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Tensors

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Encyclopedia of Nonlinear Science

TENSORS

Tensors are mathematical representations of objects that have intrinsic, geometric significances. This is a rather wider definition than many which can be found in textbooks, often referring to “sets of quantities” that “transform according to hideous formulae.” The best way to understanding is likely to follow a few examples.

Readers interested in learning more will find the books of Simmon (1994) and Dodson & Poston (1991) helpful, as well as that of Bishop & Goldberg (1968). The main caveat, at the risk of repetition, is to beware of books that define tensors as some set of quantities tied to a coordinate system and then describe how they transform—tensors are there whether or not you have coordinates.

Suppose you have some apples on a table. The quantity of apples present is described completely by one number, and that number is well defined and meaningful no matter how you might orient any coordinate axes in the room. The fact that it is meaningful independent of the coordinates makes it a tensor, and the fact that it is the same no matter what coordinates might be introduced makes it a scalar or zeroth-rank tensor—the simplest kind of tensor.

Now, consider the weight of one of the apples. This weight is a force, equal to the mass of the apple, m, times the gravitational acceleration (g=9.8m/s²), and it is directed downwards. That is, its weight is mg directed downwards towards the table top. Let us call this weight W, which has a clear physical meaning independent of coordinates, although it is helpful to describe its direction as “downwards.” The fact that W is well defined even in the absence of coordinates makes it a tensor, and because it has a direction, it is called a vector, or first-rank tensor. If I introduce a set of coordinates x, y, z, I may place these axes in any way I choose. If I put the z-axis pointing downwards, the vector will have components relative to that coordinate system, with only the z component nonzero, and I might describe it as “the vector (0, 0, mg).” This (old) point of view is legitimate but carries with it the need to understand that the components are referred to as a coordinate system and as such have no intrinsic meaning.

More complicated objects can be easily imagined. For example, there could be a wind blowing across the table, and that could be described by another vector V describing its velocity. Referred to a coordinate system, it would be described by three components, and the presence of these two vectors is clearly something of intrinsic geometric significance. We can think of a product of the two vectors, meaning simply all the information needed to describe them. In terms of coordinates, if the weight is described by components ω_i with i ranging from 1 to 3 and the wind velocity by υ_j with j ranging from 1 to 3, we can think of a single geometric object called the tensor product of V and W with 9 components The object is called a second-rank tensor, and this construction can be repeated to create more and more complex objects. (The reason for the upper placement of indices will be made clear shortly.)

The example given above was chosen to make it clear that tensors can turn up anytime one has vectors, but it is easy to find more physically motivated examples. Suppose one has a perfect fluid of density ρ moving with velocity V with components υⁱ. Density is a scalar if we allow only “proper” rotations which cannot change the signs of volumes. The flux or mass per unit volume per unit area perpendicular to direction i flows has components ρυⁱ. The flux of momentum in direction j is ρυⁱυ^j. This motivates defining the second-rank “momentum tensor” tensor with components T^ij=ρυⁱ υ^j.

All the tensors of a given rank have the structure offs a vector space which they inherit from the operations allowed on vectors: they can be added, or multiplied by scalars. In fact, all the machinery of linear algebra carries over directly to them.

There has been an implicit indication that tensors here have something to do with symmetry or a concept of allowed transformation of the coordinates. The class of acceptable coordinate systems defines the geometry in any given situation, and we can speak then of tensors with respect to a symmetry group. For example, if we allow arbitrary rotations of a given orthogonal coordinate system, we can speak of “Cartesian tensors” or tensors under the group of orthogonal transformations in three dimensions. This means that one can pass from a representation of a tensor in terms of components with respect to one coordinate system to another by making the coordinates transform according to a matrix which represents that coordinate change.

This final step in thinking clarifies the notion of what a tensor must be. Given some space, one considers the geometry to be defined by the group of transformations which leave it invariant. For example, in flat Euclidean three-dimensional space, one might take the group of rotations, which are orthogonal matrices of determinant one, or SO(3). The vectors and tensors we have been talking about so far are then elements of spaces in which SO(3) acts linearly, i.e., they lie in representation spaces of SO(3).

Depending on the space in question and the structures imposed upon it, special tensors and operations can be defined. For example, in three-dimensional flat space, we have an operation that takes two vectors and produces from them a (coordinate-independent!) scalar called the “dot product,” “inner product,” or “scalar product.” This is defined using the Kronecker delta δ_ij which is defined to have the value 1 when i=j and 0 otherwise. It is a geometric object with a well-defined meaning independent of coordinates and is, thus, a tensor but of a different kind. One says that while are components of a “contravariant tensor,” δ_ij are the components of a “covariant tensor.” The expression is the scalar product of υ and ω and is a scalar. The terms covariant and contravariant are historical and come from the idea that for a quantity such as to be invariant, the components with upper and lower indices should transform differently under a change of coordinates. The summation signs are often omitted, with a summation over repeated indices implied following the so-called “Einstein summation convention.” An inner product automatically provides a dual space to that of the tensors, and one can write υ_i=δ_ijυ^j and speak of “covariant” as opposed to “contravariant” components of υ, and of the use of δ_ij to “lower indices.”

The concept of a tensor is independent of any notion of a dot product, but many spaces of interest have such a product naturally present. In special relativity, for example, we consider “four-vectors” labeling differences in space and time and requiring four components. In units where the speed of light is unity, the dot product is provided by the Minkowski metric tensor η_ab: (a, b running from 0 to 3) where η_ab is 1 when a=b=0, −1 when a=b≠0, and zero otherwise. This dot product is preserved by a larger group than just rotations, and this group is the Lorentz group of rotations and boosts, also called SO(3, 1). Note that it is not positive-definite, so it is not an inner product in the strict sense of the word, and it defines a pseudo-Euclidean metric. In this case, we have SO(3, 1) vectors as opposed to SO(3) vectors.

The most general (pseudo-)Riemannian geometry provides a dot product in terms of a metric tensor g_ab which is similar to η_ab but unrestricted other than to be nonsingular. In general relativity, we allow all invertible linear transformations in four dimensions, the symmetry group is GL(4), and we have GL(4) tensors for physical quantities.

JOHN DAVID SWAIN

Tensors

TENSORS

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