Metric tensor

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Tenseur métrique (Fr); Tensore metrico (It); 計量テンソル (Ja).

Contents

Definition

Given a basis ei of a Euclidean space, En, the metric tensor is a rank 2 tensor the components of which are:

gij = ei . ej = ej.ei = gji.

It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, x = xi ei and y = yj ej is written:

x . y = xi ei . yj ej = gij xi yj.

In a three-dimensional space with basis vectors a, b, c, the coefficients gij of the metric tensor are:

g11, = a2; g12 = a . b; g13 = a . c;
g21 = b . a; g22 = b2; g23 = b . c;
g31 = c . a; g32 = c . b; g33 = c2;

The inverse matrix of gij, gij, (gikgkj = δkj, Kronecker symbol, = 0 if ij, = 1 if i = j) relates the dual basis, or reciprocal space vectors ei to the direct basis vectors ei through the relations:

ej = gij ej

In three-dimensional space, the dual basis vectors are identical to the reciprocal space vectors and the components of gij are:

g11 = a*2; g12 = a* . b*; g13 = a* . c*;
g21 = b* . a*; g22 = b*2; g23 = b* . c*;
g31 = c* . a*; g32 = c* . b*; g33 = c*2;

with:

g11 = b2c2 sin2 α/ V2; g22 = c2a2 sin2 β/ V2; g33 = a2b2 sin2 γ/ V2;

g12 = g21 = (abc2/ V2)(cos α cos β - cos γ); g23 = g32 = (a2bc/ V2)(cos β cos γ - cos α); g31 = g13 = (ab2c/ V2)(cos γ cos α - cos β)

where V is the volume of the unit cell (a, b, c).

Change of basis

In a change of basis the direct basis vectors and coordinates transform like:

e'j = Aj i ei; x'j = Bi j x i,

where Aj i and Bi j are transformation matrices, transpose of one another. According to their definition, the components gij, of the metric tensor transform like products of basis vectors:

g'kl = AkiAljgij.

They are the doubly covariant components of the metric tensor.

The dual basis vectors and coordinates transform in the change of basis according to:

e'j = Bi j ei; x'j = Aj ixi,

and the components gij transform like products of dual basis vectors:

g'kl = Aik Ajl gij.

They are the doubly contravariant components of the metric tensor.

The mixed components, gij = δij, are once covariant and once contravariant and are invariant.

Properties of the metric tensor

  • The tensor nature of the metric tensor is demonstrated by the behaviour of its components in a change of basis. The components gij and gij are the components of a unique tensor.
  • The squares of the volumes V and V* of the direct space and reciprocal space unit cells are respectively equal to the determinants of the gij 's and the gij 's:

V 2 = Δ (gij) = abc(1 - cos 2 α - cos 2 β - cos2 γ + 2 cos α cos α cos α)

V*2 = Δ (gij) = 1/ V 2.

  • One changes the variance of a tensor by taking the contracted tensor product of the tensor by the suitable form of the metric tensor. For instance:

gimt ij..kl.. = t j..klm..

Multiplying by the doubly covariant form of the metric tensor increases the covariance by one, multiplying by the doubly contravariant form increases the contravariance by one.

See also