8.23.3 Tensor Index Notation Transformation Boundary

Tensor Index Notation Transformation Boundary is the set of limits on the standard tensor transformation law — one factor of the Jacobian for each upper index, one factor of its inverse for each lower index — marking where a change of coordinates fails to be smoothly invertible, where the transformation law must be supplemented by an extra determinant factor as for tensor densities, or where the map relating two coordinate systems does not even have matching dimension, so that the ordinary rule for how components change under a change of basis no longer applies as stated. It identifies exactly where "transform by the Jacobian" stops being a sufficient description of how a quantity behaves under a change of coordinates.

The Boundary at a Non-Invertible or Non-Smooth Jacobian

Singular Points of the Coordinate Map

The standard transformation law presumes the Jacobian matrix ∂x̄ⁱ/∂xʲ is invertible at the point in question; at a point where the Jacobian degenerates — becomes singular, with zero determinant — the inverse Jacobian factors needed to transform lower indices do not exist, and the transformation law cannot be applied there even though it may work perfectly well at every nearby point. Such points, often coinciding with coordinate singularities, sit directly at this boundary.

Coordinate Changes That Are Not Differentiable

The transformation law's Jacobian factors are, by definition, partial derivatives of the new coordinates with respect to the old; a proposed change of coordinates that is continuous but not differentiable at some point (or is not differentiable enough times for the tensor quantity in question, as with second-derivative-dependent objects like curvature) places any tensor field evaluated there outside the domain where the ordinary transformation law can be invoked, since the very factors the law requires fail to exist.

The Boundary at Tensor Densities and Extra Determinant Factors

Quantities That Transform With an Additional Jacobian Determinant

Certain quantities closely related to tensors, called tensor densities, transform according to the ordinary tensor rule multiplied by an additional power of the Jacobian determinant; the volume element √|g| d^n x used in integration is a standard example, picking up one factor of the Jacobian determinant beyond what a plain scalar would require. Applying the plain tensor transformation law to such a density, omitting the extra determinant factor, misdescribes how it actually changes under a coordinate transformation, placing densities just outside the strict boundary of ordinary tensor transformation even though they are built from and closely tied to genuine tensors.

for a density of weight w, showing the extra determinant power that the plain tensor transformation law does not include.

Pseudotensors and Orientation-Dependent Behavior

A pseudotensor picks up an additional sign flip under orientation-reversing coordinate changes beyond the ordinary transformation rule, so its transformation law depends not just on the Jacobian's entries but on the sign of the Jacobian determinant; this orientation-sensitivity is a further departure from the plain tensor transformation law and marks another way an indexed quantity can sit at, without falling entirely outside, this boundary.

The Boundary at Mismatched or Non-Bijective Coordinate Maps

Transformations Between Spaces of Different Dimension

The standard transformation law implicitly assumes the old and new coordinate systems describe the same underlying space with the same dimension; a map between coordinate systems of different dimension — such as a projection or an embedding — is not invertible in the sense the transformation law requires, and index notation's ordinary transformation rule does not directly apply to relate tensor components across such a map without additional structure (such as a pullback or pushforward construction) supplied separately.

Multi-Valued or Non-Bijective Coordinate Relations

If the correspondence between two coordinate systems is not one-to-one — multiple points in one system corresponding to the same point in another, or vice versa — the Jacobian-based transformation law, which presumes a well-defined, invertible local correspondence, cannot be unambiguously applied, and the transformation boundary is crossed until the coordinate relation is restricted to a region where it is genuinely bijective.

Diagram of the Transformation Law's Domain and Its Edges

Practical Handling at the Transformation Boundary

Restricting to the Well-Behaved Region

The standard remedy for a singular or non-differentiable Jacobian is to restrict attention to the open region where the coordinate map is smooth and invertible, treating the boundary points themselves separately (often by switching to a different, non-singular coordinate chart there), rather than attempting to force the ordinary transformation law to apply exactly at the problematic points.

Extending the Law Rather Than Abandoning It

For tensor densities, pseudotensors, and other near-tensorial objects, the transformation boundary is handled not by discarding index notation but by explicitly recording the extra determinant or sign factor the object requires, so that a modified, but still precisely stated, transformation law governs the object correctly; this is the same general strategy used at every boundary of index notation — supplementing the notation with exactly the additional structure needed, rather than treating the boundary as a dead end.