1.6.5 Tensor Equation Invariance

Tensor Equation Invariance explores how tensor equations remain unchanged under coordinate transformations, preserving physical laws across different frames.

Tensor Equation Invariance is the property that an equation asserting the equality of two tensors of the same type, once verified in a single basis or coordinate system, automatically holds in every other basis or coordinate system reachable by an admissible transformation, without needing to be separately checked or reformulated in each one. It is distinct from the invariance of a single tensor object or the invariance of a transformation law; it concerns the logical status of a relation connecting several tensors, and it is the principle that allows a tensor equation established once to be treated as universally valid.

The Structure of a Tensor Equation

Equality Between Two Tensors of Matching Type

A tensor equation has the form of one tensor of type (p, q) set equal to another tensor of the same type (p, q), expressed in components as the equality of their component arrays in a common basis.

S_{j_{1} \dots}^{i^{1} \dots} = T_{j_{1} \dots}^{i^{1} \dots}

Matching Type Is Required

The equation is only meaningful, and only eligible for the invariance property, when both sides are tensors of the identical type; an expression equating a (1, 0) array to a (0, 1) array is not a tensor equation and carries no invariance guarantee at all.

Proof of the Invariance Property

Applying the Transformation Law to Both Sides

Because S and T are tensors of the same type, both of their component arrays obey the identical transformation law when the basis changes, using the same factors of the change-of-basis matrix and its inverse for corresponding indices.

S' = A \dots B \dots S and T' = A \dots B \dots T

The Equality Transfers Automatically

Since S = T in the original basis, applying the same linear transformation to both sides of the equality preserves it, so S′ = T′ in the new basis follows immediately, with no further computation or verification required.

S = T \Rightarrow A \dots B \dots S = A \dots B \dots T \Rightarrow S' = T'

Failure of Invariance for Non-Tensorial Equations

Equations Involving Non-Tensor Quantities

An equation relating quantities that do not individually obey the tensor transformation law, such as an equality holding among ordinary partial derivatives of components or among Christoffel symbols alone, need not be preserved under a change of basis, since the extra terms produced by the transformation of such quantities generally do not cancel between the two sides.

Verifying Only One Side Is Not Sufficient

Confirming that a proposed relation holds numerically in one particular, often conveniently chosen, coordinate system does not establish that it holds in general unless both sides of the relation are independently known to be tensors of the same type; otherwise the agreement may be an artifact specific to that coordinate system.

Practical Use of the Principle

Proving General Statements from a Single Case

Tensor equation invariance allows a general claim to be established by verifying it in whatever basis makes the computation simplest, often a highly symmetric or orthonormal basis, with the assurance that the resulting equation, being an equality between tensors of matching type, extends to every other admissible basis without additional work.

The Principle of General Covariance

In physical theories formulated with tensors, requiring the fundamental equations to be tensor equations is the technical content of the principle of general covariance: physical laws expressed this way hold in every coordinate system, since the equality between the tensors representing the two sides of the law is preserved automatically under any admissible change of coordinates.

Distinguishing Related Notions

Equation Invariance Presupposes Object Invariance

Tensor equation invariance relies on each individual tensor already possessing object invariance, existing as one fixed entity across bases; what equation invariance adds is that an equality connecting two such objects, once established, transfers along with them to every other basis.

Equation Invariance Versus Coordinate Transformation Invariance on a Manifold

Where coordinate transformation invariance concerns tensor fields whose Jacobian varies from point to point across a manifold, tensor equation invariance is the underlying algebraic fact, holding pointwise at every fixed location, that makes the field-theoretic version of the principle work.

Diagrammatic Summary

The diagram contrasts a genuine tensor equation, which transfers intact to any new basis because both sides share the same transformation law, with a relation among non-tensorial quantities, which carries no such guarantee and may hold in one basis while failing in another.