6.11.4 Tensor Zero Zero Coordinate Behavior

Tensor Zero Zero Coordinate Behavior is the precise description of how a type zero-zero tensor responds, or rather fails to respond at all in any nontrivial way, to a change of coordinate system, this behavior consisting entirely of the value at a given point remaining exactly the same number no matter which coordinate chart is used to describe that point. Unlike tensors of positive type, whose components are recomputed through Jacobian factors when coordinates change, a type zero-zero tensor's coordinate behavior is characterized by the complete absence of any recomputation: the same value is simply read off again in the new system.

Invariance Under Arbitrary Coordinate Changes

Behavior Under General, Not Just Linear, Transformations

The coordinate behavior of a type zero-zero tensor holds under any admissible change of coordinates, whether that change is a simple linear substitution or a general, possibly nonlinear, diffeomorphism relating one coordinate chart to another. Because the general transformation pattern assigns zero Jacobian factors of either kind to a type zero-zero tensor, no matter how complicated the Jacobian matrix of the coordinate change itself might be, the absence of any factor to apply means the value is unaffected regardless of the nature of the underlying transformation.

Behavior at a Point Versus Behavior as a Field

At any single fixed point, the coordinate behavior of a type zero-zero tensor is simply that its one value there is chart-independent. When the tensor varies from point to point as a scalar field, this same invariance holds independently at every point of the domain: the value assigned to a given point does not depend on which chart was used to name that point's coordinates, even though the numerical value generally does depend on which point is being considered.

Distinguishing Invariant Values From Coordinate-Dependent Ones

Coordinate Functions Themselves Are Not Type Zero-Zero

An individual coordinate function, the value of one particular coordinate at a given point, is not a type zero-zero tensor, since its numerical value changes under a change of coordinate chart by definition; a coordinate value in one chart generally differs from the coordinate value assigned to the same point in another chart. This distinguishes genuine scalar quantities, which are type zero-zero tensors, from the coordinate labels used merely to name points, which are not tensors of any type at all.

Only Full Contractions Reliably Produce This Behavior

An expression built from tensors of positive type generally exhibits type zero-zero coordinate behavior, complete chart independence, only once every index on every tensor involved has been fully contracted away, since any remaining uncontracted index would still carry its own Jacobian factor and thus its own dependence on the coordinate chart. Partial contractions or bare components of higher-type tensors do not display type zero-zero coordinate behavior even though they may be single numbers at a glance.

Composition Behavior Across Successive Coordinate Changes

Transitivity of the Trivial Transformation

Applying the type zero-zero coordinate behavior across two successive changes of chart, first from an original chart to an intermediate one and then from the intermediate chart to a final one, reproduces exactly the same value obtained by comparing the original chart directly to the final one. Since no factor is applied at any stage, this transitivity is automatic and requires no separate verification, unlike the corresponding transitivity property for tensors of positive type, which relies on the chain rule composition of their Jacobian factors.

Coordinate Behavior Along a Path

When a scalar field is evaluated along a curve or path that is itself described using changing coordinate charts along its length, the value of the scalar field at each point along the path remains determined solely by the point itself, never by which chart happened to be in use at that particular point of the path. This makes type zero-zero coordinate behavior the natural foundation for defining path integrals and other constructions that accumulate scalar values continuously across a domain covered by multiple overlapping charts.