Environmental Engineering Reference
In-Depth Information
Time
Time
Space
B
Space
Space
B
y
A
A
x
(b)
(a)
Figure 1.5 Different structures of space and time: (a) absolute space where points
A
and
B
are the same point in space; (b) a fibre-bundle structure where each moment in time has
its own space.
relativity challenges this picture by rejecting the notion of absolute space and
replacing it with the idea that space is defined relative to some chosen set of axes
at a given instant in time. This is more like the picture in Figure 1.5(b), a structure
that mathematicians call a fibre bundle. The same events
A
and
B
now lie in differ-
ent spaces and the connection between them is no longer obvious. The fibre bundle
is a more abstract structure to deal with than the space
time structure of (a). Imag-
ine, for example, trying to calculate the displacement from
A
to
B
.Todothiswe
have to assume some additional structure of space-time that allows us to compare
points
A
and
B
. It is as if space is erased and redefined at each successive instant
and we have no automatic rule for saying how the 'new' space relates to the 'old'
one. Notice that this view still treats time as absolute; observers at different points
in the
x
×
y
plane agree on the common time
t
. In later chapters we will recon-
sider the geometry of space and time when we come to study the theory of Special
Relativity, where universal time will be rejected in favour of a new space-time
geometry in which observers at different positions each have their own local time.
−
1.2 VECTORS AND CO-ORDINATE SYSTEMS
As far as we can tell, space is three-dimensional, which means that three numbers
are required to define a unique position. How we specify the three position-giving
numbers defines what is known as the co-ordinate system. The co-ordinate system
therefore introduces a sort of invisible grid or mesh that maps every point in
space onto a unique ordered set of three real numbers. Figure 1.6 shows two
commonly-used 3-dimensional co-ordinate systems. The Cartesian system is named
after the French philosopher and mathematician Rene Descartes (1596 - 1650), who
is reputed to have invented it from his bed while considering how he might specify
the position of a fly that was buzzing around his room. This co-ordinate system
consists of three mutually perpendicular axes, labelled
x
,
y
and
z
, that intersect at
the point
O
, called the origin. The position of a particle at
P
may be specified
by giving the set of three distances
(x,y,z)
. Another frequently used co-ordinate
system, the spherical-polar system, is obtained when the position of the particle is
given instead by the distance from the origin
r
and two angles: the polar angle
