# Metric tensor

### From Online Dictionary of Crystallography

Tenseur métrique (*Fr*). Metrischer Tensor (*Ge*). Tensore metrico (*It*). Метрический тензор (*Ru*). 計量テンソル (*Ja*). Tensor métrico (*Sp*).

## Definition

A metric tensor is used to measure distances in a space. In crystallography the spaces considered are vector spaces with
*Euclidean* metrics, *i.e.* ones for which the rules of Euclidean geometry apply. In that case,
given a basis **e _{i}** of a

*Euclidean space*,

*E*, the metric tensor is a rank 2 tensor the components of which are:

^{n}*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** = *x ^{i}*

**e**and

_{i}**y**=

*y*

^{j}**e**is written:

_{j}**x** . **y** = *x ^{i}*

**e**.

_{i}*y*

^{j}**e**=

_{j}*g*

_{ij}*x*

^{i}*y*.

^{j}In a three-dimensional space with basis vectors **a**, **b**, **c**, the coefficients *g _{ij}* of the metric tensor are:

*g _{11}* =

**a**;

^{2}*g*=

_{12}**a . b**;

*g*=

_{13}**a . c**;

*g*=

_{21}**b . a**;

*g*=

_{22}**b**;

^{2}*g*=

_{23}**b . c**;

*g*=

_{31}**c . a**;

*g*=

_{32}**c . b**;

*g*=

_{33}**c**.

^{2}Because the metric tensor is symmetric, *g _{12}* =

*g*,

_{21}*g*=

_{13}*g*, and

_{31}*g*=

_{13}*g*. Thus there are only six unique elements, often presented as

_{31}*g _{11}*

*g*

_{22}*g*

_{33}*g*

_{23}*g*

_{13}*g*

_{12}or, multiplying the second row by 2, as a so-called G^{6} ("G" for Gruber) vector

( **a ^{2}**,

**b**,

^{2}**c**, 2

^{2}**b . c**, 2

**a . c**, 2

**a . b**)

The inverse matrix of *g _{ij}*,

*g*, relates the dual basis, or reciprocal space vectors

^{ij}**e**

^{i}to the direct basis vectors

**e**

_{i}, through the relations

**e**^{j} = *g ^{ij}*

**e**

_{j}.

Note that *g ^{ik}g_{kj}* = δ

*, where δ*

^{k}_{j}*is the Kronecker symbol, equal to 0 if*

^{k}_{j}*i*≠

*j*, and equal to 1 if

*i*=

*j*.

In three-dimensional space, the dual basis vectors are identical to the reciprocal space vectors and the components of *g ^{ij}* are:

*g ^{11}* =

**a***;

^{2}*g*=

^{12}**a* . b***;

*g*=

^{13}**a* . c***;

*g*=

^{21}**b* . a***;

*g*=

^{22}**b***;

^{2}*g*=

^{23}**b* . c***;

*g*=

^{31}**c* . a***;

*g*=

^{32}**c* . b***;

*g*=

^{33}**c***;

^{2}with:

*g ^{11}* =

*b*

^{2}

*c*

^{2}sin

^{2}α/ V

^{2};

*g*=

^{22}*c*

^{2}

*a*

^{2}sin

^{2}β/ V

^{2};

*g*=

^{33}*a*

^{2}

*b*

^{2}sin

^{2}γ/ V

^{2};

*g ^{12}* =

*g*= (

^{21}*abc*

^{2}/ V

^{2})(cos α cos β - cos γ);

*g ^{23}* =

*g*= (

^{32}*a*/ V

^{2}bc^{2})(cos β cos γ - cos α);

*g ^{31}* =

*g*= (

^{13}*ab*/ V

^{2}c^{2})(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*are transformation matrices, transpose of one another. According to their definition, the components

_{i}^{ j}*g*of the metric tensor transform like products of basis vectors:

_{ij},*g' _{kl}* =

*A*.

_{k}^{i}A_{l}^{j}g_{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}**e**;

^{i}*x'*=

_{j}*A*,

_{j}^{ i}x_{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, *g ^{i}_{j}* = δ

^{i}

_{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*and_{ij}*g*are the components of a^{ij}*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*'s and the_{ij}*g*'s:^{ij}

*V*^{ 2} = Δ (*g _{ij}*) =

*abc*(1 - cos

^{2}α - cos

^{2}β - cos

^{2}γ + 2 cos α cos β cos γ)

*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 _{im}t^{ 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 No. 10 of the International Union of Crystallography)- Chapter 1.1.3 of
*International Tables for Crystallography, Volume B* - Chapter 1.1.2 of
*International Tables for Crystallography, Volume D*