# Metric tensor

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

- | + | '''e'<sub>j</sub>''' = ''A<sub>j</sub><sup> i</sup>'' '''e<sub>i</sub>'''; ''x'<sup>j</sup>'' = ''B<sub>i</sub><sup> j</sup>'' ''x<sup> i</sup>'', | |

- | where ''A | + | where ''A<sub>j</sub><sup> i</sup>'' and ''B<sub>i</sub><sup> j</sup>'' are transformation matrices, transpose of one another. According to their |

- | definition, the components ''g | + | definition, the components ''g<sub>ij</sub>,'' of the metric tensor transform like products of basis vectors: |

- | + | ''g'<sub>kl</sub>'' = ''A<sub>k</sub><sup>i</sup>A<sub>l</sub><sup>j</sup>g<sub>ij</sub>''. | |

They are the doubly covariant components of the metric tensor. | They are the doubly covariant components of the metric tensor. | ||

- | The [ | + | The [[dual basis]] vectors and coordinates transform in the change of basis according to: |

- | + | '''e'<sup>j</sup>''' = ''B<sub>i</sub><sup> j</sup>'' '''e<sup>i</sup>'''; ''x'<sub>j</sub>'' = ''A<sub>j</sub><sup> i</sup>x<sub>i</sub>'', | |

- | and the components ''g | + | and the components ''g<sup>ij</sup>'' transform like products of dual basis vectors: |

- | + | ''g'<sup>kl</sup>'' = ''A<sub>i</sub><sup>k</sup> A<sub>j</sub><sup>l</sup> g<sup>ij</sup>''. | |

They are the doubly contravariant components of the metric tensor. | They are the doubly contravariant components of the metric tensor. | ||

- | The mixed components, ''g | + | The mixed components, ''g<sup>i</sup><sub>j</sub>'' = δ<sup>i</sup><sub>j</sub>, are once covariant and once contravariant and are invariant. |

== Properties of the metric tensor == | == Properties of the metric tensor == |

## Revision as of 14:45, 25 January 2006

## 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*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

* 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*