1.2.3 Tensorial Quantity Definition

Tensorial quantities are mathematical objects that transform under coordinate changes, crucial for expressing physical laws consistently.

Tensorial Quantity Definition is the characterization of a tensorial quantity as any physical or mathematical quantity that can be validly represented as a tensor of some definite type, meaning that its numerical description in any given basis obeys the tensor transformation law when the basis is changed. It distinguishes quantities that possess a genuine, basis-independent tensorial character from those that merely happen to be expressed using multiple numerical components without satisfying this transformation requirement.

From Abstract Tensors to Tensorial Quantities

The formal definition of a tensor characterizes it as an element of a tensor product space or as a multilinear map satisfying a universal property. A tensorial quantity is the applied counterpart of this abstract notion: it is a quantity — drawn from physics, engineering, or another applied discipline — that can be modeled as a tensor because its components, once a coordinate system or basis has been fixed, change from one coordinate system to another according to the same transformation law that governs the components of an abstract tensor.

This distinction matters because not every collection of numbers arranged with several indices deserves to be called tensorial. A table of numbers indexed by two subscripts might represent a rank-two tensor, or it might simply be a matrix of unrelated data with no coordinate-transformation behavior at all. Calling a quantity tensorial is therefore a substantive claim about how its description behaves under a change of reference frame, not merely a statement about how many indices are used to write it down.

The Defining Criterion

A quantity qualifies as tensorial of type (p, q) if, when its components are computed in two different bases related by a known linear transformation, the components in the new basis can be obtained from the components in the old basis by applying that transformation once for each contravariant index and its inverse once for each covariant index. This criterion is precise and checkable: given any proposed set of components and any change of basis, one can verify directly whether the transformation law holds.

T_{j'}^{i'} = A_{i}^{i'} B_{j'}^{j} T_{j}^{i}

The expression above states the general transformation law for a mixed tensor of type (1, 1), where the coefficients relate the components in the primed basis to the components in the original basis. A quantity is tensorial precisely when its components obey a law of this form.

Examples of Tensorial Quantities

Physical Quantities

Many quantities encountered in physics are tensorial by this criterion. A displacement or velocity is a rank-one contravariant tensorial quantity, transforming as a vector. The stress within a deformable body is a rank-two tensorial quantity, since the force per unit area across a surface depends on both the orientation of the surface and the direction of the force, and its numerical description transforms correctly when the coordinate axes are rotated. The curvature of spacetime in general relativity is described by a rank-four tensorial quantity, the Riemann curvature tensor, whose transformation behavior under a change of coordinates is essential to the theory's requirement that physical laws take the same form in every coordinate system.

Non-Tensorial Quantities

Not every multi-indexed physical quantity is tensorial. The Christoffel symbols used in differential geometry to describe how a coordinate basis itself changes from point to point fail to transform according to the tensor law, acquiring an additional inhomogeneous term under a change of coordinates; consequently, despite being written with indices in the same style as tensors, they are explicitly classified as non-tensorial quantities.

Why the Distinction Matters

Recognizing whether a quantity is genuinely tensorial has direct consequences for how it may be used in calculation and in physical reasoning. Equations built entirely from tensorial quantities, combined using tensor operations such as addition, tensor product, and contraction, automatically hold in every coordinate system once verified in one, because both sides of the equation transform consistently under a change of basis. This property, sometimes called covariance, is why physical laws are conventionally expressed in tensorial form: it guarantees that the law does not depend on an arbitrary choice of coordinates, but reflects a relationship that holds independently of how it is described.