Environmental Engineering Reference
In-Depth Information
The speed
c
has entered into the way we define the co-ordinates in space-time
but at this stage we stress that it is simply an entirely arbitrary constant speed
introduced purely to make
g
00
dimensionless, i.e. it is needed because we choose
to measure one of the space-time co-ordinates in different units to the others. At
this stage in our considerations,
g
00
is to be viewed simply as a dimensionless
constant characteristic of the space-time.
What have we assumed? Certainly the matrix we have written down looks very
special with all of its entries along the diagonal and it is true that the most general
distance measure for what is called in mathematics a Riemannian space
3
would
allow much more general 4
4 matrices. Clearly the choice of matrix is very
intimately connected with the geometry of the space for it tells us how to compute
the distance between neighbouring points. Since this matrix is so important, it
has a name: it is called the 'metric' of the space. Perhaps before we answer the
question posed at the start of this paragraph we should get better acquainted with
the idea of a metric.
×
Example 13.2.1
What is the metric that determines distances on the surface of a
sphere of radius R? You should work in spherical polar co-ordinates.
Solution 13.2.1
The surface of a sphere is a two dimensional space. As illustrated
in Figure 13.6, neighbouring points A and B are separated by a distance ds which
satisfies
(
d
s)
2
(R
d
θ)
2
(R
sin
θ
d
ϕ)
2
.
=
+
(13.6)
This result is valid in the limit of vanishing distance since in that limit the relevant
portion of the sphere looks flat (i.e. Euclidean) and we can use Pythagoras' Theorem.
R
dq
A
dq
B
R
sinq dj
dj
Figure 13.6
The distance between two points on the surface of a sphere of radius
R
.
3
A space is Riemannian if the squared distance between two neighbouring points is of the form
(
d
s)
2
=
g
ij
d
x
i
d
x
j
.
