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 g,,ij,, of the metric tensor are:

```g,,11,, = a^2^; g,,12,, = a . b; g,,13,, = a . c;BR
g,,21,, = b . a; g,,22,, = b^2^; g,,23,, = b . c;BR
g,,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 ij, = 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*;BR
g^21^ = b* . a*; g^22^ = b*^2^; g^23^ = b* . c*;BR
g^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

```* 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.