Environmental Engineering Reference
In-Depth Information
It is now easy enough to read off the metric for this space:
R
2
0
g
=
(13.7)
R
2
sin
2
θ
0
in the (θ, ϕ) basis. As an aside, notice that we compute distances on the surface of a
sphere by integrating the distance measure, e.g. the circumference can be obtained
by integrating ds along the curve ϕ
=
constant,
0
<θ<
2
π :
sin
2
θ
d
ϕ
d
θ
2
π
circumference
=
d
s
=
R
d
θ
1
+
(13.8)
0
2
π
=
=
R
d
θ
2
πR.
(13.9)
0
The metric of ordinary three dimensional Euclidean space when expressed in
Cartesian co-ordinates is just the unit matix, whilst in spherical polar co-ordinates
it is given by
10 0
0
R
2
0
00
R
2
sin
2
θ
g
=
(13.10)
in the
(r,θ,ϕ)
basis. The metric is an inherent geometrical feature of the space and
it is therefore a tensor
4
, so although for any given space the matrix representation
of the metric depends upon the chosen co-ordinate basis the metric itself remains
unchanged. Notice that whilst the metric of Euclidean space written in Eq. (13.10)
describes a flat space (i.e. a space in which Pythagoras' Theorem always works) the
same cannot be said of the metric written in Eq. (13.7) which describes a curved
space, e.g. right-angled triangles drawn on the surface of a sphere do not satisfy
Pythagoras' Theorem. Equivalently, it is not possible to identify a two-dimensional
co-ordinate basis in which the metric of Eq. (13.7) is represented by the unit matrix.
We're now ready to return to space-time and the metric of Eq. (13.5). If we
assume that the metric is constant, i.e. that space-time has the same geometry at
all points, then its diagonal form follows rather generally since any non-singular
matrix can be diagonalised by an appropriate change of basis. The presence of the
diagonal entries equal to
−
1 arises if we insist that the space should be Euclidean
if we take slices through it of constant time. That we chose
1
(or any other number) is a matter of convention. For example if in Euclidean
space we chose a metric equal to minus the unit matrix then all distances would
be multiplied by the square root of minus one, which is not very economical but
otherwise wholly acceptable. In summary, we have established that Eq. (13.5) is
in fact the most general metric which satisfies the constraint that slices of constant
−
1 rather than
+
4
See Section 10.2 for a discussion of the moment of inertia tensor.
