11.11.4 Tensor Index Lowering Component Formula

The Tensor Index Lowering Component Formula transforms upper indices to lower indices using the metric tensor in tensor algebra.

Tensor Index Lowering Component Formula is the explicit expression, written entirely in terms of indexed components and the summation convention, that specifies exactly how each individual component of the lowered tensor is computed from the components of the contravariant source and the covariant metric tensor.

Statement of the Formula

General Single-Index Form

The formula states that the lowered component with a given lower index value is obtained by summing, over every possible value of the shared summation index, the product of the corresponding metric component and the corresponding contravariant source component.

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

Expanded Form in a Finite-Dimensional Space

In a space of finite dimension, the summation implicit in the formula can be written out explicitly as a finite sum of terms, one for each value that the summed index can take, making concrete the number of individual products that must be computed and added together.

A_{1} = g_{11} A^{1} + g_{12} A^{2} + \dots + g_{1 n} A^{n}

Extension to Multiple Indices

Formula for a Mixed Tensor With Several Indices

When the source tensor carries additional indices beyond the one being lowered, the component formula simply carries those additional indices through unchanged on both sides of the equation, attaching them to the same position they occupied on the source tensor.

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

Simultaneous Lowering of More Than One Index

If more than one upper index of the same tensor is to be lowered, the component formula applies one copy of the metric factor for each index being lowered, with each copy using its own independent summation variable, matching the structure used for the mixed variance transformation law.

Correctness and Verification

Consistency With the Symmetry of the Metric

Because the covariant metric tensor is symmetric in its two lower indices, the component formula gives the same result regardless of which of the two metric indices is regarded as being contracted with the source and which is regarded as the free index of the result.

Verification Through the Inverse Relation

The correctness of the component formula can be checked by substituting the lowered component back into the corresponding raising formula, using the contravariant metric tensor, and confirming that the original contravariant source component is recovered exactly, since the two metric tensors are mutual inverses.

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

Role Within Tensor Algebras

Computational Basis for the Abstract Operation

The component formula is what turns the abstract description of index lowering into a concrete recipe that can be evaluated numerically or symbolically, given specific numerical values for the metric tensor and the contravariant source at a particular point.

Template for Related Index Manipulations

The structure of the component formula, one metric factor multiplied against one source component and summed over a shared index, serves as the template for the analogous component formula used in the complementary index raising operation, differing only in which metric tensor is used and which index position changes.