11.11 Tensor Index Lowering Operation

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

Definition and Basic Form

The Contraction Formula

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

Application to a Single Index of a Larger Tensor

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

Geometric Meaning

Converting a Vector Into a Covector

Applied to a vector, the index lowering operation produces the covector that is metrically associated with it, meaning the covector whose evaluation on any other vector, computed directly, reproduces the value that the metric would assign to the pairing of the two vectors.

Dependence on the Ambient Metric Structure

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

Properties of the Operation

Linearity

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

Reversibility Through the Raising Operation

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

Role Within Tensor Algebras

Bridge Between Contravariant and Covariant Representations

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

Dependence on Metric Availability

Index lowering 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.