Metric tensor

Tenseur métrique (Fr). Tensore metrico (It)

Definition

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

g_ij = e_i . e_j = e_j.e_i = g_ji.

It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, x = xⁱ e_i and y = y^j e_j is written:

x . y = xⁱ e_i . y^j e_j = g_ij xⁱ y^j.

In a three-dimensional space with basis vectors a, b, c, the coefficients g_ij 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 g_ij, g^ij, (g^ikg_kj = δ^k_j, Kronecker symbol, = 0 if i ≠ j, = 1 if i = j) relates the dual basis, or reciprocal space vectors eⁱ to the direct basis vectors e_i through the relations:

e^j = g^ij e_j

In three-dimensional space, the dual basis vectors are identical to the reciprocal space vectors and the components of g^ij 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 = A_j ⁱ e_i; x'^j = B_i ^j x ⁱ,

where A_j ⁱ and B_i ^j are transformation matrices, transpose of one another. According to their definition, the components g_ij, of the metric tensor transform like products of basis vectors:

g'_kl = A_kⁱA_l^jg_ij.

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 = B_i ^j eⁱ; x'_j = A_j ⁱx_i,

and the components g^ij transform like products of dual basis vectors:

g'^kl = A_i^k A_j^l g^ij.

They are the doubly contravariant components of the metric tensor.

The mixed components, gⁱ_j = δⁱ_j, 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 g_ij and g^ij 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 g_ij 's and the g^ij 's:

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

V*² = Δ (g^ij) = 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:

g_imt ^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

dual basis
reciprocal space
Metric Tensor and Symmetry Operations in Crystallography (Teaching Pamphlet of the International Union of Crystallography)
Section 1.1.3 of International Tables of Crystallography, Volume B
Section 1.1.2 of International Tables of Crystallography, Volume D

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