11.10.4 Tensor Index Raising Component Formula

The Tensor Index Raising Formula uses the metric tensor to convert lower to upper indices in tensor components.

Tensor Index Raising 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 raised tensor is computed from the components of the covariant source and the contravariant metric tensor.

Statement of the Formula

General Single-Index Form

The formula states that the raised component with a given upper 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 covariant 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 raised, 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_{k}^{i} = \sum_{j} g^{i j} A_{j k}

Simultaneous Raising of More Than One Index

If more than one lower index of the same tensor is to be raised, the component formula applies one copy of the metric factor for each index being raised, 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 contravariant metric tensor is symmetric in its two upper 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 raised component back into the corresponding lowering formula, using the covariant metric tensor, and confirming that the original covariant 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 raising into a concrete recipe that can be evaluated numerically or symbolically, given specific numerical values for the metric tensor and the covariant 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 lowering operation, differing only in which metric tensor is used and which index position changes.