8.19.5 Tensor Contraction Scalar Case

Tensor Contraction Scalar Case is the special instance of tensor contraction in which every upper index of a tensor is paired with and summed against every lower index, leaving no free indices at all, so that the result is a single number — a scalar — rather than a tensor of some smaller nonzero order. It is the terminal point of the general order-reduction process: whereas an ordinary contraction lowers order by two while typically leaving free indices behind, the scalar case is reached precisely when the contracted tensor had equal numbers of upper and lower indices and all of them have been consumed.

When the Scalar Case Arises

Equal Numbers of Upper and Lower Indices

A tensor of type (p, q) can be reduced to a scalar by contraction only if its indices can be exhausted in matched upper-lower pairs, which requires p = q. Contracting all p upper indices against all q = p lower indices, in some assignment of pairs, removes every index and leaves order zero:

after p successive single contractions, each pairing one upper index with one lower index until none remain.

The Simplest Instance: Rank-2 Trace

The most elementary scalar case is the full contraction of a (1,1) tensor, A^i_i, which is precisely the trace of the linear map or matrix that A represents. Every higher-order scalar case is built from repeating this same one-index-pair-at-a-time reduction, so the rank-2 trace is the base case from which all fully contracted scalar invariants are constructed.

Basis Independence of the Result

Why the Scalar Does Not Depend on Coordinates

Each single contraction in the chain leading to a scalar cancels one Jacobian factor against one inverse-Jacobian factor under a change of basis, as is true for any valid upper-lower contraction. After all indices have been eliminated, no Jacobian factors remain at all, so the resulting number is identical in every coordinate system:

where T denotes the fully contracted scalar computed in one basis and the overbar denotes the same computation carried out after an arbitrary change of basis. This coordinate independence is exactly what qualifies the scalar case as producing a true invariant rather than merely a basis-dependent number.

Contrast With Partially Contracted Results

A tensor left with any free indices still transforms nontrivially under a change of basis, following the transformation rule appropriate to its remaining type (p', q'). Only in the scalar case, with p' = q' = 0, does the transformation rule become trivial, since there are no remaining Jacobian or inverse-Jacobian factors to apply; this is why scalars, and only scalars among contracted tensors, are directly comparable as plain numbers across different coordinate systems without any further conversion.

Order of Pairing Can Matter

Different Complete Pairings May Give Different Scalars

When p = q > 1, there is generally more than one way to pair all upper indices against all lower indices, and distinct complete pairings can produce different scalar values unless the tensor possesses symmetries that force them to coincide. For a (2,2) tensor T^{ij}_{\ \ kl}, contracting i with k and j with l gives T^{ij}_{\ \ ij}, while contracting i with l and j with k gives T^{ij}_{\ \ ji}, and these two scalars need not agree.

Symmetric Tensors Collapse the Ambiguity

If a tensor is fully symmetric among its upper indices and fully symmetric among its lower indices, every complete pairing yields the same scalar, since permuting the paired indices merely permutes identical terms in the sum. This is why, for tensors built from symmetric objects such as the metric and its inverse, a single fully contracted scalar is unambiguous even though multiple pairings could in principle be written down.

Diagram of Complete Reduction to a Scalar

Named Examples of the Scalar Case

Scalar Curvature

The scalar curvature R of a Riemannian manifold is obtained by fully contracting the Ricci tensor R_{ij} (itself already a partial contraction of the Riemann tensor) against the inverse metric g^{ij}, R = g^{ij} R_{ij}, a two-step chain of contractions ending in a single coordinate-independent number at each point of the manifold.

Squared Norm and Inner Products

The squared norm of a vector, g_{ij} v^i v^j, and more generally the inner product of two vectors, g_{ij} v^i w^j, are scalar cases obtained by contracting the metric's two lower indices against the upper indices of the vectors involved, illustrating that the familiar dot product of ordinary vector algebra is itself an instance of the tensor contraction scalar case.