8.14.3 Tensor Index Range Coordinate Domain

Tensor Index Range Coordinate Domain is the specific set of coordinate labels — determined by the choice of coordinate system on a manifold or space — over which a tensor index is understood to range, distinguishing this coordinate-driven notion of range from a purely numerical or dimension-driven one. Where dimension dependence fixes how many values an index takes, the coordinate domain fixes which labeled coordinate directions those values correspond to, tying the abstract integer range of an index directly to the named coordinate axes of a particular chart or coordinate patch.

Coordinates as the Source of the Domain

Indexing the Coordinate Functions

A coordinate system on an $n$-dimensional space consists of $n$ coordinate functions, typically written $x^{1}, x^{2}, \dots, x^{n}$, and the index attached to each coordinate function is precisely what an index range in tensor notation is tracking. The coordinate domain of an index $i$ is the set of labels

identifying which of the $n$ coordinate functions $x^{i}$ a given component refers to.

Domain as a Property of the Chart, Not the Manifold Alone

Because a manifold can be covered by multiple coordinate charts, each with its own set of coordinate functions, the coordinate domain of an index is a property of the specific chart in use, not an intrinsic property of the underlying manifold. Switching from one coordinate chart to another — for instance, from Cartesian to spherical coordinates on the same three-dimensional space — leaves the dimension $n$ unchanged but replaces the coordinate labels the index domain refers to.

Coordinate Domain in Named Coordinate Systems

Cartesian Coordinates

In an $n$-dimensional Cartesian coordinate system, the coordinate domain of an index is the ordered list $x^{1}, x^{2}, \dots, x^{n}$, typically corresponding to directions such as $x, y, z$ in three dimensions. Every tensor component $A^{i}$ or $A_{i}$ is understood to refer to one of these specific Cartesian directions, according to the value assigned to $i$.

Curvilinear and Spherical Coordinates

In curvilinear systems such as spherical or cylindrical coordinates, the coordinate domain still consists of $n$ labels, but those labels correspond to different geometric quantities — radius, polar angle, azimuthal angle, for instance, in the three-dimensional spherical case. The numerical range ${1, 2, 3}$ is unchanged from the Cartesian case, but the coordinate domain now identifies $x^{1} = r$, $x^{2} = \theta$, $x^{3} = \phi$ rather than three linear directions.

Spacetime Coordinate Domains

In relativistic contexts, the coordinate domain of a Greek index conventionally spans ${0, 1, 2, 3}$, with $x^{0}$ identified as a time coordinate and $x^{1}, x^{2}, x^{3}$ identified as spatial coordinates. This labeling convention is itself part of the coordinate domain, since it fixes not just how many values an index takes but which physical coordinate each value names.

Restriction of the Coordinate Domain

Sub-Domains for Coordinate Subsets

When only a subset of the full coordinate system is relevant to a particular expression, the coordinate domain of the governing index is restricted accordingly. Isolating the spatial part of a four-dimensional spacetime index, for example, restricts the coordinate domain from ${0,1,2,3}$ to ${1,2,3}$, and this restriction must be indicated explicitly, since the full coordinate domain is otherwise assumed by default.

Domain Boundaries in Bounded Coordinate Systems

Some coordinate systems have inherently restricted coordinate domains due to the geometry they describe — an angular coordinate such as $\theta$ in spherical coordinates has a bounded domain distinct from the unbounded domain of a Cartesian coordinate, even though both are indexed by an integer label drawn from the same abstract range ${1, \dots, n}$. The coordinate domain in this sense carries information about the coordinate's own admissible values, layered on top of the purely combinatorial index range.

Coordinate Domain and Component Interpretation

Correct Association of Components to Directions

The coordinate domain is what allows a bare tensor component such as $A^{2}$ to be interpreted concretely — as the polar-angle component in spherical coordinates, or the $y$-component in Cartesian coordinates, depending entirely on which coordinate domain governs the index in that context. Without a declared coordinate domain, the index range alone supplies only an abstract slot number, not a geometric meaning.

Transformation Between Coordinate Domains

Passing from one coordinate system to another transforms not only the numerical values of tensor components but also reassigns the coordinate domain each index label refers to. The Jacobian matrices used in tensor transformation laws are precisely the objects that translate component values from one coordinate domain to another, while the underlying index range ${1, \dots, n}$ remains fixed throughout the transformation.

Role Within Index Range Notation

The coordinate domain is the concrete, geometrically meaningful layer that sits beneath the purely numerical range notation used in abstract tensor algebra. While an index range specifies how many values an index may take, the coordinate domain specifies what those values actually name within a chosen coordinate system, providing the interpretive link between symbolic tensor index notation and the specific geometric coordinates of the space in which a computation is being carried out.