# Metric tensor

### From Online Dictionary of Crystallography

## Contents |

# Metric tensor

### Other languages

Tenseur métrique (*Fr*).

## Definition

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;BRg,,21,,=b . a;g,,22,,=b^2^;g,,23,,=b . c;BRg,,31,,=c . a;g,,32,,=c . b;g,,33,,=c^2^;BR

The inverse matrix of *g,,ij,,*, *g^ij^*, (*g^ik^g,,kj,,* = δ*^k^,,j,,*, Kronecker symbol, = 0 if *i* ≠ *j*, = 1 if *i* = *j*) relates the ["dual basis"], or ["reciprocal space"] vectors **e^i^** 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:

g^11^=a*^2^;g^12^=a* . b*;g^13^=a* . c*;BRg^21^=b* . a*;g^22^=b*^2^;g^23^=b* . c*;BRg^31^=c* . a*;g^32^=c* . b*;g^33^=c*^2^;BR

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^;BR

g^12^=g^21^= (abc^2^/ V^2^)(cos α cos β - cos γ);g^23^=g^32^= (a^2^bc/ V^2^)(cos β cos γ - cos α);g^31^=g^13^= (ab^2^c/ V^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,,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,,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

* Thetensor natureof the metric tensor is demonstrated by the behaviour of its components in a change of basis. The componentsg,,ij,,andg^ij^are the components of auniquetensor.

* Thesquares of the volumesVandV*of the direct space and reciprocal space unit cells are respectively equal to the determinants of theg,,ij,,'s and theg^ij^'s:

V^2^ = Δ (g,,ij,,) =abc(1 - cos ^2^ α - cos ^2^ β - cos ^2^ γ + 2 cos α cos α cos α)

*V**^{2} = Δ (*g ^{ij}*) = 1/

*V*

^{2}.

* One changes thevariance of a tensorby 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

Section 1.1.3 of *International Tables of Crystallography, Volume B*

Section 1.1.2 of *International Tables of Crystallography, Volume D*