1.2.4 Tensorial Relation Definition

Tensorial Relation Definition is the characterization of a tensorial relation as an equation or correspondence between tensorial quantities that remains valid in every admissible coordinate system, because every term appearing in it transforms consistently under a change of basis. It specifies the condition an expression must satisfy to count as a genuinely coordinate-independent statement about tensors, rather than a coincidence that happens to hold only in one particular frame of reference.

What Makes a Relation Tensorial

A relation between mathematical or physical quantities is tensorial when it is built entirely from tensors of definite types, combined using operations — addition, scalar multiplication, tensor product, and contraction — that are themselves compatible with the tensor transformation law. Because each such operation preserves tensorial character, an equation constructed exclusively from tensors and these operations will, if it holds in one basis, automatically hold in every other basis reachable by a linear change of coordinates.

This property follows directly from how tensors transform: since both sides of a tensorial equation transform by the same rule when the basis changes, the equality between them is preserved. If, instead, an equation held between quantities that transformed differently — for instance, one side tensorial and the other not — the equation could hold in one coordinate system purely by coincidence and fail in another, disqualifying it as a tensorial relation.

Formal Criterion

To verify that a proposed relation is tensorial, it suffices to check two things: that every quantity appearing in the relation is itself a tensor of some definite type, and that the operations combining these quantities are among those known to preserve tensorial character. If both conditions are met, the relation is guaranteed to be form-invariant under any admissible change of basis, meaning that the same relation, written with primed indices and the transformed components, follows automatically from the original relation together with the transformation law.

The expression above relates a force covector to a stress tensor contracted with a normal vector. Because every quantity involved — the force, the stress tensor, and the normal vector — is tensorial, and contraction preserves tensorial character, this relation holds in any coordinate system once verified in one.

Tensorial Relations Versus Coordinate-Dependent Statements

Not every true statement expressed using indexed quantities is a tensorial relation. A statement that holds only because of a special property of one particular coordinate system — for example, an equation that is valid only in coordinates where the basis vectors happen to be orthonormal — is not tensorial, since it will generally fail once expressed in a different, non-orthonormal basis. Distinguishing genuinely tensorial relations from such coordinate-dependent statements is essential in any discipline that uses tensor formalism, since only tensorial relations are guaranteed to express a fact about the underlying objects themselves, rather than an artifact of a particular descriptive choice.

Significance in Physics and Geometry

The requirement that physical laws be expressible as tensorial relations underlies the principle of general covariance central to modern physics: a law of nature, to be considered fundamental rather than an artifact of a chosen coordinate system, should be capable of being written as a relation among tensors. Einstein's field equations of general relativity, the stress-energy relations of continuum mechanics, and the source equations of electromagnetism are all formulated as tensorial relations precisely so that they hold in every coordinate system, reflecting facts about physical reality rather than facts about a particular, arbitrarily chosen frame of description.

This is also why establishing that a given physical law can be recast as a tensorial relation is treated as a significant achievement: it demonstrates that the law reflects an intrinsic, geometric or algebraic fact rather than depending on the incidental features of the coordinates used to express it.