9.6.3 Tensor Noncoordinate Basis Component Assignment

Tensor Noncoordinate Basis Component Assignment is the act of computing, for a given tensor, the numerical entries of its component array by pairing that tensor with the frame and coframe members of a noncoordinate basis, following the same slot-by-slot pairing procedure used for any coordinate basis, but producing components indexed relative to a frame that carries no coordinate origin; it converts an abstract tensor into a concrete array of numbers exactly as component assignment does for a coordinate basis, while relying on a frame whose members do not commute.

Carrying Out the Assignment

Pairing the Tensor With Frame and Coframe Members

For a noncoordinate basis {e_a} with coframe {θ^a}, each entry of a tensor's component array is obtained by substituting the appropriate frame or coframe members into the tensor's argument slots, in their fixed order, and evaluating.

The Procedure Does Not Reference the Bracket

Component assignment for a single tensor at a single point is a purely algebraic pairing and makes no reference whatsoever to whether the frame members commute; the noncoordinate character of the basis affects later operations on the resulting components, not the assignment procedure itself.

What Distinguishes This From Coordinate Component Assignment

The Indices No Longer Refer to Coordinate Directions

Because the frame does not arise from coordinate functions, an index such as a in T^a labels a frame direction rather than a direction of increasing some coordinate x^a; reading such an index as if it referred to a coordinate would be a misinterpretation specific to the noncoordinate setting.

Structure Functions Travel Alongside the Assigned Components

Whenever these noncoordinate components are used in a calculation that involves differentiating along the frame, the structure functions of the noncoordinate basis must be supplied together with the assigned components, since the components alone do not carry information about how the frame members fail to commute.

Reassignment Under a Change of Frame

Frame Transformations Replace Coordinate Transformations

Passing from one noncoordinate basis to another is governed by an invertible matrix relating the two frames directly, playing the role that the Jacobian matrix plays for coordinate bases, even though no coordinate functions underlie either frame.

Components Transform Using This Same Matrix

Tensor components assigned relative to a noncoordinate basis transform under a change of frame using this frame transformation matrix and its inverse, applied according to whether each index is upper or lower, exactly paralleling the rule used for coordinate transformations but built from the frame relation rather than from partial derivatives of coordinate functions.

Diagram of Noncoordinate Component Assignment

Consequences of Noncoordinate Component Assignment

Ordinary Component Algebra Remains Available

Because the assignment procedure itself is unchanged, addition, scalar multiplication, and contraction of tensors expressed in a noncoordinate basis proceed exactly as with coordinate components, so long as no differentiation along the frame is involved in the calculation.

Differentiated Quantities Require the Structure Functions to Be Reintroduced

Any quantity obtained by differentiating noncoordinate components along the frame must have the relevant structure functions reintroduced explicitly, since the plain component assignment, considered alone, carries no memory of the frame's commutation behavior.