Environmental Engineering Reference
In-Depth Information
Eqs. (14.40) and (14.41) are none other than the equations for the Coriolis and
centrifugal forces embodied in the equation we derived in Chapter 8, i.e.
x
=−
2
ω
×
x
−
ω
×
(
ω
×
x
).
(14.42)
Eq. (14.35) has a very interesting interpretation in the mathematics of curved
spaces. It is the equation that describes the curved space generalisation of a straight
line and it is often referred to as the 'geodesic equation'. The idea of a straight line
in a curved space may not be immediately intuitive but the notion is well defined
mathematically. One can imagine sliding a tangent vector on the surface along its
length. For example, great circles are straight lines on the surface of a sphere. Eq.
(14.35) is also our replacement for Newton's First Law. It says that free particles
always follow geodesics through space-time and only in inertial frames do these
correspond to Euclidean straight lines in space.
This sets the standard: our goal should always be to write all of the laws of
physics in a manifestly co-ordinate independent way. It is very important to realise
that the space-time about which we have been speaking so far in this topic is in all
cases Minkowski space-time. Changing co-ordinates to an accelerating frame does
not change space-time, it merely makes the geodesic equation more complicated.
The mathematics of curved spaces is however also the mathematics of General
Relativity: Einstein's theory of gravitation. As we shall shortly discover, in this
case the space-time need no longer be Minkowskian.
14.2
A GLIMPSE OF GENERAL RELATIVITY
Newton's Law of Gravitation states that a body of mass
m
has an acceleration
a
which is directed towards a body of mass
M
,i.e.
GMm
ˆ
r
2
.
a
=−
m
(14.43)
At first glance this equation seems fairly unremarkable. However it really is quite
astonishing that the mass
m
on the left hand side is the same as that on the right
hand side. It means that all bodies fall with the same acceleration in a gravitational
field. This is surprising; what has the mass in Newton's Second Law got to do
with the mass appearing in the law of gravitation? There are certainly no other
forces in Nature that act upon particles but which induce an acceleration that does
not depend upon any intrinsic property of the particle. For example, accelerations
in electrodynamics depend upon the ratio
q/m
. That all bodies fall with the same
acceleration in a gravitational field is known as the Equivalence Principle and its
consequences are, as we shall very soon see, profound.
Now if at some point in space-time a body experiences a particular acceleration
then it is always possible to change co-ordinates so that, at that particular point,
the acceleration disappears and, in the infinitesimal neighbourhood of the point,
space-time is Minkowskian. If the acceleration is due to gravity then since all bodies
experience the same acceleration it follows that we can eliminate the gravitational
force at a point by suitably changing co-ordinates.
