11.10 Tensor Index Raising Operation

Tensor Index Raising Operation is the metric-dependent algebraic procedure by which a lower index of a tensor is converted into an upper index, achieved by contracting the tensor's lower index with the contravariant metric tensor, thereby producing a new tensor whose type has one fewer lower index and one more upper index than the original.

Definition and Basic Form

The Contraction Formula

The index raising operation is carried out by multiplying the tensor's covariant component by the contravariant metric tensor and summing over the shared index, converting the lower position into an upper one on the resulting object.

Application to a Single Index of a Larger Tensor

When a tensor carries multiple indices, the raising operation can be applied to just one selected lower index while leaving the remaining indices, whether upper or lower, entirely untouched, producing a new mixed tensor with the selected index moved to the upper position.

Geometric Meaning

Converting a Covector Into a Vector

Applied to a covector, the index raising operation produces the vector that is metrically associated with it, meaning the vector whose pairing with any other vector, computed through the metric, reproduces the value the covector would have assigned directly to that vector.

Dependence on the Ambient Metric Structure

Because the operation depends entirely on the contravariant metric tensor, the specific vector produced by raising a given covector will differ depending on which metric is in use, so the raising operation is meaningful only relative to a fixed choice of metric on the space under consideration.

Properties of the Operation

Linearity

The index raising operation is linear, meaning that raising the index of a sum of two covariant components produces the same result as raising each component separately and then adding, and raising the index of a scalar multiple of a covariant component scales the raised result by the same scalar.

Reversibility Through the Lowering Operation

Raising a lower index and then immediately applying the complementary lowering operation to the resulting upper index returns the original tensor exactly, since the contravariant metric tensor used for raising is the matrix inverse of the covariant metric tensor used for lowering.

Role Within Tensor Algebras

Bridge Between Covariant and Contravariant Representations

The index raising operation, together with its inverse lowering operation, provides the mechanism by which a single geometric object can be represented in either covariant or contravariant form, depending on which representation is more convenient for a given calculation.

Dependence on Metric Availability

Index raising is only defined on spaces equipped with a non-degenerate metric tensor, distinguishing it from the coordinate transformation laws of tensors, which require no metric at all; a space without a metric permits tensors to be classified by type but does not permit converting between upper and lower indices.