8.10.4 Tensor Einstein Contraction Expression

The Tensor Einstein Contraction Expression simplifies tensor operations by summing over indices, central to general relativity and tensor algebra applications.

Tensor Einstein Contraction Expression is the general class of tensor expressions that rely on the Einstein summation convention rule to represent an implicit contraction, characterized by containing one or more repeated indices, each satisfying the exact-count and opposite-variance requirements of the convention, and thereby denoting a specific combination of tensors reduced in rank through implicit summation.

Anatomy of a Contraction Expression

Combining Factors with Repeated Indices

A contraction expression consists of one or more tensor factors, written adjacent to one another, in which at least one index letter is shared between factors, or within a single factor, satisfying the convention's requirement of occurring exactly twice in opposite variance.

A_{i} B^{i}

Distinguishing the Contracted from the Free Portion

Any complete contraction expression can be decomposed into two conceptual parts: the contracted portion, consisting of the repeated indices that vanish through summation, and the free portion, consisting of whatever indices remain and determine the rank of the final result.

R_{k} = A_{i k} B^{i}

Here the contracted portion is the pairing of (i) across (A) and (B), while the free portion is the surviving index (k), which alone determines that the result (R) is rank one.

Varieties of Contraction Expressions

Single Contraction Between Two Factors

The simplest contraction expression involves exactly one repeated index linking exactly two tensor factors, corresponding to the most elementary form of tensor contraction, such as an inner product between a vector and a covector.

Multiple Simultaneous Contractions

A more elaborate contraction expression may involve several distinct repeated indices, each independently satisfying the convention, linking multiple factors together in a chain or a more complex network, as occurs when contracting a higher-rank tensor against several other tensors at once.

s = A_{i j} B^{i j}

Self-Contraction Within a Single Tensor

A contraction expression may also involve only a single tensor factor, when that factor itself carries both an upper and a lower slot bearing the same repeated letter, producing a trace as the simplest possible self-contained contraction expression.

t = T_{i}^{i}

Reading and Constructing Contraction Expressions

Verifying Well-Formedness

Before interpreting a contraction expression, every repeated index within it should be checked against the full set of convention requirements, exact occurrence count of two and opposite variance, to confirm that the expression is genuinely well formed rather than containing a notational error.

Predicting the Output from the Expression Alone

Given a well-formed contraction expression, the rank and variance of the resulting object can be predicted directly by identifying the surviving free indices, without needing to carry out the summation numerically, since the structural pattern of repeated and free indices alone determines this outcome.

Practical Illustration

Recognizing a tensor expression as an Einstein contraction expression, and correctly separating its contracted portion from its free portion, is the key analytical step that allows the rank, variance, and structural meaning of even a complex multi-factor tensor expression to be understood at a glance.