Environmental Engineering Reference
In-Depth Information
As we have already stated, a uniform field is one in which a particle released
anywhere in it feels an acceleration which is the same regardless of when or where
the particle was released. The last section revealed that there is no way to arrange
this using a single co-ordinate system in Minkowski space. However it can be
arranged if we distort space-time accordingly. We start by assuming that the metric
yields the following invariant distance:
( d s) 2
f(x) 2 (c d t) 2
( d x) 2
( d y) 2
( d z) 2 .
=
(14.48)
This metric has the virtue that the corresponding co-ordinates are rigid in the sense
that the distance between any two points is independent of t . All that we demand is
that f(x) be chosen such that any particle released from rest accelerates at the same
rate throughout the frame. We can make use of the geodesic equation, Eq. (14.35),
to solve the problem for us for it describes the trajectory of a free particle released
from rest. The metric is quite simple and, setting i
=
1 yields the equation
c d t
d τ
2
d 2 x
d τ 2
1
2
∂g 00
∂x
=
0 ,
(14.49)
f(x) 2 and τ is the proper time measured on the particle. Now an
analysis identical to that leading up to Eq. (14.24) tells us that
=
where g 00
d t
d τ =
1
f(x)
(14.50)
for a particle at rest (i.e. d x/ d τ
0). Thus the acceleration felt by a particle at rest
in a uniform gravitational field is given by
=
d 2 x
d τ 2
c 2 1
f
d f
d x .
=−
(14.51)
We want this to be a constant over the whole space and hence
c 2 1
f
d f
d x =
g,
exp (gx/c 2 )
i.e. f(x)
=
(14.52)
and we have arbitrarily chosen f( 0 )
1. To recap, we have succeeded in identify-
ing a space-time that is not Minkowskian but which does respresent a uniform grav-
itational field in which particles released at rest remain equidistant. The invariant
distance in this space is
=
( d s) 2
exp ( 2 gx/c 2 )(c d t) 2
( d x) 2
( d y) 2
( d z) 2 .
=
(14.53)
The first thing to notice is that this space-time interval is approximately equal to
that of the uniformly accelerating frame which we presented in Eq. (14.29) for
sufficiently small values of gx/c 2 . This is not too surprising on reflection since we
would expect that for a sufficiently weak uniform gravitational field there should
be an approximation in which the space-time is Minkowski flat.
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