Metric tensor

(Difference between revisions)
 Revision as of 14:28, 25 January 2006 (view source)← Older edit Revision as of 14:37, 25 January 2006 (view source) (→Definition)Newer edit → Line 28: Line 28: In three-dimensional space, the dual basis vectors are identical to the [[reciprocal space]] vectors and the components of ''gij'' are: In three-dimensional space, the dual basis vectors are identical to the [[reciprocal space]] vectors and the components of ''gij'' are: - ''g^11^'' = '''a*^2^'''; ''g^12^'' = '''a* . b*'''; ''g^13^'' = '''a* . c*''';[[BR]] + ''g11'' = '''a*2'''; ''g12'' = '''a* . b*'''; ''g13'' = '''a* . c*''';
- ''g^21^'' = '''b* . a*'''; ''g^22^'' = '''b*^2^'''; ''g^23^'' = '''b* . c*''';[[BR]] + ''g21'' = '''b* . a*'''; ''g22'' = '''b*2'''; ''g23'' = '''b* . c*''';
- ''g^31^'' = '''c* . a*'''; ''g^32^'' = '''c* . b*'''; ''g^33^'' = '''c*^2^''';[[BR]] + ''g31'' = '''c* . a*'''; ''g32'' = '''c* . b*'''; ''g33'' = '''c*2'''; with: with: - ''g^11^'' =  ''b''^2^''c''^2^ sin^2^ α/ V^2^; + ''g11'' =  ''b''2''c''2 sin2 α/ V2; - ''g^22^'' =  ''c''^2^''a''^2^ sin^2^ β/ V^2^; + ''g22'' =  ''c''2''a''2 sin2 β/ V2; - ''g^33^'' =  ''a''^2^''b''^2^ sin^2^ γ/ V^2^;[[BR]] + ''g33'' =  ''a''2''b''2 sin2 γ/ V2; - ''g^12^'' = ''g^21^'' = (''abc''^2^/ V^2^)(cos α cos β - cos γ); + ''g12'' = ''g21'' = (''abc''2/ V2)(cos α cos β - cos γ); - ''g^23^'' = ''g^32^'' = (''a^2^bc''/ V^2^)(cos β cos γ - cos α); + ''g23'' = ''g32'' = (''a2bc''/ V2)(cos β cos γ - cos α); - ''g^31^'' = ''g^13^'' = (''ab^2^c''/ V^2^)(cos γ cos α - cos β). + ''g31'' = ''g13'' = (''ab2c''/ V2)(cos γ cos α - cos β) - where ''V'' is the volume of the unit cell ('''a''', '''b''', '''c''') + where ''V'' is the volume of the unit cell ('''a''', '''b''', '''c'''). == Change of basis == == Change of basis ==

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,,  = 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

```* 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.