11.16.5 Tensor Coordinate Change Invariance Result

The Tensor Coordinate Change Invariance Result ensures mathematical properties remain consistent across different coordinate systems in tensor algebra.

Tensor Coordinate Change Invariance Result is the conclusion, following from the coordinated covariant and contravariant transformation rules, that certain quantities built from tensors, particularly full contractions between upper and lower indices, retain identical numerical values regardless of which coordinate system is used to compute them, establishing tensors as the natural language for expressing coordinate-independent physical and geometric statements.

Foundational Setting

The Motivation Behind Invariance

Coordinate systems are human-chosen labeling schemes with no independent physical significance. A meaningful physical or geometric quantity should not depend on which labeling scheme happens to be used, and the invariance result formalizes exactly this requirement for quantities expressed through tensors.

Recalling the Transformation Laws

The result depends on the fact that contravariant components transform with the Jacobian factor $J$ while covariant components transform with the inverse-direction Jacobian factor $K$ , where these two matrices satisfy:

\sum_{j} K_{i}^{j} J_{j}^{k} = δ_{i}^{k}

The Core Invariance Statement

Scalar Contraction of a Vector and Covector

For a vector with contravariant components $v^{i}$ and a covector with covariant components $ω_{i}$ , the invariance result states:

\sum_{i} {\tilde{ω}}_{i} {\tilde{v}}^{i} = \sum_{i} ω_{i} v^{i}

Proof Sketch

Substituting the transformation laws for each factor and using the identity above between $J$ and $K$ collapses the double sum back to the original untransformed contraction, demonstrating that the opposing transformation directions of covariant and contravariant indices exist precisely to produce this cancellation.

Generalization to Higher-Rank Tensors

Full Contraction of Mixed Tensors

The invariance result extends to any tensor expression in which every upper index is paired with a matching lower index through summation. For a mixed tensor $T_{j}^{i}$ , its trace is invariant:

\sum_{i} {\tilde{T}}_{i}^{i} = \sum_{i} T_{i}^{i}

Products of Multiple Tensors

More generally, any product of several tensors in which all indices are fully contracted in matched upper-lower pairs yields a scalar invariant, since each contraction independently produces the cancellation described above.

Visual Interpretation

The Cancellation Mechanism

Physical and Geometric Significance

Coordinate-Independent Laws

Because contracted tensor expressions are invariant, physical laws expressed as such contractions, such as the invariant length of a vector computed through a metric tensor, hold true in every admissible coordinate system without modification.

s^{2} = \sum_{i, j} g_{i j} v^{i} v^{j}

Distinguishing Invariant Quantities from Components

The invariance result applies to fully contracted scalar quantities, not to the individual components of a tensor, which do change value under coordinate transformation even though the tensor as a whole, and any complete contraction built from it, does not.

Summary of Key Traits

Defining Characteristics

Invariance arises from the mutually inverse relationship between contravariant and covariant transformation factors.
Fully contracted expressions pairing every upper index with a matching lower index remain unchanged under coordinate change.
The result generalizes to tensors of arbitrary rank and to products of multiple tensors.
Individual tensor components are not invariant; only complete contractions are.