11.11.3 Tensor Index Lowering Covariant Result

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

Definition and Identification

What the Result Represents

The covariant result is the output of contracting the contravariant source with the covariant 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 covariant component would be used.

Distinguishing the Result From an Independently Defined Covariant Tensor

Although the covariant result behaves in every formal respect like an ordinary covariant tensor, it is distinguished conceptually by the fact that it was produced through the metric from a specific contravariant 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 covariant result obeys the ordinary covariant transformation law under a change of coordinates, transforming with the inverse Jacobian factor exactly as any covariant tensor would, since it is, by construction, a genuine tensor of covariant type once the lowering operation has been completed.

Recoverability of the Source From the Result

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

Consequences of the Result's Structure

Usability in Covariant Contexts

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

Dependence on the Same Metric Used for Lowering

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

Role Within Tensor Algebras

Completing the Lowering and Raising Cycle

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

Basis for Further Index Manipulation

Once obtained, the covariant result can itself serve as the starting object for further operations, such as lowering an additional upper 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.