# 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**;

*g*=

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

*g*=

_{22}**b**;

^{2}*g*=

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

*g*=

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

*g*=

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

*g*=

_{33}**c**;

^{2}The inverse matrix of *g _{ij}*,

*g*, (

^{ij}*g*= δ

^{ik}g_{kj}*, Kronecker symbol, = 0 if*

^{k}_{j}*i*≠

*j*, = 1 if

*i*=

*j*) relates the dual basis, or reciprocal space vectors

**e**to the direct basis vectors

^{i}**e**through the relations:

_{i}**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***;

*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,,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*