11.11.2 Tensor Index Lowering Contravariant Source

Tensor index lowering via contravariant source transforms upper indices to lower ones using metric tensor in curved spacetime.

Tensor Index Lowering Contravariant Source is the original contravariant tensor component that serves as the starting object supplied to the index lowering operation, providing the upper index that will be contracted against the covariant metric tensor to produce a new component carrying a lower index in its place.

Definition and Identification

The Role of the Source Object

The contravariant source is identified as whichever tensor component currently carries the upper index targeted for conversion, and it is this component, not the metric, that determines the specific numerical values entering the contraction that defines the lowering operation.

A_{i} = g_{i j} A^{j}

Distinguishing the Source From the Result

The contravariant source and the resulting covariant component are related but distinct objects: the source retains its upper index and its original set of numerical values, while the result is a newly computed set of values carrying a lower index, obtained only after the contraction with the metric has been performed.

Requirements on the Source

Any Valid Contravariant Tensor Qualifies

The contravariant source may be any tensor component that carries at least one upper index, including a simple vector, a single upper index of a higher-rank mixed tensor, or one of several upper indices present on a purely contravariant tensor, provided the index being lowered is clearly identified.

The Source Must Belong to the Same Coordinate System as the Metric

For the contraction defining the lowering operation to be valid, the contravariant source and the covariant metric tensor must be expressed in the same coordinate system at the same point, since contracting components described in different coordinate systems would not produce a meaningful result.

Behavior of the Source Under the Operation

Preservation of Untouched Indices

If the contravariant source carries additional indices beyond the one being lowered, those additional indices, whether upper or lower, pass through the operation unchanged and appear in exactly the same position on the resulting tensor as they did on the source.

A_{i k}^{} = g_{i j} A_{k}^{j}

Recoverability of the Source

Because the lowering operation is invertible through the corresponding raising operation, the contravariant source can always be recovered exactly from the result of the lowering operation by contracting the result once more with the contravariant metric tensor, confirming that no information present in the source is lost.

Role Within Tensor Algebras

Anchoring the Operation to a Concrete Object

Identifying the contravariant source clearly is what makes the abstract description of index lowering into a concrete, computable procedure, since the operation is defined as an action performed on this specific object rather than as an operation that could be applied without reference to any starting tensor.

Relationship to Chains of Index Manipulation

When several indices of a tensor are raised or lowered in sequence, each successive operation takes as its contravariant source the result produced by the previous operation, so tracking the contravariant source at each stage is essential for following a multi-step chain of index manipulations correctly.