8.19.1 Tensor Contraction Index Pair

Tensor Contraction Index Pair is the specific selection of one upper (contravariant) index slot and one lower (covariant) index slot within a tensor expression that are designated to be summed against each other under the Einstein summation convention. While the general contraction pattern establishes that a repeated upper-lower index triggers summation, the index pair is the concrete choice — among possibly several available upper and lower slots — of exactly which two positions participate in a given contraction, a choice that must be made explicit whenever a tensor carries more than one index of each variance.

Identifying a Valid Pair

Variance Requirement

A valid contraction index pair always consists of one upper index and one lower index. Two upper indices or two lower indices cannot be paired directly; such a pairing is only possible after one member is converted in variance using the metric tensor, g_{ij} to lower an upper index or g^{ij} to raise a lower one. The index pair itself, once formed, is always a mixed pair by the time contraction is carried out.

showing the index j first converted from upper to lower by the metric, after which it forms a genuine mixed pair available for contraction against another tensor's upper j index.

Position Versus Label

The pair is defined by position within the index list, not merely by shared letter: a tensor R^i_{\ jkl} written with its four slots in a fixed order has its single upper slot available to pair with any one of its three lower slots, and the pair actually used must be specified, since R^i_{\ ijl}, R^i_{\ jil}, and R^i_{\ jli} designate three different index pairs and generally yield three different rank-2 tensors.

Slot Notation and Disambiguation

Explicit Slot Marking

When ambiguity is possible, the index pair is disambiguated by writing dots or spacing to preserve the original slot order, so that R^i_{\ jkl} distinguishes the upper index's original position from the three lower positions, and the chosen pair for contraction, for example indices one and two, is written by repeating the same letter only in those two slots while keeping the others distinct.

Abstract Index Notation

In abstract index notation, each slot carries a fixed abstract label independent of any coordinate system, and a contraction index pair is indicated by literally using the identical abstract label in the two slots to be paired, for instance T^{ab}_{\ \ bc} where the repeated label b marks the pair consisting of the second upper slot and the first lower slot, while a and c remain free.

Multiplicity of Possible Pairs

Counting Available Pairings

For a tensor of type (p, q), the number of distinct single contraction index pairs available is p · q, since any one of the p upper slots may be paired with any one of the q lower slots. Each such pairing is, in general, a distinct operation producing a distinct (p-1, q-1) tensor.

The Ricci Contraction as a Named Example

The Ricci tensor is obtained from the Riemann curvature tensor R^a_{\ bcd} by a specific, conventionally fixed index pair: the upper index a is contracted against the third lower index c (equivalently, after applying the symmetries of the Riemann tensor, against d up to a sign), namely R_{bd} = R^a_{\ bad}. Choosing a different available pair, such as a with b, gives zero identically because of the antisymmetry of the Riemann tensor in its first two lower indices, illustrating that the outcome of a contraction depends critically on which index pair is selected.

Diagram of Multiple Candidate Pairs

Consistency Conditions on the Pair

Well-Definedness Under Symmetry

If a tensor is symmetric in several of its lower indices, all index pairs formed by contracting the upper index against any one of those symmetric lower indices yield the same result, collapsing what would otherwise be several distinct pairs into a single effective contraction. Antisymmetry, conversely, can force certain pairs to vanish identically, as with the Riemann example above.

Preservation of Tensorial Character

Regardless of which valid upper-lower pair is chosen, the contraction of that pair always yields a genuine tensor of type (p-1, q-1), because the Jacobian factor associated with the contracted upper slot and the inverse-Jacobian factor associated with the contracted lower slot cancel under a change of coordinates. This cancellation depends only on the pair being one upper and one lower index, not on which particular slots among several were chosen, so every valid index pair produces a coordinate-independent operation, even though different pairs generally produce different tensors.