8.19.4 Tensor Contraction Order Reduction

Tensor Contraction Order Reduction simplifies tensor expressions by reducing contraction order through index pairing and summation.

Tensor Contraction Order Reduction is the general fact that summing a matched upper-lower index pair out of a tensor lowers its order — the total count of its indices — by exactly two, one upper and one lower, converting a tensor of order p + q (with p upper and q lower indices) into a tensor of order (p-1) + (q-1). It is the structural consequence that gives the contraction pattern its purpose: contraction is not merely a bookkeeping notation for a sum, but a genuine operation that maps tensors of one order into tensors of a strictly smaller order, and repeated application drives arbitrarily high-order tensors down toward scalars.

The Reduction Step

From Order p+q to Order p+q-2

Given a tensor T of type (p, q), contracting one specified upper slot against one specified lower slot produces a tensor C(T) of type (p-1, q-1):

C : T_{q}^{p} \to T_{q - 1}^{p - 1}

Order here means p + q, the total number of indices regardless of variance, so the order strictly decreases by two with every single contraction performed, independent of the summed dimension or of which particular slots were chosen.

Order Reduction Is Linear

The contraction map C is linear: contracting a sum of tensors of the same type equals the sum of the contractions, and contracting a scalar multiple of a tensor equals that same scalar multiple of the contraction. Order reduction is therefore compatible with the vector space structure of the space of tensors of a given type, not merely a pointwise numerical operation on components.

Iterated Reduction to Lower Orders

Repeated Contraction

Order reduction can be applied repeatedly: contracting a tensor of type (p, q) down to type (p-1, q-1), then contracting the result down to (p-2, q-2), and so on, is possible as long as at least one upper and one lower index remain. After k successive contractions, a tensor originally of order p + q has order p + q - 2k.

Full Reduction to a Scalar

If a tensor has equal numbers of upper and lower indices, p = q, then p successive contractions — pairing up all the upper indices against all the lower indices — reduce it all the way to order zero, a single scalar invariant of the tensor. This is exactly how a fully contracted expression such as T^{ij}_{\ \ ij} is understood as an iterated order reduction: each contraction removes one upper and one lower index until none remain.

T_{i j}^{i j} \in T_{0}^{0} = F

where F denotes the base field of scalars, the order-zero tensor type.

What Order Reduction Does Not Change

Rank of the Underlying Vector Space Is Untouched

Order reduction changes only how many indices a tensor carries, not the dimension n of the vector space the indices range over. A contracted tensor built from an n-dimensional space still has each of its remaining free indices ranging over the same n values; only the count of indices has decreased.

Free Indices Retain Their Variance

Contraction removes exactly one upper and one lower index and leaves every other index, whether upper or lower, in its original slot and variance. Order reduction is therefore selective: it is not a uniform shrinking of the tensor but the specific removal of one matched pair, so a (3, 2) tensor contracted once becomes a (2, 1) tensor, never a (2, 2) or a (3, 1) tensor.

Diagram of Successive Order Reduction

Distinction From Other Order-Changing Operations

Contraction Versus Raising and Lowering

Raising an index with g^{ij} or lowering one with g_{ij} changes the variance of an index but leaves the total order unchanged, since one index is converted, not removed; this contrasts with contraction, which strictly decreases order by two. Order reduction refers only to the contraction operation, not to variance-changing operations, even though both use the metric tensor when non-mixed pairs are involved.

Contraction Versus the Tensor Product

The tensor product of a type (p, q) tensor and a type (r, s) tensor increases order to (p+r) + (q+s), the opposite direction from contraction. Many tensor computations alternate the two operations, first forming a tensor product to combine two tensors into one object of higher order, then contracting selected index pairs to reduce the order back down, which is precisely the index-notation description of operations like matrix multiplication, A^i_j B^j_k, viewed as a product followed by a contraction.