The metric tensor

26.15 THE METRIC TENSOR
second-order tensor T. Using the outer product notation in (26.23), we may write
T in three different ways:
T = T ij ei ⊗ ej = T ij ei ⊗ e j = Tij ei ⊗ e j ,
where T ij , T ij and Tij are called the contravariant, mixed and covariant components of T respectively. It is important to remember that these three sets of
quantities form the components of the same tensor T but refer to different (tensor)
bases made up from the basis vectors of the coordinate system. Again, if we are
using Cartesian coordinates then all three sets of components are identical.
We may generalise the above equation to higher-order tensors. Components
carrying only superscripts or only subscripts are referred to as the contravariant
and covariant components respectively; all others are called mixed components.
26.15 The metric tensor
Any particular curvilinear coordinate system is completely characterised at each
point in space by the nine quantities
gij = ei · ej ,
(26.56)
which, as we will show, are the covariant components of a symmetric second-order
tensor g called the metric tensor.
Since an infinitesimal vector displacement can be written as dr = dui ei , we find
that the square of the infinitesimal arc length (ds)2 can be written in terms of the
metric tensor as
(ds)2 = dr · dr = dui ei · du j ej = gij dui du j .
(26.57)
It may further be shown that the volume element dV is given by
dV =
√
g du1 du2 du3 ,
(26.58)
where g is the determinant of the matrix [ gij ], which has the covariant components
of the metric tensor as its elements.
If we compare equations (26.57) and (26.58) with the analogous ones in section
10.10 then we see that in the special case where the coordinate system is orthogonal
(so that ei · ej = 0 for i = j) the metric tensor can be written in terms of the
coordinate-system scale factors hi , i = 1, 2, 3 as
#
h2i i = j,
gij =
0 i = j.
Its determinant is then given by g = h21 h22 h23 .
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TENSORS
Calculate the elements gij of the metric tensor for cylindrical polar coordinates. Hence
find the square of the infinitesimal arc length (ds)2 and the volume dV for this coordinate
system.
As discussed in section 10.9, in cylindrical polar coordinates (u1 , u2 , u3 ) = (ρ, φ, z) and so
the position vector r of any point P may be written
r = ρ cos φ i + ρ sin φ j + z k.
From this we obtain the (covariant) basis vectors:
∂r
= cos φ i + sin φ j;
∂ρ
∂r
e2 =
= −ρ sin φ i + ρ cos φ j;
∂φ
∂r
e3 =
= k.
∂z
Thus the components of the metric tensor [gij ] = [ei · ej ] are found to be


1 0 0
2

0 ,
0 ρ
G = [gij ] =
0 0 1
e1 =
(26.59)
(26.60)
from which we see that, as expected for an orthogonal coordinate system, the metric tensor
is diagonal, the diagonal elements being equal to the squares of the scale factors of the
coordinate system.
From (26.57), the square of the infinitesimal arc length in this coordinate system is given
by
(ds)2 = gij dui du j = (dρ)2 + ρ2 (dφ)2 + (dz)2 ,
and, using (26.58), the volume element is found to be
√
dV = g du1 du2 du3 = ρ dρ dφ dz.
These expressions are identical to those derived in section 10.9. We may also express the scalar product of two vectors in terms of the metric
tensor:
a · b = ai ei · b j ej = gij ai b j ,
(26.61)
where we have used the contravariant components of the two vectors. Similarly,
using the covariant components, we can write the same scalar product as
a · b = ai ei · bj e j = g ij ai bj ,
(26.62)
where we have defined the nine quantities g ij = ei ·e j . As we shall show, they form
the contravariant components of the metric tensor g and are, in general, different
from the quantities gij . Finally, we could express the scalar product in terms of
the contravariant components of one vector and the covariant components of the
other,
a · b = ai ei · b j ej = ai b j δji = ai bi ,
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(26.63)
26.15 THE METRIC TENSOR
where we have used the reciprocity relation (26.54). Similarly, we could write
a · b = ai ei · bj e j = ai bj δij = ai bi .
(26.64)
By comparing the four alternative expressions (26.61)–(26.64) for the scalar
product of two vectors we can deduce one of the most useful properties of
the quantities gij and g ij . Since gij ai b j = ai bi holds for any arbitrary vector
components ai , it follows that
gij b j = bi ,
which illustrates the fact that the covariant components gij of the metric tensor
can be used to lower an index. In other words, it provides a means of obtaining
the covariant components of a vector from its contravariant components. By a
similar argument, we have
g ij bj = bi ,
so that the contravariant components g ij can be used to perform the reverse
operation of raising an index.
It is straightforward to show that the contravariant and covariant basis vectors,
ei and ei respectively, are related in the same way as other vectors, i.e. by
ei = g ij ej
and
ei = gij e j .
We also note that, since ei and ei are reciprocal systems of vectors in threedimensional space (see chapter 7), we may write
ei =
ej × ek
,
ei · (ej × ek )
for the combination of subscripts i, j, k = 1, 2, 3 and its cyclic permutations. A
similar expression holds for ei in terms of the ei -basis. Moreover, it may be shown
√
that |e1 · (e2 × e3 )| = g.
Show that the matrix [g ij ] is the inverse of the matrix [gij ]. Hence calculate the contravariant components g ij of the metric tensor in cylindrical polar coordinates.
Using the index-lowering and index-raising properties of gij and g ij on an arbitrary vector
a, we find
δki ak = ai = g ij aj = g ij gjk ak .
But, since a is arbitrary, we must have
g ij gjk = δki .
ij
(26.65)
Denoting the matrix [gij ] by G and [g ] by Ĝ, equation (26.65) can be written in matrix
form as ĜG = I, where I is the unit matrix. Hence G and Ĝ are inverse matrices of each
other.
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