Environmental Engineering Reference
In-Depth Information
Figure 14.1 A fleet of tiny rocket ships defines a uniformly accelerating frame of reference.
The formation of the ships remains the same for all time as measured by an observer on
any one of the ships. However, this does not imply that observers on two different ships
feel the same acceleration.
x
-direction in such a way that the distance between any two of them does not vary
according to an observer on any other rocket in the fleet (i.e. any observer in
S
).
This is how we define our uniformly accelerating frame of reference.
We already know how the co-ordinates of the rocket that defines the origin in
S
relate to the corresponding co-ordinates in the inertial frame since this might be
the rocket occupied by the astronaut twin of the previous section:
g
sinh
gt
c
=
t
c
cosh
gt
c
.
c
2
g
and
x
=
(14.15)
The second of these equations is obtained by integrating Eq. (14.10), i.e.
x
t
gt
d
t
=
d
x
1
g
2
t
2
/c
2
.
(14.16)
+
0
0
Using Eq. (14.12) we can re-express the right-hand-side as an integral over
t
. Thus
sinh
gt
/c
cosh
gt
/c
d
t
cosh
(gt
/c)
t
x
=
c
0
cosh
gt
c
c
2
g
=
(14.17)
as claimed. We can say that the origin in
S
is located at the space-time point
O
=
g
sinh
gt
/c
,
cosh
gt
/c
,
0
,
0
in the basis of an inertial observer in
S
.
c
2
