1.2.52 Type Two Zero Tensor Definition

A Type Two Zero Tensor is a rank-two tensor with all components zero, acting as the additive identity in tensor algebra.

Type Two Zero Tensor Definition is the characterization of a tensor with exactly two contravariant factors and no covariant factors, the case in which the type $(r, s)$ equals $(2, 0)$ . A type $(2, 0)$ tensor is an element of the two-fold tensor product of a vector space with itself and, under the identification of tensors with multilinear maps, corresponds to a bilinear form defined not on vectors but on covectors, making it the contravariant counterpart to the more commonly encountered covariant bilinear form.

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual space $V^{*}$ . Setting $r = 2$ and $s = 0$ gives the tensor product space

T_{0}^{2} (V) = V \otimes V

An element $T$ of this space is a type $(2, 0)$ tensor, written in components as $T^{i j}$ with both indices as superscripts. Under the correspondence between tensors and multilinear maps, $T$ is equivalently a bilinear map

T : V^{*} \times V^{*} \to F

taking two covector arguments and returning a scalar, linear in each argument separately.

Contrast with Type (0, 2)

A type $(2, 0)$ tensor should not be confused with a type $(0, 2)$ tensor, the familiar bilinear form on $V$ , even though both are order-two tensors with two indices. The two are built from different factor spaces and act on different kinds of arguments: a type $(0, 2)$ tensor takes two vector arguments and its components $T_{i j}$ carry two subscripts, whereas a type $(2, 0)$ tensor takes two covector arguments and its components $T^{i j}$ carry two superscripts. Only with a metric tensor available on $V$ can the two types be canonically related by raising both indices of the covariant version, and even then the resulting type $(2, 0)$ tensor is derived data, not intrinsically identical to the original.

Symmetric and Alternating Cases

Symmetric Type (2, 0) Tensors

A symmetric type $(2, 0)$ tensor satisfies $T^{i j} = T^{j i}$ , and the most important example of such a tensor is the inverse metric tensor, the type $(2, 0)$ tensor whose components are the matrix inverse of the components of a symmetric type $(0, 2)$ metric tensor. The inverse metric is precisely the tool used to raise indices, converting covariant tensors into contravariant ones.

Alternating Type (2, 0) Tensors

An alternating type $(2, 0)$ tensor satisfies $T^{i j} = - T^{j i}$ and represents a bivector, an element of the second exterior power $⋀^{2} V$ , used to encode oriented two-dimensional planar elements spanned by pairs of vectors, dual to the alternating type $(0, 2)$ two-forms that measure such planar quantities.

Transformation Law

The components of a type $(2, 0)$ tensor transform under a change of basis with matrix $A$ by applying the inverse matrix to each of the two indices independently,

{\tilde{T}}^{i j} = \sum_{k, l} {(A^{-1})}_{k}^{i} {(A^{-1})}_{l}^{j} T^{k l}

which extends the single-index contravariant transformation law used for type $(1, 0)$ tensors to each of the two independent indices of the type $(2, 0)$ case.

Role Within Tensor Algebra

Type $(2, 0)$ tensors occupy the purely contravariant, order-two slot of the type-graded tensor algebra, generalizing single vectors to pairs of vector-like factors and providing the natural home for inverse metrics, bivectors, and any bilinear operation on covectors. Together with type $(0, 2)$ and type $(1, 1)$ tensors, it completes the full classification of order-two tensors by type, illustrating how the same total order can house three structurally distinct kinds of tensor depending on how that order is distributed between contravariant and covariant slots.