8.19 Tensor Index Contraction Pattern

Tensor Index Contraction Pattern is the recurring notational structure in which one upper (contravariant) index and one lower (covariant) index of a tensor expression are written with the same letter and are understood, under the Einstein summation convention, to be summed over the full range of that index, thereby producing a new tensor of rank two less than the original uncontracted expression. The pattern is the mechanism by which index notation encodes operations such as matrix trace, inner products, matrix-vector multiplication, and the divergence of a tensor field, all as special cases of a single repeated-index summation rule.

The Defining Rule

Repeated Indices Signal Summation

In the pattern, whenever an index letter appears exactly twice in a single term of an expression — once as a superscript and once as a subscript — summation over that index is implied without an explicit summation sign. For a tensor T with one upper and one lower index, the contraction is written

where n is the dimension of the underlying vector space. The index i on the left-hand side is called a dummy index: it does not appear in the final result and its letter may be renamed freely without changing the meaning of the expression.

Free Indices Versus Dummy Indices

Any index that appears only once in a term is a free index, and it survives into the result, ranging independently over all its values. The contraction pattern is precisely the rule that distinguishes these two roles: a repeated upper-lower pair is consumed by summation, while single, unpaired indices remain as the labels of the output tensor's components. An expression such as A^i_j B^j_k has j as a dummy (contracted) index and i, k as free indices, producing a rank-2 result indexed by i and k.

Rank Reduction Mechanics

Why Contraction Lowers Rank by Two

A tensor of type (p, q) has p upper indices and q lower indices, for a total rank p + q. Contracting one upper index against one lower index removes one instance of each, so the resulting object has type (p - 1, q - 1) and rank p + q - 2. This is a structural, not merely notational, fact: contraction is a linear map

from the space of (p, q) tensors to the space of (p-1, q-1) tensors, defined by evaluating the tensor on a basis covector and the corresponding dual basis vector for the chosen slots and summing.

Slot Selection

When a tensor has more than one upper index and more than one lower index, the contraction pattern must also specify which upper slot is paired with which lower slot, since contracting different slot pairs generally produces different results. A tensor R^i_{\ jkl} admits a contraction on its single upper index against any of its three lower indices, and these three contractions — R^i_{\ ijl}, R^i_{\ jil}, R^i_{\ jli} — are in general distinct tensors, as is the case for the contractions of the Riemann curvature tensor that produce the Ricci tensor.

Contraction Requiring the Metric

Same-Variance Index Pairs

The pattern as stated applies directly only to an upper-lower pair. When both indices to be paired are upper (or both lower), the metric tensor g_{ij} or its inverse g^{ij} must first be used to lower or raise one of them before contraction can proceed:

This composite operation — lowering an index with the metric and then contracting — is how inner products between two contravariant vectors are formed in index notation, and it explains why the metric tensor is indispensable for defining lengths and angles once tensor components are expressed in an arbitrary basis.

Consistency With the Kronecker Delta

The mixed Kronecker delta δ^i_j acts as the identity for contraction: contracting it against any tensor on one of its indices simply relabels that index, δ^i_j T^j = T^i. This property is what allows the metric and its inverse to satisfy g^{ik} g_{kj} = δ^i_j, tying the raising and lowering operations back to the basic contraction pattern.

Canonical Examples of the Pattern

Matrix Trace

For a (1,1) tensor A^i_j representing a linear map, contracting its single upper index with its single lower index, A^i_i, reproduces the trace of the corresponding matrix, a basis-independent scalar.

Matrix Multiplication and Vector Action

The product of two linear maps A^i_j B^j_k contracts the shared index j, reproducing ordinary matrix multiplication; contracting a (1,1) tensor against a vector, A^i_j v^j, reproduces the action of a matrix on a vector.

Divergence of a Vector Field

In a coordinate basis, the divergence of a vector field V^i is the contraction of the covariant derivative's upper and lower indices, ∇_i V^i, illustrating that the pattern extends from purely algebraic tensors to differential objects such as the covariant derivative ∇_i V^j, a (1,1) tensor at each point.

Diagram of an Index Pairing

Coordinate Independence of the Result

Transformation of the Contracted Object

Under a change of basis, the components of a (p, q) tensor transform with p factors of the Jacobian and q factors of its inverse. When an upper and a lower index are contracted, the corresponding Jacobian factor and inverse-Jacobian factor combine into an identity, since

which is exactly why the contraction pattern is only valid, and only produces a genuine tensor, when it pairs one upper with one lower index: the transformation factors cancel precisely in that combination and in no other.

Consequence for Physical and Geometric Invariants

Because the Jacobian factors cancel, the contracted object transforms as a tensor of the reduced rank, and any fully contracted expression, such as T^i_i, transforms as a scalar — a single number that is the same in every coordinate system. This is the underlying reason contraction is used throughout physics and geometry to extract coordinate-independent invariants, such as the trace of the stress-energy tensor or the scalar curvature obtained by contracting the Ricci tensor with the inverse metric.