6.17.3 Tensor p q Total Order

Tensor p q Total Order is the sum p + q of the contravariant count and the covariant count of a tensor, giving a single integer that measures the overall number of indices carried by the tensor regardless of how those indices are split between upper and lower positions. This quantity is the coarsest classification available for tensors, coarser than the full type (p, q), and it governs structural facts such as the total number of components, the depth of nested contraction possible, and the way tensors of different total orders relate to each other through tensor products and contractions.

Defining the Total Order

Sum of Contravariant and Covariant Counts

For a tensor of type (p, q), written with components T^{i_1 ... i_p}_{j_1 ... j_q}, the total order is simply:

A scalar has total order zero, since p = q = 0; a vector or a covector has total order one; and every type (2,0), (1,1), or (0,2) tensor has total order two, despite representing three structurally different objects, a bilinear form on covectors, a linear operator, and a bilinear form on vectors, respectively.

Total Order Does Not Determine the Split

Because the total order only records p + q, tensors sharing the same total order can have very different behavior: a total order of two is shared by the type (2,0), (1,1), and (0,2) classifications, which transform differently under a change of basis and play entirely different algebraic roles, from bilinear forms on the dual space to operators to bilinear forms on the original space.

Total Order and the Number of Components

Component Count Depends on Order but Not Only on Order

For a vector space of dimension n, the number of independent components of a type (p, q) tensor is n^{p+q}, which depends only on the total order and the dimension n, not on how the total order is split between p and q. This means a type (2,0) tensor and a type (1,1) tensor in a three-dimensional space both have nine components, even though those components transform under entirely different rules.

Growth Rate with Increasing Order

As the total order increases by one, the component count multiplies by a factor of n, so tensors of high total order become extremely large very quickly; a total order four tensor in a four-dimensional space, such as the Riemann curvature tensor, already has 4^4 = 256 raw components before any symmetries are used to reduce that count.

Total Order Under Tensor Operations

Tensor Product Adds Total Orders

If S has total order p_1 + q_1 and T has total order p_2 + q_2, their tensor product S ⊗ T has total order (p_1 + p_2) + (q_1 + q_2), which is exactly the sum of the two original total orders. This additive behavior under tensor product is the most basic quantitative fact about how tensors combine to form higher-order objects.

Contraction Reduces Total Order by Two

A single contraction, pairing one upper index with one lower index and summing over it, always reduces the total order by exactly two, since it removes one instance of p and one instance of q simultaneously. Repeated contraction can therefore only ever reduce the total order in steps of two, and a tensor with an odd total order composed entirely of paired contravariant and covariant indices in equal number can be fully contracted down to a scalar only if p = q, since each contraction consumes one of each.

Diagram of Total Order Across Common Tensors

Why the Total Order Alone Is Insufficient for Full Classification

The Need for the Full Type Specification

Because tensors of the same total order can play entirely different roles, the total order is used mainly as a quick descriptive summary or as a bookkeeping tool for predicting the outcome of tensor products and contractions, while the full type (p, q) remains the correct and complete classification whenever the precise transformation behavior or algebraic role of a tensor needs to be determined.

Practical Use in Estimating Complexity

Despite its incompleteness as a classification, the total order remains a practically useful measure of computational complexity, since it directly determines the exponent in the component count formula n^{p+q} and therefore the amount of storage and computation required to work with a given tensor numerically, independent of the finer details of how its indices are split between contravariant and covariant slots.