9.16.2 Tensor Basis Dependent Component Indexing

Tensor Basis Dependent Component Indexing is the practice of labeling each component of a tensor with indices that refer to a specific, fixed basis, so that the index itself carries an implicit dependence on which basis is currently in force. It concerns how the labels attached to components relate to the particular basis vectors and dual basis covectors of the moment, rather than to any basis-independent numbering scheme.

The Nature of Basis Dependent Indices

Indices Refer to Positions in a Specific Basis

An index such as i in a component does not name an intrinsic property of the tensor; it names the position of a particular basis vector or dual basis covector within the currently chosen basis. The same index value, i equal to one for instance, refers to entirely different directions under different bases.

Indices Do Not Travel with the Tensor

Because indices are tied to the basis rather than to the tensor, the same numerical index used in two different bases does not identify the same directional content of the tensor; only the pairing of an index together with its associated basis carries a well-defined meaning.

Indexing Across a Basis Change

Reassigning the Same Index Labels

When a basis is changed, the same index letters are typically reused to label the components relative to the new basis, even though the basis vectors and dual basis covectors those indices now refer to are entirely different from before.

Distinguishing Old and New Indexing

To avoid ambiguity when both the old and new bases must be discussed together, a distinguishing mark, such as placing a bar over the symbol, is used to indicate that an index or component belongs to the indexing scheme of the new basis rather than the old one.

Consequences of Basis Dependent Indexing

Index Position Does Not Fix Numerical Value

Two components sharing the same index value under two different bases generally hold different numerical values, since basis dependent indexing means the index specifies a position relative to a basis rather than a fixed numerical identity.

Correct Interpretation Requires Basis Context

Reading a component's index without knowing which basis is in force leaves the meaning of that index incomplete, since the index alone does not specify which basis vector or dual basis covector it refers to.

Practical Handling of Basis Dependent Indices

Fixing Notation Within a Single Calculation

Within a single, self-contained calculation, it is standard practice to fix one basis and its associated indexing scheme at the outset and use it consistently throughout, so that index labels can be interpreted unambiguously for the duration of that calculation.

Explicit Reindexing When Switching Bases

When a calculation must switch between two different bases, basis dependent indexing requires that the transition be made explicit, typically through distinguishing notation and the transformation rule connecting the two indexing schemes, rather than silently reusing the same index labels to mean two different things.