tensor

Overview

A tensor is the mathematical idealization of a geometric orphysical quantity whose analytic description, relative to a fixedframe of reference, consists of an array of numbers. Some well knownexamples of tensors in geometry are quadratic forms, and the curvaturetensor. Examples of physical tensors are the energy-momentum tensor,and the polarization tensor.

Geometric and physical quantities may be categorized by consideringthe degrees of freedom inherent in their description. The scalarquantities are those that can be represented by a single number —speed, mass, temperature, for example. There are also vector-likequantities, such as force, that require a list of numbers for theirdescription. Finally, quantities such as quadratic formsnaturally require a multiply indexed array for their description.These latter quantities can only be conceived of as tensors.

Actually, the tensor notion is quite general, and applies to all ofthe above examples; scalars and vectors are special kinds oftensors. The feature that distinguishes a scalar from a vector, anddistinguishes both of those from a more general tensor quantity isthe number of indices in the representing array. This number iscalled the rank of a tensor. Thus, scalars are rank zero tensors (noindices at all), and vectors are rank one tensors.

It is also necessary to distinguish between two types of indices,depending on whether the corresponding numbers transform covariantlyor contravariantly relative to a change in the frame of reference.Contravariant indices are written as superscripts, while thecovariant indices are written as subscripts. The valenceof a tensor is the pair $(p,q)$, where $p$ is the number contravariantand $q$ the number of covariant indices, respectively.

It is customary to represent the actual tensor, as a stand-aloneentity, by a bold-face symbol such as $𝖠$. The corresponding arrayof numbers for a type $(p,q)$ tensor is denoted by the symbol$A_{j_1\mathrm{\dots }{j}_q}^{i_1\mathrm{\dots }{i}_p},$ where the superscripts andsubscripts are indices that vary from $1$ to $n$. This number $n$, therange of the indices, is called the dimension of the tensor. Thetotal degrees of freedom required for the specification of aparticular tensor is the product of the tensor’s rank and its dimension.

Again, it must be emphasized that the tensor $𝖠$ and therepresenting array $A_{j_1\mathrm{\dots }{j}_p}^{i_1\mathrm{\dots }{i}_q}$ are not thesame thing. The values of the representing array are given relativeto some frame of reference, and undergo a linear transformation whenthe frame is changed.

Finally, it must be mentioned that most physical and geometricapplications are concerned with tensor fields, that is to saytensor valued functions, rather than tensors themselves. Some care isrequired, because it is common to see a tensor field called simply atensor. There is a difference, however; the entries of a tensor array$A_{j_1\mathrm{\dots }{j}_p}^{i_1\mathrm{\dots }{i}_q}$ are numbers, whereas the entriesof a tensor field are functions. The present entry treats the purelyalgebraic aspect of tensors. Tensor field concepts, which typicallyinvolved derivatives of some kind, are discussed elsewhere.

Definition.

The formal definition of a tensor quantity begins with afinite-dimensional vector space $U$, which furnishes the uniform“building blocks” for tensors of all valences. In typicalapplications, $U$ is the tangent space at a point of a manifold; theelements of $U$ represent velocities and forces. The space of$(p,q)$-valent tensors, denoted here by ${𝒯}_{p,q}(U)$ is obtained bytaking the tensor product of $p$ copies of $U$, and $q$ copies ofthe dual vector space $U^{*}$. To wit,

$${𝒯}_{p,q}(U)=\stackrel{p\text{ times}}{\overbrace{U\otimes \mathrm{\dots }\otimes U}}\otimes \stackrel{q\text{ times}}{\overbrace{U^{*}\otimes \mathrm{\dots }\otimes U^{*}}}.$$

In order to represent a tensor by a concrete array of numbers, werequire a frame of reference, which is essentially a basis of $U$,say$e_1,\mathrm{\dots },e_n\in U.$Every vector in $U$ can be“measured” relative to this basis, meaning that for every$𝐯\in U$ there exist unique scalars $v^i$, suchthat (note the use of the Einstein summation convention)

$$𝐯=v^i{e}_i.$$

These scalars are called the components of $𝐯$ relative to the framein question.

Let ${\epsilon }^1,\mathrm{\dots },{\epsilon }^n\in U^{*}$ be the corresponding dualbasis, i.e.,

$${\epsilon }^i(e_j)={\delta }_j^i,$$

where the latter is the Kronecker delta array. For every covector$\alpha \in U^{*}$ there exists a unique array of components ${\alpha }_i$ suchthat

$$\alpha ={\alpha }_i{\epsilon }^i.$$

More generally, every tensor $𝖠\in {𝒯}_{p,q}(U)$ has a uniquedescription in terms of components. That is to say, there exists aunique array of scalars $A_{j_1\mathrm{\dots }{j}_p}^{i_1\mathrm{\dots }{i}_q}$ such that

$$𝖠=A_{j_1\mathrm{\dots }{j}_p}^{i_1\mathrm{\dots }{i}_q}{e}_{i_1}\otimes \mathrm{\dots }\otimes e_{i_q}\otimes {\epsilon }^{j_1}\otimes \mathrm{\dots }\otimes {\epsilon }^{j_p}.$$

Transformation rule.

Next, suppose that a change is made to a different frame ofreference, say${\widehat{e}}_1,\mathrm{\dots },{\widehat{e}}_n\in U.$Any two frames are uniquely related byan invertible transition matrix $X_j^i$, having the property that forall values of $j$ we have

$${\widehat{e}}_j=X_j^i{e}_i.$$(1)

Let $𝐯\in U$ be a vector, and let $v^i$and ${\widehat{v}}^i$ denote the corresponding component arrays relative tothe two frames. From

$$𝐯=v^i{e}_i={\widehat{v}}^i{\widehat{e}}_i,$$

and from (1) we infer that

$${\widehat{v}}^i=Y_j^i{v}^j,$$(2)

where $Y_j^i$ is the matrix inverse of $X_j^i$, i.e.,

$$X_k^i{Y}_j^k={\delta }_j^i.$$

Thus, thetransformation rule for a vector’s components (2) iscontravariant to the transformation rule for the frame of reference(1). It is for this reason that the superscriptindices of a vector are called contravariant.

To establish (2), we note that the transformation rulefor the dual basis takes the form

$${\widehat{\epsilon }}^i=Y_j^i{\epsilon }^j,$$

and that

$$v^i={\epsilon }^i(𝐯),$$

while

$${\widehat{v}}^i={\widehat{\epsilon }}^i(𝐯).$$

The transformation rule for covector components is covariant. Let$\alpha \in U^{*}$ be a given covector, and let ${\alpha }_i$ and${\widehat{\alpha }}_i$ be the corresponding component arrays. Then

$${\widehat{\alpha }}_j=X_j^i{\alpha }_i.$$

The above relation is easily established. We need only remark that

$${\alpha }_i=\alpha (e_i),$$

and that

$${\widehat{\alpha }}_j=\alpha ({\widehat{e}}_j),$$

and then use (1).

In light of the above discussion, we see that the transformation rulefor a general type $(p,q)$ tensor takes the form

$${\widehat{A}}_{j_1\mathrm{\dots }{j}_p}^{i_1\mathrm{\dots }{i}_q}=X_{k_1}^{i_1}\mathrm{\cdots }{X}_{k_q}^{i_q}{Y}_{j_1}^{l_1}\mathrm{\cdots }{Y}_{j_p}^{l_1}{A}_{l_1\mathrm{\dots }{l}_p}^{k_1\mathrm{\dots }{k}_q}.$$