11.5.4 Tensor Covariant Object Coordinate Dependence
Tensors maintain invariance under coordinate changes, ensuring physical laws remain consistent across different frames.
Tensor Covariant Object Coordinate Dependence is the precise characterization of which aspects of a covariant object change when the coordinate system is changed and which aspects remain fixed, separating the coordinate-dependent numerical component array from the coordinate-independent underlying dual-space element that those components describe.
What Depends on the Coordinate System
The Numerical Component Values
The specific numbers making up a covariant object's component array are entirely coordinate dependent, changing according to the inverse-Jacobian-factor transformation rule whenever the coordinate system is changed, so no single component value can be quoted as meaningful without also specifying the coordinate system it belongs to.
The Associated Dual Basis
The specific dual basis one-forms used to expand a covariant object also depend on the coordinate system, since they are constructed directly from the coordinate functions of whichever coordinate system is in use, changing along with the coordinates themselves.
What Remains Independent of the Coordinate System
The Underlying Dual-Space Element
The covariant object itself, understood as a specific element of the dual vector space, does not depend on the coordinate system at all; it is the same linear functional regardless of which coordinates are used to describe it numerically.
Any Full Contraction Against a Contravariant Vector
Because a covariant object's measurement action on a fixed contravariant vector produces a scalar, and scalars are coordinate independent, the numerical result of pairing a covariant object with any specific vector remains fixed, even though the component arrays of both the covariant object and the vector individually depend on the coordinate system.
The Precise Nature of the Dependence
A Linear, Not Arbitrary, Dependence
The coordinate dependence of a covariant object's components is not an arbitrary distortion but a precisely linear relationship governed by the inverse Jacobian factor, meaning the dependence is fully predictable once the coordinate transformation relating the two systems is known.
Local Dependence at a Single Point
For a covariant tensor field defined throughout a region, the coordinate dependence of its components at any one point involves only the Jacobian factor evaluated at that same point, not at neighboring points, so the coordinate dependence discussed here is a pointwise phenomenon rather than one requiring information about the field's behavior nearby.
Consequences of Recognizing This Dependence
Avoiding Misinterpretation of Raw Component Values
Recognizing that component values alone are coordinate dependent prevents the common error of comparing or combining covariant components taken from calculations performed in two different, unstated coordinate systems, since such a comparison is meaningless without first transforming both sets of components into a common coordinate system.
Justifying Coordinate-Free Statements of Physical Law
Because the underlying dual-space element and any full contraction against a vector remain coordinate independent, physical laws expressed in terms of a full contraction of a covariant object, such as work computed from force and displacement, can be stated without reference to any particular coordinate system, relying on this coordinate dependence analysis to guarantee the statement's validity in every admissible coordinate system.