13.9 Tensor Inner Product Contraction Case

Tensor Inner Product Contraction Case is the specific instance of tensor contraction in which two vectors, or two tensors of matching rank, are combined by contracting each corresponding pair of indices against the other, using a metric tensor where necessary to pair indices of the same variance, so as to produce the scalar known as their inner product. It identifies the contraction pattern that generalizes the familiar dot product of vectors to arbitrary tensors, distinguishing it from trace contraction, which acts within a single tensor, by instead linking indices belonging to two separate tensorial objects.

Conceptual Basis

From Dot Product to Tensor Contraction

The ordinary dot product of two vectors in Euclidean space sums the products of corresponding components. The inner product contraction case expresses this same operation in tensor language by contracting the vectors' indices, using a metric tensor to convert a vector's naturally contravariant index into a covariant one where required for the contraction to be valid.

Role of the Metric Tensor

Because a valid contraction always pairs one contravariant index with one covariant index, and two vectors are typically both represented with contravariant indices, the inner product contraction case commonly proceeds by first using the metric tensor to lower one vector's index, after which the resulting covariant index is contracted against the other vector's contravariant index.

Symmetric Bilinear Structure

The inner product contraction case produces a result that is linear in each of the two tensors being combined, reflecting the bilinear nature of the underlying pairing, and is symmetric in the two arguments whenever the metric tensor itself is symmetric.

Formal Description

Vector Inner Product via the Metric

For two vectors with contravariant components $$u^i$$ and $$v^j$$, and a metric tensor $$g_{ij}$$, the inner product contraction case is written:

where both index pairs $$i$$ and $$j$$ are summed, fully exhausting the indices of both vectors and the metric.

Direct Contraction Without an Explicit Metric

If one vector is already presented in covariant form as $$w_i$$, the inner product contraction case reduces to a single direct contraction:

matching the ordinary contraction of one covariant and one contravariant index.

Extension to Higher-Rank Tensors

For two tensors of matching rank, such as $$A^{ij}$$ and $$B^{kl}$$, the inner product contraction case extends by contracting each index of one tensor against the corresponding index of the other, typically through repeated application of the metric tensor:

Properties

Bilinearity

The inner product contraction case is linear in each argument separately, meaning it distributes over sums of tensors and factors out scalar multiples from either argument, a direct consequence of the linearity of contraction and of the metric tensor itself.

Symmetry

When the metric tensor is symmetric, the inner product contraction case satisfies $$⟨u,v⟩=⟨v,u⟩$$, mirroring the symmetry of the ordinary dot product.

Positive Definiteness in Special Cases

For metrics that are positive definite, the inner product contraction case applied to a vector with itself yields a nonnegative scalar, vanishing only when the vector itself is zero, recovering the familiar notion of squared length from ordinary Euclidean geometry.

Applications

Defining Norms and Angles

The inner product contraction case underlies the definition of vector norms, obtained by taking the inner product of a vector with itself, and of angles between vectors, derived from the ratio of their inner product to the product of their norms.

Orthogonality Conditions

Two tensors are said to be orthogonal with respect to a given metric precisely when their inner product contraction case evaluates to zero, a condition used extensively in decomposing tensors into independent components.

Generalization to Field Theories

In tensor field theories, the inner product contraction case is applied pointwise to tensor fields to construct scalar fields, such as the squared magnitude of a vector field or the pairing of a field with its dual, that carry coordinate-independent physical meaning at each point.