Metric Tensor

Relevant parts to questions...

• Using $g_{ik}={\mathbf{e}}_i\cdot {\mathbf{e}}_k$ and $g^{ik}={\mathbf{e}}^i\cdot {\mathbf{e}}^k$ to compute components from a basis.
• Using $g_i^k={\delta }_i^k$ for the mixed metric.
• Using the arc length formula $(ds{)}^2=g_{ik}d{x}^{i}d{x}^k$.
• Using the metric coefficients $h_i=\sqrt{g_{ii}}$ for orthogonal bases.
• Using $g_{ik}$ to lower an index and $g^{ik}$ to raise one (see Index Raising and Lowering).

The Metric Tensor determines the geometry of a space - distances, angles, and the conversion between Covariant and Contravariant Components. It has three flavours of components, all part of a single second-rank tensor:

• Covariant::$g_{ik}={\mathbf{e}}_i\cdot {\mathbf{e}}_k$ (original basis).
• Contravariant::$g^{ik}={\mathbf{e}}^i\cdot {\mathbf{e}}^k$ (dual basis).
• Mixed::$g_i^k={\mathbf{e}}_i\cdot {\mathbf{e}}^k={\delta }_i^k$ - exactly the Kronecker Delta.

The defining equation is the arc-length formula:

$$\boxed{(ds{)}^2=g_{ik}d{x}^{i}d{x}^k=g^{ik}d{x}_{i}d{x}_k=d{x}_{i}d{x}^i}$$

Properties

• Symmetric::$g_{ik}=g_{ki}$ (since the dot product is commutative); same for $g^{ik}$.
• Mutually inverse matrices::$[g^{ik}]=[g_{ik}{]}^{-1}$, so $g^{ij}{g}_{jk}={\delta }_k^i$.
• Orthogonal-basis shortcut::if ${\mathbf{e}}_i\cdot {\mathbf{e}}_k=0$ for $i\ne k$, the metric is diagonal and $g_{ii}=1/g^{ii}$ (no sum).
• Cartesian special case::$g_{ik}={\delta }_{ik}$, so covariant and contravariant components coincide - this is why earlier lectures did not distinguish them.
• It genuinely is a tensor::$g_{ik}^{\prime }=L_{i^{\prime }}^{\ell }{L}_{k^{\prime }}^m{g}_{\ell m}$ obeys the Tensor Transformation Rule for covariant rank 2.

Deriving the Metric from a Position Vector

Given a parametrisation $\mathbf{r}(x^1,x^2,x^3)$, the basis vectors are

$${\mathbf{e}}_i=\frac{\partial \mathbf{r}}{\partial x^i}$$

and the metric follows directly:

$$\boxed{g_{ij}={\mathbf{e}}_i\cdot {\mathbf{e}}_j=\frac{\partial r^{\ell }}{\partial x^i}\frac{\partial r^{\ell }}{\partial x^j}}$$

Metric Coefficients

For an orthogonal basis, the metric coefficients (or scale factors) are

$$h_i=\sqrt{g_{ii}}(\text{no sum on }i)$$

and the arc length simplifies to $(ds{)}^2=\sum _{i=1}^3(h_{i}d{x}^i{)}^2$.

Applications

1. Raising and lowering indices, via $A_i=g_{ik}{A}^k$ and $A^i=g^{ik}{A}_k$ (see Index Raising and Lowering).
2. Arc length in any coordinate system, via $(ds{)}^2=g_{ik}d{x}^{i}d{x}^k$.
3. Dot and cross products in curvilinear coordinates::$\mathbf{A}\cdot \mathbf{B}=g_{ik}{A}^i{B}^k=A^i{B}_i$.
4. Identifying associated tensors, by combining $g_{ik}$ (and $g^{ik}$) with a tensor to produce Associated Tensors.

Cylindrical coordinates

For $(r,\theta ,z)$, $\mathbf{r}=r\cos \theta {\mathbf{i}}_1+r\sin \theta {\mathbf{i}}_2+z{\mathbf{i}}_3$. Differentiating:

${\mathbf{e}}_1=\cos \theta {\mathbf{i}}_1+\sin \theta {\mathbf{i}}_2$, ${\mathbf{e}}_2=-r\sin \theta {\mathbf{i}}_1+r\cos \theta {\mathbf{i}}_2$, ${\mathbf{e}}_3={\mathbf{i}}_3$.

Dot products give $g_{11}=1$, $g_{22}=r^2$, $g_{33}=1$, so $h_1=1$, $h_2=r$, $h_3=1$. Hence $(ds{)}^2=\boxed{(dr{)}^2+(rd\theta {)}^2+(dz{)}^2}$.

Basis vectors dot, metric out.