10.16.3 Tensor Passive Transformation Object Preservation

Tensor Passive Transformation Object Preservation ensures invariance under coordinate changes through algebraic structure maintenance in tensor algebra.

Tensor Passive Transformation Object Preservation is the principle that the abstract, multilinear tensor object itself is left entirely unaltered by a passive coordinate transformation, so that every valid change of basis produces only a re-expression of the tensor's already-existing components and never a distinct tensor, an invariance that underlies why tensor equations retain the same form in every coordinate system.

What Is Preserved

The Tensor as a Multilinear Map

A tensor of a given rank is fundamentally a multilinear map taking a fixed number of vectors and covectors as arguments and producing a scalar, and this map does not reference any particular coordinate system in its definition; passive transformation object preservation is the statement that this map, considered abstractly, is precisely what stays fixed while its numerical components, computed relative to a chosen basis, are recomputed:

T = \sum_{i}^{n} \sum_{j}^{n} T^{ij} e_{i} \otimes e_{j} = \sum_{k}^{n} \sum_{l}^{n} {\bar{T}}^{kl} {\bar{e}}_{k} \otimes {\bar{e}}_{l}

Preservation of Action on Arguments

Because the tensor is preserved as a map, feeding it the same set of vector and covector arguments, described consistently in either basis, must yield the same scalar output, regardless of which basis was used to store the tensor's own components or to describe the arguments themselves.

Consequences of Object Preservation

Invariance of Scalar Contractions

The clearest consequence of object preservation is that a fully contracted scalar, formed by pairing every upper index of a tensor with a lower index of another tensor or covector, evaluates to the same number in every coordinate system:

\sum_{i}^{n} V^{i} W_{i} = \sum_{j}^{n} {\bar{V}}^{j} {\bar{W}}_{j}

Invariance of Tensor Equations

If a tensor equation, such as one component tensor being equal to another, holds in one coordinate system, object preservation guarantees it holds in every other coordinate system reachable by a valid passive transformation, since transforming both sides of the equation by the identical Jacobian factors preserves the equality; this is the reason physical laws expressed in fully tensorial form are automatically valid in any coordinate system once verified in one.

Preservation Under Repeated Transformation

Composed Passive Transformations

Applying two passive transformations in succession, first from a source chart to an intermediate chart and then from the intermediate chart to a final target chart, preserves the same underlying tensor throughout, and the composed transformation of components agrees exactly with the single direct transformation between the source and final target charts, as guaranteed by the chain rule relating the intermediate Jacobians.

Round-Trip Recovery

Transforming a tensor's components from the source chart to a target chart and then back to the source chart, using the forward and inverse Jacobian in sequence, recovers the original components exactly, which is only possible because the tensor object itself was never altered at any intermediate step, only its coordinate description.

Diagram of the Preserved Object

One Object, Many Descriptions

What Is Not Preserved

Numerical Components Change

Object preservation says nothing about the individual numbers making up the tensor's components, which generally change, often substantially, from one coordinate system to another; only the abstract object and any fully contracted scalars built from it remain numerically or structurally invariant.

Coordinate-Dependent Quantities

Quantities that are not themselves tensors, such as an individual Christoffel symbol used in covariant differentiation, do not enjoy object preservation in the same sense: they are defined relative to a coordinate system and transform with an extra inhomogeneous term beyond the ordinary tensor transformation law, precisely because they are not the components of a genuine preserved tensor object.

Role in Justifying Coordinate-Free Notation

Basis for Abstract Index and Coordinate-Free Formulations

Object preservation is the conceptual justification for writing tensor equations in coordinate-free notation, referring to the tensor itself rather than to any particular set of components, since the preserved object is what such notation is meant to denote, with the passive transformation machinery available whenever an explicit numerical computation in some chosen basis is required.