10.19.4 Tensor Non Tensorial Object Failure

Tensor Non Tensorial Object Failure is the deeper conceptual breakdown underlying every non-tensorial transformation pattern, namely the failure of a collection of coordinate components to correspond to any single, coordinate-independent geometric object at all, so that there is no fixed multilinear map, analogous to a genuine tensor, that the various coordinate descriptions could be said to jointly represent.

What Object Preservation Would Require

The Missing Invariant Entity

For a genuine tensor, passive transformation object preservation guarantees that a single abstract multilinear map exists, with its components in any basis obtained by evaluating that one fixed map against the basis vectors of that coordinate system; a non-tensorial quantity has no such underlying map, since no single abstract object exists whose evaluation against different bases would reproduce the actual, extra-term-carrying transformation law observed:

For a non-tensorial quantity, no analogous single sum exists that is simultaneously consistent with the components in every coordinate system, since the extra transformation term prevents such a sum from being well defined independent of the coordinate system chosen.

Consequence for the Notion of Sameness

Without such an underlying invariant object, the very question "is this the same quantity described in two coordinate systems, or two different quantities" loses its ordinary tensorial answer, since the mechanism that would normally guarantee sameness, namely a single fixed multilinear object being merely re-expressed, is absent for a non-tensorial quantity.

Manifestations of the Object Failure

No Coordinate-Free Formula Exists

A direct symptom of object failure is the absence of any coordinate-free, geometric definition of the quantity that reproduces its behavior in every coordinate system; the Christoffel symbols, for instance, are defined through an explicit coordinate-dependent formula involving derivatives of the metric components, with no way to state their definition purely in terms of vectors and covectors without reference to a specific coordinate system.

Arbitrary Value Achievable at a Point

A further manifestation, closely tied to the object failure, is that the quantity's value at a single point can be set arbitrarily, including to zero, by an appropriate local coordinate choice, which would be impossible for a genuine invariant object, since an invariant object's vanishing or non-vanishing at a point is a fact about the object itself, not about the coordinates used to describe it.

Diagram of the Missing Object

Tensor Case Versus Non-Tensor Case

Recovering an Object From a Failed Quantity

Extracting a Genuine Tensor

Although the quantity itself fails to correspond to a single object, certain operations performed on it can produce a genuine object where object failure no longer applies, such as forming the difference of two connections sharing the same coordinate-dependent quantity, or building the Riemann curvature tensor from derivatives and products of the connection coefficients, both of which do correspond to well-defined invariant objects despite being built from ingredients that individually suffer object failure.

Local Approximate Recovery

At a single point, after choosing coordinates that make the non-tensorial quantity vanish there, the quantity can be locally treated as if it carried no information at that point, but this is a coordinate artifact rather than a genuine recovery of an invariant object, since the same quantity generally reappears as non-zero at every neighboring point in the same coordinate system.

Why Object Failure Does Not Disqualify Practical Use

Utility Without Invariance

Despite lacking a corresponding invariant object, a quantity exhibiting object failure remains useful and well defined within a fixed coordinate system, serving a specific computational role, such as correcting an ordinary derivative to produce a covariant one, provided its non-invariant character is properly tracked and never mistaken for the behavior of a genuine tensor when combining it with other quantities in an equation.