13.9.1 Tensor Inner Product Metric Input

The Tensor Inner Product Metric Input defines how tensors interact via inner products, setting a framework for geometric computations.

Tensor Inner Product Metric Input is the metric tensor supplied to an inner product contraction as the auxiliary object that converts one or more contravariant indices into covariant form, thereby making it possible to pair those indices against the remaining contravariant indices of the tensors being combined. It identifies the metric's role specifically as an input to the inner product operation, rather than as an independent object of study, emphasizing the function it performs within the contraction rather than its own intrinsic properties.

Conceptual Basis

Why an Auxiliary Input Is Needed

An inner product between two vectors, both naturally expressed with contravariant components, cannot be formed by direct contraction alone, since contraction requires one contravariant and one covariant index. The metric input supplies the missing covariant structure, acting as a bridge that allows two same-variance indices to be joined through an intermediate contraction with the metric.

The Metric as a Rank-Two Covariant Tensor

The metric input is itself a tensor, typically of type $(0, 2)$ , meaning it possesses two covariant indices, each of which is available to be contracted against a contravariant index supplied by one of the vectors or tensors entering the inner product.

Distinguishing the Metric's General Role From Its Use Here

While the metric tensor also defines notions such as distance and angle throughout a geometric space, its role as an inner product metric input is narrower: it is simply the object invoked at the moment of contraction to supply the covariant indices needed to complete the pairing.

Formal Description

Supplying One Covariant Pair

For two vectors $u^{i}$ and $v^{j}$ , the metric input $g_{i j}$ is introduced and contracted against both:

⟨ u, v ⟩ = g_{i j} u^{i} v^{j}

so that all three objects, the two vectors and the metric input, have their indices fully exhausted in the resulting scalar.

Metric Input for Higher-Rank Tensors

When forming the inner product of two rank-two tensors $A^{i j}$ and $B^{k l}$ , two copies of the metric input are required, one for each pair of indices being matched:

⟨ A, B ⟩ = g_{i k} g_{j l} A^{i j} B^{k l}

Redundancy When Covariant Indices Are Already Present

If one of the tensors entering the inner product already carries a covariant index, such as $w_{i}$ , no metric input is required for that index, since direct contraction against the corresponding contravariant index of the other tensor suffices.

Properties

Symmetry Inherited From the Metric

If the metric input is symmetric, meaning $g_{i j} = g_{j i}$ , the resulting inner product is symmetric in its two vector arguments, since exchanging the vectors and relabeling the summed indices leaves the sum unchanged.

Dependence of the Result on the Chosen Metric Input

The scalar produced by the inner product depends directly on which metric input is used, so changing the metric while keeping the same underlying vectors generally produces a different inner product value, reflecting a different notion of pairing between the same vectors.

Requirement of Non-Degeneracy

For the metric input to support a well-defined and invertible correspondence between contravariant and covariant indices more generally, it is typically required to be non-degenerate, meaning no nonzero vector is mapped to a covector that vanishes when paired against every other vector.

Practical Considerations

Choosing the Metric Input in Context

The specific metric input used depends on the geometric or physical context of the problem, such as a Euclidean metric for ordinary space or a pseudo-Euclidean metric in contexts involving indefinite forms, and this choice directly affects which pairs of vectors are treated as orthogonal.

Omission When the Metric Is the Identity

In settings where the metric input coincides with the identity matrix in the chosen basis, the explicit appearance of the metric is often suppressed in notation, since contracting against the identity leaves the remaining indices numerically unchanged.

Consistency Across Repeated Use

When the same metric input is used across multiple inner products within a larger computation, consistency in which metric is applied is essential, since substituting a different metric partway through a derivation would introduce an inconsistency in the geometric interpretation of the results.