ
A
=
g
+
(14.26)
and hence
d
t
d
τ
4
c
4
c
2
x
g
2
g(x
)
2
−
=−
+
,
g
d
t
d
τ
2
g
1
.
gx
c
2
i.e.
g(x
)
=
+
(14.27)
Using Eq. (14.24) then gives our final answer:
g
g(x
)
=
gx
/c
2
.
(14.28)
1
+
It is simply not possible to build
S
out of a fleet of rockets such that they all
accelerate at the same rate in their own rest frame and preserve the distance between
each rocket. To do that, as we shall discuss in the next section, means going beyond
Minkowski space-time.
We can determine the form of the metric for the accelerating frame. The fleet of
rockets is moving through Minkowski space-time hence the interval between any
