Environmental Engineering Reference
In-Depth Information
Generally speaking since the acceleration due to gravity varies over space-time
we can't eliminate it everywhere by a single change of co-ordinates but we can
eliminate it everywhere if we change co-ordinates differently at different space-time
points. This leads to the fascinating possibility that all effects of gravity can
be entirely eliminated by a suitable change of co-ordinates. The effect of grav-
ity can therefore be converted entirely into a specification of the geometry of
space-time. That the geometry is no longer Minkowskian, but some more general
curved space-time follows from the fact that the transformation from a particular
co-ordinate basis to a locally inertial (i.e. Minkowski) co-ordinate basis is different
at different points in space-time. Put another way, there exists no single co-ordinate
transformation that is able to convert the metric tensor into Minkowski form.
Thus gravitation dictates that space-time is locally Minkowskian but glob-
ally curved. To illustrate the geometrical ideas involved let us consider the
two-dimensional surface of a sphere. We can imagine chopping the surface up
into a very large number of small patches. Each patch is approximately flat, with
the approximation becoming better the smaller the size of the patch. Physics
in the vicinity of any one patch can be described using Euclidean geometry.
However, physics that extends over more than one patch is clearly not Euclidean.
The curved nature of the sphere is manifest by the fact that it is not possible to
represent it by a single Euclidean patch. Free particles will follow straight lines on
the surface of the sphere, or more precisely they follow geodesics. Over any patch
the path of a free particle is a Euclidean straight line but Eq. (14.35) is needed in
order to determine the path of a free particle over a larger portion of the sphere.
The same can be said of gravitation and so Eq. (14.35) tells us how particles
move not only in the absence of any external forces but also in the presence of
gravity. Conveniently, the geodesic equation is an equation expressed in terms of
a single co-ordinate system (i.e. not in terms of one co-ordinate system for each
point in space-time). It is the space-time dependence of the metric that tells us
how, at any particular space-time point, we can transform co-ordinates so that we
are in a locally inertial frame.
To summarize, we have explained how the Equivalence Principle can be used
to express the influence of all gravitational fields in terms of the geometry of
space-time. Mathematically this information is encoded in the metric tensor
g
. What
we have not yet explained is how one should compute
. Clearly the distribution
of matter must play an important role in fixing the space-time geometry. To say
more than that takes us beyond the scope of this topic but we do hope to have
whetted the reader's appetite to study further Einstein's theory of gravitation.
g
14.2.1
Gravitational fields
As we discussed in the previous section, it is possible to express the invariant
distance between a pair of neighbouring events in terms of co-ordinates corre-
sponding to a rigid frame of reference which is accelerating uniformly. The result
is given in Eq. (14.29), i.e.
1
c
2
2
(c
d
t)
2
gx
(
d
s)
2
(
d
x)
2
(
d
y)
2
(
d
z)
2
,
=
+
−
−
−
(14.44)
