# Metric tensor

(Difference between revisions)
 Revision as of 14:37, 25 January 2006 (view source) (→Definition)← Older edit Revision as of 14:45, 25 January 2006 (view source) (→Change of basis)Newer edit → Line 48: Line 48: 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',,j,, ''' = ''A,,j,, ^i^'' '''e,,i,, '''; ''x'^j^'' = ''B,,i,, ^j^'' ''x^i^'', + '''e'j''' = ''Aj i'' '''ei'''; ''x'j'' = ''Bi j'' ''x i'', - where ''A,,j,, ^i^'' and ''B,,i,, ^j^'' are transformation matrices, transpose of one another. According to their + where ''Aj i'' and ''Bi 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: + definition, the components ''gij,'' of the metric tensor transform like products of basis vectors: - ''g',,kl,,'' = ''A,,k,, ^i^A,,l,, ^j^g,,ij,,''. + ''g'kl'' = ''AkiAljgij''. They are the doubly covariant components of the metric tensor. They are the doubly covariant components of the metric tensor. - The ["dual basis"] vectors and coordinates transform in the change of basis according to: + 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,,'', + '''e'j''' = ''Bi j'' '''ei'''; ''x'j'' = ''Aj ixi'', - and the components ''g^ij^'' transform like products of dual basis vectors: + and the components ''gij'' transform like products of dual basis vectors: - ''g'^kl^'' = ''A,,i,,^k^ A,,j^l^ g^ij^''. + ''g'kl'' = ''Aik Ajl gij''. They are the doubly contravariant components of the metric tensor. 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. + The mixed components, ''gij'' = δij, are once covariant and once contravariant and are invariant. == Properties of the metric tensor == == Properties of the metric tensor ==

# Metric tensor

### Other languages

Tenseur métrique (Fr).

## Definition

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

gij = ei . ej = ej.ei = gji.

It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, x = xi ei and y = yj ej is written:

x . y = xi ei . yj ej = gij xi yj.

In a three-dimensional space with basis vectors a, b, c, the coefficients gij 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 gij, gij, (gikgkj = δkj, Kronecker symbol, = 0 if ij, = 1 if i = j) relates the dual basis, or reciprocal space vectors ei to the direct basis vectors ei through the relations:

ej = gij ej

In three-dimensional space, the dual basis vectors are identical to the reciprocal space vectors and the components of gij 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 = Aj i ei; x'j = Bi j x i,

where Aj i and Bi j are transformation matrices, transpose of one another. According to their definition, the components gij, of the metric tensor transform like products of basis vectors:

g'kl = AkiAljgij.

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 = Bi j ei; x'j = Aj ixi,

and the components gij transform like products of dual basis vectors:

g'kl = Aik Ajl gij.

They are the doubly contravariant components of the metric tensor.

The mixed components, gij = δij, 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 - cos ^2^ γ + 2 cos α cos α cos α)
```

V*2 = Δ (gij) = 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.