11.10.3 Tensor Index Raising Contravariant Result

Tensor Index Raising Contravariant Result is the newly produced tensor component that emerges from the index raising operation, carrying an upper index in the position where the covariant source previously carried a lower index, and representing the same underlying geometric object as the source but expressed now in contravariant form relative to the chosen metric.

Definition and Identification

What the Result Represents

The contravariant result is the output of contracting the covariant source with the contravariant metric tensor, and it stands as a fully formed tensor component in its own right, ready to be used in further calculations exactly as any other contravariant component would be used.

Distinguishing the Result From an Independently Defined Contravariant Tensor

Although the contravariant result behaves in every formal respect like an ordinary contravariant tensor, it is distinguished conceptually by the fact that it was produced through the metric from a specific covariant source, so that its numerical values are tied to that source and to the choice of metric, rather than being assigned independently.

Behavior of the Result

Correct Transformation Under Coordinate Change

The contravariant result obeys the ordinary contravariant transformation law under a change of coordinates, transforming with the direct Jacobian factor exactly as any contravariant tensor would, since it is, by construction, a genuine tensor of contravariant type once the raising operation has been completed.

Recoverability of the Source From the Result

Applying the corresponding index lowering operation to the contravariant result, using the covariant metric tensor, reproduces the original covariant source exactly, confirming that the raising operation and the result it produces are fully reversible.

Consequences of the Result's Structure

Usability in Contravariant Contexts

Because the contravariant result behaves identically to any other tensor of contravariant type, it can be freely contracted with covariant tensors, combined with other contravariant tensors through tensor products, or transformed across coordinate systems, without any special treatment arising from the fact that it was produced by raising rather than assigned directly.

Dependence on the Same Metric Used for Raising

Since the contravariant result is computed using a specific contravariant metric tensor, changing the metric used in the raising operation, while keeping the covariant source fixed, would generally produce a different contravariant result, so the result is meaningful only in reference to the metric that produced it.

Role Within Tensor Algebras

Completing the Raising and Lowering Cycle

The contravariant result, together with the covariant source and the metric input, completes the three essential components of the index raising operation, illustrating the operation as a well-defined map that takes a covariant tensor and a metric and produces a specific contravariant tensor.

Basis for Further Index Manipulation

Once obtained, the contravariant result can itself serve as the starting object for further operations, such as raising an additional lower index if the original tensor had more than one, or being contracted with other tensors to form new invariant scalars or tensors of different rank.