8.6.2 Tensor Free Index Equation Preservation

Tensor Free Index Equation Preservation ensures algebraic consistency by maintaining equation validity across tensor index manipulations.

Tensor Free Index Equation Preservation is the requirement that a free index, once introduced on one side of a tensor equation, must remain present, with unchanged name, unchanged variance, and unchanged position, throughout every subsequent manipulation of that equation, including addition, contraction with other tensors, differentiation, and substitution of one expression for another.

Preservation Under Algebraic Manipulation

Preservation Through Addition

When two tensor expressions are added, any free index shared between them is preserved unchanged in the sum, retaining its variance and its role as a placeholder for an unspecified but fixed component.

S_{i} = P_{i} + Q_{i}

The free index (i) is preserved on the left-hand side exactly as it appears on the right, since addition does not alter which slot of the resulting tensor a component equation describes.

Preservation Through Multiplication by a Scalar or Contraction with Another Tensor

Multiplying a tensor expression by a scalar, or contracting it against another tensor using a dummy index, preserves any existing free index untouched, since the operation introduces or removes only dummy indices, never free ones.

Y_{i} = λ A_{i j} B^{j}

Here the free index (i) survives the introduction of the scalar (\lambda) and the contraction over the dummy index (j), remaining the sole free index of the resulting expression.

Preservation Under Substitution

Substituting an Equivalent Expression

Free index equation preservation guarantees that when one tensor expression is substituted for another within a larger equation, because the two are known to be equal, the free indices of the substituted expression must match, in name and variance, the free indices of the expression it replaces.

V_{i} = g_{i j} V^{j}

If this expression for (V_i) is substituted into a larger equation wherever (V_i) appears, the free index (i) carried by the substituted expression must align exactly with the free index (i) it is replacing, so that the overall equation remains consistent term by term.

Renaming Without Loss of Preservation

A free index may be renamed throughout an entire equation, provided the renaming is applied consistently to every occurrence and does not collide with another index already in use, and preservation is maintained because the renamed index still occupies the identical position and variance as before.

Preservation Under Differentiation

Free Indices in Derivative Expressions

When a tensor expression is differentiated, whether by an ordinary or covariant derivative, any free indices already present in the original expression are preserved in the result, and the differentiation typically introduces one additional free index corresponding to the direction of differentiation.

{\nabla V}_{i k} = \partial_{k} V_{i} - Γ_{i k}^{l} V_{l}

The free index (i) inherited from (V_i) is preserved throughout every term of the covariant derivative expression, while the new free index (k), introduced by the derivative operation, is likewise present in every term.

Consequence for Verifying Tensor Equations

Because free index equation preservation must hold at every stage of a derivation, checking that a claimed free index appears with matching name, variance, and count in every term of a final result, compared against the original expression, is a standard and reliable method for verifying that no error, such as an accidental contraction or a mismatched substitution, was introduced during the manipulation of a tensor equation.