Environmental Engineering Reference
In-Depth Information
What we'd really like is the location of a general rocket ship. For simplicity, let us
confine our attention to a rocket lying at position
x
on the
x
-axis. This rocket is
displaced from the origin in
S
by a distance
x
. But what are its co-ordinates in
S
?
To obtain this we use a neat trick, we make use of the fact that the four-velocity of
the rocket at the origin in
S
points in the time direction of space-time according
to an observer on the rocket. Hence a four-vector that is orthogonal to this must
point along the
x
axis and we can therefore locate the four-vector position of the
rocket at
x
. Let us follow this train of thinking. The four-velocity of the origin in
S
is
c
cosh
gt
c
/c
,
sinh
(gt
/c
,
0
,
0
)
U
=
(14.18)
and hence the four-vector
D
=
sinh
gt
/c
,
cosh
gt
/c
,
0
,
0
(14.19)
is a unit four-vector pointing in the
x
direction since
0. Thus an event
occurring on the little rocket at
x
, which occurs at a time
t
, is located at space-time
position
U
·
D
=
x
D
(ct, x,
0
,
0
)
=
O
+
(14.20)
and we now have the general relationship we desire:
c
2
g
x
sinh
gt
c
=
+
ct
c
2
g
+
x
cosh
gt
c
.
and
x
=
(14.21)
It is important to realise that the time
t
appearing in these two equations is the
time according to an observer at the origin in
S
. It is called the co-ordinate time,
for it is the time co-ordinate we choose to define the location of space-time events
in
S
. We are not entitled to claim that this is also equal to the time measured on
a clock located on the little rocket at
x
and indeed we shall soon see that it is not
possible for the two to be equal at all times.
The set of co-ordinates defined by the fleet of rockets constitutes a uniformly
accelerating frame of reference. We shall now demonstrate that only the little rocket
at the origin accelerates at rate
g
in its own rest frame. All other rockets at
x
=
0
accelerate at a different rate in their respective rest frames. Consider the little rocket
located at a particular value of
x
. We know that this point must have a four-velocity
of magnitude equal to
c
(see Eq. (12.8)), i.e.
c
2
=
V
·
V
(14.22)
where
c
2
g
x
sinh
gt
c
,
c
2
g
x
cosh
gt
c
,
0
,
0
d
d
τ
V
=
+
+
(14.23)
