8.21.5 Tensor Summation to Contraction Translation

Tensor Summation to Contraction Translation is the reverse of implicit sum expansion: the process of taking an explicitly written summation, complete with a summation sign and stated range, and rewriting it in the compact Einstein-convention contraction form by dropping the summation sign whenever the summed term is a product in which the summed index appears exactly once as an upper index and once as a lower index. It is the compression step that turns verbose, explicit sum notation into terse tensor index notation, valid only when the underlying term already has the specific upper-lower repeated-index structure that the convention is designed to abbreviate.

The Basic Compression Rule

Dropping a Correctly Structured Summation Sign

Given an explicit sum

the summation sign and its range can be dropped entirely, leaving the compact contraction A^i B_i, precisely because the summed index i occurs exactly twice in the summand, once as a superscript and once as a subscript. This translation is a pure notational compression: the two expressions denote exactly the same number, and nothing about the underlying computation changes, only how much of it is written explicitly.

The Structural Precondition That Must Be Checked First

Not every explicit sum qualifies for this compression. A sum such as Σᵢ AᵢBᵢ, in which both A and B carry a subscript rather than one upper and one lower index, cannot be compressed into implicit-summation form as written, because the Einstein convention specifically requires opposite variance on the repeated letter; translating such a sum correctly requires first identifying, from the transformation behavior of A and B, whether one of the two should in fact be written with an upper index (possibly after raising it with the inverse metric), before the implicit form can be applied.

Translating Sums Over Several Indices

Multiple Explicit Sums Collapse Independently

An explicit double (or higher) sum, with one summation sign for each of several indices, translates term by term: each summed index that individually satisfies the upper-lower repetition condition within the product has its own summation sign dropped independently of the others.

with both summation signs removed at once, since both i and j individually satisfy the required opposite-variance repetition.

Partial Translation When Only Some Indices Qualify

If a multi-index explicit sum has some summed indices satisfying the repetition condition and others not, only the qualifying summation signs may be dropped, and the remainder must be left as explicit sums (or resolved by first correcting the variance of the offending index, typically via the metric) before full compression into pure Einstein notation is valid; a partially compressed expression that mixes an explicit summation sign with implicit summation elsewhere is a legitimate intermediate but not the final translated form.

Handling Sums That Are Not Contractions

Sums Over a Single Variance Are Not Contractions

An explicit sum in which the repeated index appears twice with the same variance, such as Σᵢ AᵢBᵢ with both subscripted, is not a contraction in the tensorial sense and cannot be validly compressed using the Einstein convention as it stands; if A and B are genuinely tensor components of the same variance, translating this sum into proper tensor index notation requires introducing the metric explicitly, Σᵢ AᵢBᵢ = g^{ij}A_iB_j, before the implicit-summation compression can be legitimately applied to the resulting mixed pair.

Sums That Are Not Products of Tensor Components at All

Some explicit sums encountered in a broader calculation are not sums of tensor-component products at all — for instance, a sum over a discrete label indexing a list of separate tensors rather than over a tensor index proper. Summation-to-contraction translation does not apply to such sums, and attempting to compress them using tensor index notation would misrepresent them as contractions when no contraction is actually taking place.

Diagram of the Translation

Why the Translation Preserves Meaning

The Underlying Numerical Sum Is Unchanged

Whether written with an explicit summation sign or in compact contracted form, the two notations instruct exactly the same arithmetic: add together, for every value of the summed index in its stated range, the product of the corresponding components. Summation-to-contraction translation changes nothing about which numbers are added or how they are combined; it only removes a symbol that the reader is expected to supply mentally once the variance pattern of the repeated index has been confirmed.

The Value of Performing the Translation

Beyond mere brevity, converting an explicit sum into contraction notation exposes the tensorial character of a formula — its transformation behavior under a change of basis — in a way that the explicit summation sign does not, since only the contracted form directly shows which index is contravariant and which is covariant. This is the primary reason the translation is worth performing even when it saves relatively little space: it makes the coordinate-independence, or lack of it, of the underlying expression visible at a glance.