Mixed Components

Relevant parts to questions...

• A second-rank tensor has four component flavours: $A_{ik}$, $A^{ik}$, $A_i^{\cdot k}$, $A_{\cdot k}^i$.
• The dot tracks index order: in $A_i^{\cdot k}$, the first slot is lower (covariant), the second is upper (contravariant).
• Each covariant index picks up $L_{i^{\prime }}^{\ell }$; each contravariant index picks up $L_{\ell }^{i^{\prime }}$ - one factor per free index.
• Use the Metric Tensor to convert between any two flavours.

In a generalised coordinate system, a second-rank tensor has nine components, but those nine components can be written in four different ways:

• Covariant components::$A_{ik}$ (both indices down).
• Contravariant components::$A^{ik}$ (both indices up).
• Mixed components::$A_i^{\cdot k}$ (first covariant, second contravariant).
• Mixed components::$A_{\cdot k}^i$ (first contravariant, second covariant).

Dot notation

The dot marks the “gap” left by a moved index, preserving index order. Read $A_i^{\cdot k}$ as $(A_i{)}^k$ - the covariant index is in the first slot, contravariant in the second. The order matters because the transformation rule depends on which slot is raised and which is lowered.

Transformation Laws

Under a change of basis with coefficients $L_{i^{\prime }}^k$ (direct) and $L_i^{k^{\prime }}$ (inverse), each flavour transforms with one factor of $L$ per index:

$$\begin{array}{rlr}{A}_{ik}^{\prime }& =L_{i^{\prime }}^{\ell }{L}_{k^{\prime }}^m{A}_{\ell m}& \text{(both covariant)}\\ A^{\prime ik}& =L_{\ell }^{i^{\prime }}{L}_m^{k^{\prime }}{A}^{\ell m}& \text{(both contravariant)}\\ A_i^{\prime \cdot k}& =L_{i^{\prime }}^{\ell }{L}_m^{k^{\prime }}{A}_{\ell }^{\cdot m}& \text{(mixed)}\\ A_{\cdot k}^{\prime i}& =L_{\ell }^{i^{\prime }}{L}_{k^{\prime }}^m{A}_{\cdot m}^{\ell }& \text{(mixed)}\end{array}$$

The pattern is mechanical: each covariant index contracts with $L_{i^{\prime }}^{\ell }$, each contravariant index contracts with $L_{\ell }^{i^{\prime }}$.

Relationship via the Metric

All four flavours are Associated Tensors. The Metric Tensor converts between them:

$$A_{ik}=g_{i\ell }{g}_{km}{A}^{\ell m}=g_{k\ell }{A}_i^{\cdot \ell }=g_{i\ell }{A}_{\cdot k}^{\ell }$$ $$A^{ik}=g^{i\ell }{g}^{km}{A}_{\ell m}=g^{i\ell }{A}_{\ell }^{\cdot k}=g^{k\ell }{A}_{\cdot \ell }^i$$

Once you know any one flavour (and the metric), you know them all (see Index Raising and Lowering).

Properties

• Cartesian collapse::when $g_{ik}={\delta }_{ik}$, all four flavours coincide. This is why earlier lectures used only $A_{ij}$ without decorations.
• Symmetry lives at fixed index positions::$A_i^{\cdot k}$ symmetric in $i,k$ is meaningless - symmetry requires both indices at the same level. See Symmetry and Antisymmetry.
• Matrix realisation::if $A=[A_{ik}]$ and $G=[g_{ik}]$, you can compute other flavours via matrix multiplication: $[A_i^{\cdot k}]=A{G}^{-1}$, $[A_{\cdot k}^i]=G^{-1}A$, etc.
• Matrix form for a basis change::the covariant transformation $A^{\prime }=LA{L}^T$ is the quickest route numerically.

Applications

1. Computing tensor components after a basis change, by picking the flavour that simplifies the most (usually covariant via $LA{L}^T$).
2. Contracting cleanly, by pairing a raised with a lowered index::e.g. $A_i^{\cdot k}{B}_{\cdot k}^i$ is a well-defined scalar.
3. Translating between the “physical” and “generalised” formalisms: physical components ($A_{ik}$) vs. tensor components ($A^{ik}$) differ by scale factors in curvilinear coordinates.

All four flavours of a rank-2 tensor

Let $A_{ik}=A^{ik}=\left(\begin{array}{ccc}2& 1& 3\\ 2& 3& 4\\ 1& 2& 1\end{array}\right)$ in a Cartesian frame $K$, and define $K^{\prime }$ with ${\mathbf{e}}_1={\mathbf{i}}_1,{\mathbf{e}}_2={\mathbf{i}}_1+{\mathbf{i}}_2,{\mathbf{e}}_3={\mathbf{i}}_1+{\mathbf{i}}_2+{\mathbf{i}}_3$. The transformation matrix is $L=\left(\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right)$.

New covariant components: $A^{\prime }=LA{L}^T=\boxed{\left(\begin{array}{ccc}2& 3& 6\\ 4& 8& 15\\ 5& 11& 19\end{array}\right)}$. Raising indices with the metric $G=[g_{ik}]=[{\mathbf{e}}_i\cdot {\mathbf{e}}_k]$ gives the other three flavours.

One tensor, four flavours; the metric bridges them.