11.10.2 Tensor Index Raising Covariant Source

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

Definition and Identification

The Role of the Source Object

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

Distinguishing the Source From the Result

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

Requirements on the Source

Any Valid Covariant Tensor Qualifies

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

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

For the contraction defining the raising operation to be valid, the covariant source and the contravariant 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 covariant source carries additional indices beyond the one being raised, 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.

Recoverability of the Source

Because the raising operation is invertible through the corresponding lowering operation, the covariant source can always be recovered exactly from the result of the raising operation by contracting the result once more with the covariant 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 covariant source clearly is what makes the abstract description of index raising 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 covariant source the result produced by the previous operation, so tracking the covariant source at each stage is essential for following a multi-step chain of index manipulations correctly.