1.3.2 Tensorial Structure

Tensorial Structure organizes multi-dimensional data through tensor operations, enabling complex relationships in mathematics and physics.

Tensorial Structure is the property that certifies whether a collection of indexed quantities genuinely constitutes a tensor, requiring that the quantities transform under a change of basis according to the specific multiplicative rule dictated by their upper and lower indices, rather than being merely an array of numbers that happens to carry indices. An object is said to possess tensorial structure, or to be tensorial, precisely when it passes this transformation test; an indexed array that fails the test, no matter how naturally it arises in a calculation, is not a tensor and cannot be treated as one in operations that rely on basis independence.

The Tensoriality Criterion

Statement of the Test

An indexed quantity Q^{i_1...i_p}_{j_1...j_q}, computed in every admissible basis of V, has tensorial structure if and only if its values in any two bases related by a change-of-basis matrix A satisfy the standard transformation law.

{\tilde{Q}}_{l_{1} \dots l_{q}}^{k^{1} \dots k^{p}} = A_{i_{1}}^{k^{1}} \dots {(A^{- 1})}_{l_{q}}^{j^{q}} Q_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}}

for every choice of admissible bases and every corresponding change-of-basis matrix A. If this identity holds universally, Q has tensorial structure of type (p, q); if it fails for even a single change of basis, Q is not tensorial.

The Quotient Rule as a Practical Test

A convenient practical criterion for establishing tensorial structure is the quotient rule: if an indexed quantity Q, when contracted against an arbitrary tensor of appropriate type, always produces a result that is itself a tensor, then Q must itself have tensorial structure. This rule allows tensoriality to be verified by examining the outcome of contractions rather than working through the full transformation law directly.

Examples of Structures That Are Tensorial

Sums and Scalar Multiples of Tensors

If two indexed quantities both possess tensorial structure of the same type, their sum, formed by adding corresponding components, also possesses tensorial structure of that type, since the transformation law is linear and applies identically to both summands.

Tensor Products and Contractions

The tensor product of two tensorial quantities is itself tensorial, of the combined type, and the contraction of a tensorial quantity over a matched upper and lower index remains tensorial, of the reduced type, both facts following directly from substituting the individual transformation laws into the combined expression and observing that the change-of-basis factors for the contracted indices cancel.

Examples of Structures That Are Not Tensorial

The Christoffel Symbols

The Christoffel symbols, Γ^k_{ij}, which arise in the study of connections on a manifold and describe how basis vectors change from point to point, carry three indices but do not transform according to the tensorial rule; their transformation law includes an additional inhomogeneous term beyond the standard multiplicative factors, reflecting the fact that they encode information about the derivative of the basis itself rather than a purely algebraic multilinear map.

{\tilde{Γ}}_{i j}^{k} = (tensorial terms) + (extra inhomogeneous term)

The presence of this extra term is precisely what disqualifies the Christoffel symbols from tensorial structure, despite their indexed appearance.

Coordinate-Dependent Partial Derivatives of Tensor Components

The ordinary partial derivative of the components of a tensor with respect to coordinates does not generally produce a quantity with tensorial structure, because differentiating the transformation law introduces extra terms involving the second derivatives of the change-of-basis relation, which do not cancel unless the change of basis is restricted to a purely linear one.

Why the Distinction Matters

Basis-Dependent Statements Are Not Physically or Geometrically Meaningful

An equation built entirely from tensorial quantities, once verified to hold in one basis, automatically holds in every basis, because both sides transform by the identical rule. An equation involving a non-tensorial quantity carries no such guarantee: it may hold in one basis purely as an artifact of that basis and fail in another, making it unsuitable for expressing statements intended to hold independently of coordinate choice.

Restoring Tensorial Structure

Quantities that fail to be tensorial on their own, such as the Christoffel symbols, can often be combined with other non-tensorial quantities so that the inhomogeneous terms cancel, producing a genuinely tensorial result; the Riemann curvature tensor is constructed exactly this way, as a specific combination of Christoffel symbols and their derivatives chosen so that the inhomogeneous terms cancel completely.

Diagrammatic Summary

The diagram contrasts a genuinely tensorial quantity, whose transformation consists purely of the standard multiplicative factors, with a non-tensorial quantity, whose transformation law includes an additional term that breaks the pure multiplicative pattern required for tensorial structure.