9.2.1 Tensor Basis System Area

Tensor Basis System Area is the classification of the different kinds of basis systems used to express tensor components at each point of a space, distinguished not by which coordinate labels are attached to points but by how the basis vectors themselves are constructed — coordinate (holonomic) bases derived directly from partial derivatives of a chart, non-coordinate (anholonomic) bases chosen independently of any chart, and orthonormal frame (vielbein or tetrad) systems chosen to be orthonormal everywhere regardless of the underlying coordinates. It surveys the recurring families into which basis choices fall, complementing the classification of coordinate systems themselves by focusing on the basis vectors as objects in their own right.

Coordinate (Holonomic) Basis Systems

Basis Vectors Defined as Partial Derivative Operators

A coordinate basis system takes its basis vectors directly from the partial derivatives associated with some chart, {∂/∂xⁱ}; because these basis vectors arise from actual coordinate functions, they automatically commute with one another as differential operators, a property called holonomicity that is the defining feature of this basis system area.

Consequences of Holonomicity for Calculation

Because coordinate basis vectors commute, formulas involving derivatives of tensor components in a coordinate basis system take a comparatively simple form, with connection coefficients (Christoffel symbols) automatically symmetric in their lower indices; this simplification is one of the principal reasons coordinate bases are the default choice in most introductory tensor calculus.

Non-Coordinate (Anholonomic) Basis Systems

Basis Vectors Chosen Independently of Any Chart

A non-coordinate basis system consists of vector fields {ê_a} chosen for their own convenience — orthonormality, alignment with a physical structure, or simplicity of some other quantity — without being required to arise as partial derivatives of any coordinate system; such basis vectors generally do not commute, and their commutators define nonzero structure functions c^c_{ab} that appear as extra correction terms in formulas that would otherwise assume holonomicity.

Why Non-Holonomic Systems Are Chosen Despite the Extra Bookkeeping

Non-coordinate basis systems are adopted precisely when their advantage in some other respect — most commonly, achieving orthonormality everywhere even on a curved space where no single coordinate chart could produce an orthonormal coordinate basis throughout its whole domain — outweighs the added complexity of tracking nonzero structure functions in subsequent calculations.

Orthonormal Frame (Vielbein/Tetrad) Systems

Orthonormality Independent of the Metric's Coordinate Form

An orthonormal frame system fixes, at every point, a basis in which the metric takes the identity (or, in indefinite-signature settings, the standard flat) form exactly, regardless of how complicated the metric's components are in whatever coordinate system happens to be in use; the transformation between the coordinate basis and the orthonormal frame at each point is encoded in the vielbein (or tetrad, in four dimensions), a position-dependent invertible matrix relating the two basis systems.

A Distinct Layer of Indices

Because the orthonormal frame is generally non-coordinate, quantities expressed in it carry a separate set of indices (often denoted with a different alphabet, such as Latin letters for frame indices alongside Greek letters for coordinate indices) from ordinary coordinate-basis tensor components, and converting between the two requires contracting with the vielbein, treated as a mixed object carrying one coordinate index and one frame index.

Diagram Comparing the Three Basis System Areas

Moving Between Basis System Areas Within One Calculation

Converting Coordinate Components Into a Non-Coordinate Frame

A tensor's components computed in a coordinate basis system can always be converted into components in a chosen orthonormal or otherwise non-coordinate frame at the same point by contracting with the frame's defining transformation matrix (the vielbein or its inverse), a purely algebraic operation performed pointwise that does not require re-deriving the tensor field itself, only re-expressing its already-known components.

Selecting the Basis System Area to Match the Task at Hand

Because each basis system area offers a different trade-off — coordinate bases for simplicity of differentiation, orthonormal frames for uniform metric behavior, general non-coordinate bases for alignment with a specific physical or geometric structure — the choice of which system area to work in in a given part of a calculation is typically made locally, switching basis system areas as convenient in the same way that coordinate areas themselves are switched between as a calculation proceeds.