1.11.4 Basis Convention

Basis Convention is the set of agreed choices governing which type of basis, orthonormal, general, coordinate-induced, or otherwise, is assumed by default when tensor components are written down, together with the associated notational and computational simplifications that each choice of basis type licenses or forbids. Because the numerical components of a tensor depend entirely on the basis relative to which they are expressed, and because certain algebraic simplifications, such as identifying upper and lower indices, are valid only for specific basis types, fixing a basis convention is a prerequisite for interpreting any indexed tensor expression correctly.

Basis convention is distinct from, though closely related to, index placement and summation convention: it does not concern how indices are written or summed, but rather what kind of geometric frame those indices are understood to be measured against, a choice that determines which relationships among components hold automatically and which require explicit extra structure, such as a metric, to establish.

Orthonormal Versus General Bases

The Orthonormal Simplification

An orthonormal basis is one in which basis vectors are mutually perpendicular and each has unit length, which has the consequence that the metric tensor's components equal the Kronecker delta, g_ij = δ_ij. Under this convention, raising or lowering an index leaves the numerical value of each component unchanged, so the distinction between upper and lower indices becomes numerically inert, and many treatments that adopt an orthonormal basis convention by default choose to write all indices as subscripts, since no information is lost by doing so.

The General Basis Requirement

A general, non-orthonormal basis has no such guarantee: its metric components are not simply the identity, and raising or lowering an index genuinely changes the numerical values involved. Under this convention, the upper-lower distinction must be tracked carefully throughout every computation, since conflating the two, valid under the orthonormal convention, produces incorrect results whenever the basis is not orthonormal.

Coordinate Versus Non-Coordinate Bases

Coordinate-Induced Bases

A coordinate basis, or holonomic basis, is one obtained directly as the set of partial derivative vectors ∂/∂x^i associated with a chosen coordinate system, together with its dual basis of coordinate differentials dx^i. This convention ties the basis vectors directly to a specific labeling of points by coordinate values, and it is the default convention in most introductory treatments, since it allows the tensor transformation law to be derived directly from the chain rule applied to the coordinate functions.

Non-Coordinate (Anholonomic) Bases

A non-coordinate basis, sometimes chosen for convenience such as an orthonormal frame that is not aligned with any single coordinate system's partial derivative vectors, is not obtained from any coordinate labeling and generally does not satisfy the same simple transformation relationships as a coordinate basis. Adopting this convention requires additional bookkeeping, commonly through structure coefficients that record how such basis vectors fail to commute, bookkeeping that is unnecessary under the coordinate-basis convention.

Basis Convention and the Physical Interpretation of Components

Physical Components in Curvilinear Coordinates

In curvilinear coordinate systems, such as spherical or cylindrical coordinates, the coordinate basis vectors are generally not unit vectors, so the raw coordinate-basis components of a tensor do not directly correspond to physically measured quantities along the natural physical directions. Some fields adopt the convention of instead reporting "physical components," obtained by rescaling each coordinate-basis component by the corresponding basis vector's length, specifically to restore a direct correspondence with measured physical quantities.

Choosing Between Coordinate and Physical Component Conventions

Whether to report coordinate-basis components or physical components is itself a convention decision, generally settled by whether the work at hand prioritizes ease of transformation and differentiation, favoring coordinate-basis components, or direct physical interpretability of the numbers reported, favoring physical components; engineering and applied physics contexts more often adopt the latter, while differential-geometric derivations more often adopt the former.

Default Basis Assumptions by Field

When Orthonormality Is Assumed Without Statement

Some treatments, particularly those originating in engineering or elementary physics, assume an orthonormal Cartesian basis so consistently that the assumption is left unstated, which can cause confusion when a formula from such a source is applied in a context using a general or curvilinear basis, since operations that were numerically harmless under the unstated orthonormal assumption, such as freely moving indices between upper and lower position, are no longer valid.

When General Bases Are the Explicit Default

Other treatments, particularly in differential geometry and general relativity, explicitly default to a general basis convention from the outset, precisely because the phenomena under study, curved coordinate systems, non-Euclidean geometry, require the upper-lower distinction to remain meaningful throughout.

Why Basis Convention Must Be Fixed Explicitly

Silent Assumptions Propagate Errors

Because the algebraic simplifications available under an orthonormal basis convention are not available under a general basis convention, an expression correctly simplified under one convention will be silently wrong if reused under the other without adjustment; this is one of the more common sources of error when combining formulas drawn from sources that adopt differing default basis conventions.

Stating the Convention as a First Step

Just as index placement and summation convention must be identified before a tensor expression can be interpreted correctly, the basis convention in force, orthonormal or general, coordinate or non-coordinate, must likewise be identified first, since it determines which relationships among a tensor's components are guaranteed to hold and which must be independently established through the metric or structure coefficients.