Contraction

Relevant parts to questions...

• Set two indices of a tensor equal and invoke the summation convention. Each contraction drops the rank by 2.
• In generalised coordinates, contract one upper with one lower index only.
• Outer product followed by contraction = inner product.
• Orthogonality $L_{im}{L}_{in}={\delta }_{mn}$ is what makes the result a tensor.

Contraction takes a rank-$n$ tensor (with $n\ge 2$) and reduces it to a rank-$(n-2)$ tensor by summing over a pair of its indices. Setting two indices equal and invoking the summation convention is the whole operation.

For a rank-3 covariant tensor $P_{ik\ell }$, there are three distinct ways to contract:

• $P_{ii\ell }=P_{11\ell }+P_{22\ell }+P_{33\ell }$ (contract first two),
• $P_{ikk}=P_{i11}+P_{i22}+P_{i33}$ (contract last two),
• $P_{iki}=P_{1k1}+P_{2k2}+P_{3k3}$ (contract first with third).

Each is a rank-1 tensor (a vector).

Why Contraction Preserves Tensor Character

In Cartesian coordinates, $P_{ik\ell }^{\prime }=L_{im}{L}_{kn}{L}_{\ell r}{P}_{mnr}$. Contracting the first two indices by setting $i=k$:

$$P_{ii\ell }^{\prime }=L_{im}{L}_{in}{L}_{\ell r}{P}_{mnr}={\delta }_{mn}{L}_{\ell r}{P}_{mnr}=L_{\ell r}{P}_{mmr}$$

which is exactly the transformation law of a rank-1 tensor. The orthogonality relation $L_{im}{L}_{in}={\delta }_{mn}$ collapses the two $L$-factors into a Kronecker Delta, dropping the rank by 2.

Generalised Coordinates: Contract Upper with Lower

Only valid with mismatched index positions

In curvilinear coordinates, contraction only produces a tensor when one index is covariant and the other contravariant. Contracting two upper (or two lower) indices does not give a tensor, because $L_n^{k^{\prime }}{L}_r^{k^{\prime }}\ne {\delta }_{nr}$ in general.

So $A_i^{\cdot i\ell }$ is fine (one up, one down), but $A_i^{\cdot k\ell }$ contracted over $k$ and $\ell$ fails.

Repeated Contraction

Since each contraction drops the rank by 2:

• A rank-$n$ tensor can be contracted $\lfloor n/2\rfloor$ times.
• If $n$ is even, repeated contraction eventually gives a scalar.
• If $n$ is odd, it eventually gives a vector.

The Inner Product

The inner product of two tensors is an outer product followed by a contraction:

$$A_{ik}{B}^k{\lambda }_{ik\ell m}{B}^{\ell m}{A}^i{B}_i$$

All are inner products. They combine two tensors and immediately reduce the rank by contracting matching (upper-lower) index pairs.

Properties

• Rank drop::one contraction removes 2 from the rank; two contractions remove 4; etc.
• Preserves tensor character when indices are at opposite levels (upper ↔ lower).
• Familiar identities from the Kronecker Delta and Alternating Tensor::
• ${\delta }_{ij}{ϵ}_{ijk}=0$ (contract alternating against symmetric)
• ${ϵ}_{ijk}{ϵ}_{i\ell m}={\delta }_{j\ell }{\delta }_{km}-{\delta }_{jm}{\delta }_{k\ell }$
• ${ϵ}_{ijk}{ϵ}_{ijm}=2{\delta }_{km}$
• ${ϵ}_{ijk}{ϵ}_{ijk}=6$.

Applications

1. Reducing rank to produce a simpler tensor - e.g. the trace of a matrix $A_{ii}$ is the full contraction of a rank-2 tensor.
2. Forming scalars and invariants from higher-rank objects - e.g. $T_{\cdot i}^i$ is a coordinate-invariant trace.
3. Computing inner products cleanly in any coordinate system by pairing upper with lower indices.

Trace as a contraction

Contract the rank-2 tensor $[A_{ij}]=\left(\begin{array}{cc}3& 1\\ 4& 2\end{array}\right)$: $A_{ii}=A_{11}+A_{22}=3+2=\boxed{5}$. This is a scalar - the trace.

Set two indices equal; rank drops by 2; upper-with-lower in curvilinear coords.