6.24.2 Tensor Upper Index Notation

Tensor Upper Index Notation is the convention of writing a contravariant index as a superscript attached to the tensor symbol, as in Tⁱ or Tⁱʲ, so that the raised position of the letter is itself the signal that the corresponding component transforms using the change-of-basis matrix directly rather than its inverse. This notation is not a stylistic flourish; the vertical placement of the index carries precise mathematical meaning, and the entire discipline of tracking variance in tensor calculus depends on this placement being read correctly and applied consistently.

What Raising an Index Signals

Contravariant Transformation Behavior

An upper index attached to a tensor component indicates that, under a change of basis with transition matrix A (so that e′ᵢ = Σₖ Aₖᵢ eₖ), that particular component transforms with the inverse of A:

where C = A⁻¹. This is the defining behavior of a contravariant quantity, and the upper index is the typographic device that reminds the reader, every time the symbol is written, which transformation rule applies to it.

Association With the Vector Space Itself

An upper index marks a component as belonging to an expansion in the original vector space V, as opposed to its dual V*: writing v = Σᵢ vⁱeᵢ, the coefficients vⁱ carry upper indices precisely because they are the coordinates of an element of V relative to a basis of V, and this is the source of the contravariant transformation law — the coordinates must compensate for any change in the basis vectors to keep v itself fixed.

Notational Rules Governing Upper Indices

Position Consistency

Once an index is established as upper on a given tensor symbol, it remains upper throughout a calculation unless explicitly acted on by an operation — such as lowering with a metric tensor — that is designed to change its variance; casually moving an index from superscript to subscript without such an operation is a notational error, since it silently changes which transformation law is claimed to apply.

Distinguishing Index Position From Exponentiation

A recurring notational hazard is that vⁱ (an upper index) can visually resemble vⁱ as "v raised to the power i"; tensor notation resolves this by context and by convention — in any expression where Einstein summation or explicit tensor components are under discussion, a superscript letter is understood as an index, not an exponent, and true exponents are written distinctly or accompanied by explicit clarifying text when ambiguity could arise.

Diagram of Index Raising and Its Meaning

Where Upper Indices Appear Throughout Tensor Algebra

Vector Components and Contravariant Tensor Slots

The coordinates of an ordinary vector, vⁱ, and every contravariant slot of a higher-order tensor, T^{i₁...i_p}, use upper index notation; a fully contravariant tensor of type (p, 0) has only upper indices and no lower ones.

Raised Indices Produced by the Metric

When a metric tensor g is available, it can raise a lower index into an upper one via vⁱ = Σⱼ gⁱʲvⱼ, converting a covariant quantity into a contravariant one; the resulting upper index on vⁱ correctly signals that this raised quantity now transforms contravariantly, exactly as any other genuinely contravariant object would, confirming that the upper index notation tracks actual transformation behavior rather than merely the object's origin.