8.3.1 Tensor Indexed Symbol Base

Tensor Indexed Symbol Base is the main letter or glyph to which a tensor's indices are attached — the part of the notation, such as the T in T^{i}_{j}, that identifies which tensor object is being referred to, as distinct from the indices themselves, which identify which slot or component of that object is meant. It is the anchor of an indexed expression: the indices modify and specify the base symbol, but it is the base symbol that carries the tensor's actual identity.

The Role of the Base Symbol

Identity Versus Slot Reference

In an expression like T^{i}_{j}, the letter T names the specific tensor under discussion, while i and j specify which of its several components, or which of its argument slots, is being referenced; changing the base symbol from T to a different letter refers to an entirely different tensor, whereas changing only the index letters, without altering their positions or pattern, refers to the same tensor differently labeled.

Base Symbols Persist Across Index Changes

Because indices may be renamed freely when dummy, or relabeled when the convention for free indices changes, the base symbol is the one part of the notation that must remain fixed throughout a calculation for the reader to know that the same underlying tensor is still under discussion; a shift in the base symbol partway through a derivation, without explicit justification, signals that a different tensor has been introduced.

Conventional Base Symbols for Standard Tensors

Frequently Reused Letters

Certain base letters are conventionally reserved, across a wide range of texts, for specific standard tensors: g for the metric tensor, R for the Riemann curvature tensor or Ricci tensor depending on the number of indices attached, T for the stress-energy tensor in physical contexts or for a generic tensor in purely mathematical ones, and δ for the Kronecker delta.

The Levi-Civita Symbol as a Special Case

The Levi-Civita symbol, conventionally denoted ε, is a base symbol reserved for the fully antisymmetric object used to express determinants and cross products in index notation; its base symbol is fixed by nearly universal convention, so that seeing ε_{ijk} reliably signals this specific totally antisymmetric object rather than requiring separate definition each time.

Diagram of Base Symbol and Index Roles

Modifying the Base Symbol Rather Than Its Indices

Decorations Applied to the Base Symbol

Marks such as a bar, tilde, or hat placed directly over the base symbol — as in T̄, T̃, or T̂ — indicate a related but distinct tensor derived from T, such as a rescaled version, a version defined on a different but related space, or a version expressed in an orthonormal frame; this differs from placing such a mark on an index, which instead signals that a particular index refers to a different coordinate system or frame while still referring to the same base tensor.

Font and Typographic Conventions for the Base Symbol

Some conventions further distinguish the base symbol typographically: an uppercase letter for a tensor field defined globally versus a lowercase letter for its value at a single point, or a distinct font weight to distinguish an operator-valued object from an ordinary numerical tensor; these typographic choices operate entirely on the base symbol and do not interact with the rules governing the indices attached to it.

Choosing a Base Symbol in New Derivations

Avoiding Collisions with Established Conventions

When introducing a new tensor in a derivation, selecting a base symbol not already reserved by strong convention for a different, well-known tensor — avoiding, for instance, using g for something other than a metric — helps prevent a reader familiar with the standard conventions from misidentifying the newly introduced object.

Base Symbols Are Independent of Index Symbol Choices

The considerations governing which letter to use as a base symbol are largely independent of those governing which letters to use for indices: a base symbol is chosen once per tensor and persists through a derivation, while index letters may be freely renamed as dummy or reassigned as free indices change from one equation to the next, without any implication for which base symbols are in use.

Relationship to the Rest of Symbol Notation

The Base Symbol as the Foundation

All of the other conventions governing tensor indexed symbol notation — reserving certain index letters for spatial or spacetime roles, decorating indices with primes or hats to mark different coordinate systems — are built on top of an already-identified base symbol; without a clearly established base symbol, none of the indices attached to it can be correctly interpreted, since there would be no tensor object for them to be slots of.

A Prerequisite for Reading Any Indexed Expression

Correctly reading a tensor expression begins with identifying its base symbol and confirming which tensor it refers to, only after which the attached indices can be meaningfully interpreted according to their vertical position, order, and any decorations they individually carry.