11.7 Tensor Covariant Transformation Law

Tensor Covariant Transformation Law is the formal rule specifying how the components of a covariant tensor must change when the coordinate system is changed, stated as a contraction with the inverse Jacobian factor for each covariant index, serving as the defining criterion by which any candidate quantity is recognized as a genuine covariant tensor.

Statement of the Law

Single-Index Statement

For a covariant tensor of rank one, the law states that the component in the new coordinate system equals the inverse Jacobian factor contracted with the component in the old coordinate system.

General Rank Statement

For a purely covariant tensor of arbitrary rank, the law extends by attaching one inverse Jacobian factor to each covariant index, with all factors multiplied together and contracted with the original components under the summation convention.

Derivation From the Gradient of a Scalar

Chain Rule as the Motivating Case

The covariant transformation law is derived by applying the chain rule to the partial derivative of a scalar function with respect to the new coordinates, which produces exactly the inverse-Jacobian-factor structure that the general law then adopts for arbitrary covariant tensors.

Properties of the Law

Homogeneity

The law is homogeneous, meaning the transformed component depends linearly on the original component with no additive term, and this homogeneity is precisely what distinguishes a genuine covariant tensor from a superficially similar object, such as a connection coefficient, that fails to satisfy the law exactly.

Consistency Under Composition

Applying the law to pass from a first coordinate system to a second, and then from the second to a third, produces the same result as applying the law directly from the first system to the third, a consistency guaranteed by the chain rule acting on the composed Jacobian factors.

Compatibility With the Reciprocity Identity

The law is compatible with the reciprocity identity relating the inverse and direct Jacobian factors, ensuring that applying the law and then applying it again in reverse restores the original components exactly.

Role as a Verification Criterion

The Substitution Test

The covariant transformation law serves as the operative test for verifying that a candidate quantity is a genuine covariant tensor: substituting the candidate's definition into both sides of the law and confirming exact agreement is the standard method of certifying covariant tensorial status.

Distinguishing Genuine Tensors From Similar Objects

Any quantity that fails to satisfy this law exactly, picking up an extra term under a change of basis, is excluded from covariant tensor status regardless of how closely its index notation resembles that of a genuine covariant tensor.

Practical Application

Foundation for Coordinate-Independent Physical Statements

The covariant transformation law underlies the practice of expressing physical laws involving gradients, forces, and other naturally covariant quantities in a form guaranteed to hold in every admissible coordinate system, since any two coordinate descriptions related by this law describe the same underlying covariant object.