Tensors

A tensor of rank
in an
-dimensional space possesses
components
which are, in general, functions of position in that space. A tensor of
rank zero has one component,
, and is called a *scalar*. A tensor
of rank one has
components,
, and is called
a *vector*. A tensor of rank two has
components, which can be
exhibited in matrix format. Unfortunately, there is no convenient way of
exhibiting a higher rank tensor. Consequently, tensors are usually
represented by a typical component: for instance,
the tensor
(rank 3), or the tensor
(rank 4),
et cetera. The suffixes
are always understood to range from
1 to
.

For reasons that will become apparent later on, we shall
represent tensor components using both
superscripts and subscripts. Thus, a typical tensor might look like
(rank 2), or
(rank 2), et cetera. It is convenient to
adopt the *Einstein summation convention*. Namely, if any suffix appears twice
in a given term, once as a subscript and once as a superscript, a summation
over that suffix (from 1 to
) is implied.

To distinguish between various different coordinate systems, we shall use primed and multiply primed suffixes. A first system of coordinates can then be denoted by , a second system by , et cetera. Similarly, the general components of a tensor in various coordinate systems are distinguished by their suffixes. Thus, the components of some third rank tensor are denoted in the system, by in the system, et cetera.

When making a coordinate transformation from one set of coordinates, , to another, , it is assumed that the transformation is non-singular. In other words, the equations that express the in terms of the can be inverted to express the in terms of the . It is also assumed that the functions specifying a transformation are differentiable. It is convenient to write

(1664) | ||

(1665) |

Note that

(1666) |

by the chain rule, where (the

The formal definition of a tensor is as follows:

- An entity having components
in the
system and
in the
system is said to
behave as a
*covariant tensor*under the transformation if

- Similarly,
is said to behave as a
*contravariant tensor*under if

- Finally,
is said to behave as
a
*mixed tensor*(contravariant in and covariant in ) under if

When an entity is described as a tensor it is generally understood
that it behaves as a tensor under
all non-singular differentiable transformations of the relevant
coordinates. An entity that only behaves as a tensor under a
certain subgroup of non-singular differentiable coordinate transformations
is called a *qualified tensor*, because its name is conventionally
qualified by an adjective recalling the subgroup in question.
For instance, an entity that only exhibits tensor behavior under
Lorentz transformations is called a *Lorentz tensor*, or, more commonly, a
*4-tensor*.

When applied to a tensor of rank zero (a scalar), the previous definitions imply that . Thus, a scalar is a function of position only, and is independent of the coordinate system. A scalar is often termed an invariant.

The main theorem of tensor calculus is as follows:

If two tensors of the same type are equal in one coordinate system then they are equal in all coordinate systems.

The simplest example of a contravariant vector (tensor of rank one) is provided by the differentials of the coordinates, , because

(1670) |

The coordinates themselves do not behave as tensors under all coordinate transformations. However, because they transform like their differentials under linear homogeneous coordinate transformations, they do behave as tensors under such transformations.

The simplest example of a covariant vector is provided by the gradient of a function of position , because if we write

(1671) |

then we have

(1672) |

An important example of a mixed second-rank tensor is provided by the Kronecker delta introduced previously, because

(1673) |

Tensors of the same type can be added or subtracted to form new tensors. Thus, if and are tensors, then is a tensor of the same type. Note that the sum of tensors at different points in space is not a tensor if the 's are position dependent. However, under linear coordinate transformations the 's are constant, so the sum of tensors at different points behaves as a tensor under this particular type of coordinate transformation.

If
and
are tensors, then
is a tensor of the type indicated by the suffixes. The
process illustrated by this example is called *outer multiplication*
of tensors.

Tensors can also be combined by *inner multiplication*, which implies
at least one dummy suffix link. Thus,
and
are tensors of the type indicated by the suffixes.

Finally, tensors can be formed by *contraction* from tensors of
higher rank. Thus, if
is a tensor then
and
are tensors of the
type indicated by the suffixes. The most important type of contraction
occurs when no free suffixes remain: the result is a scalar. Thus,
is a scalar provided that
is a tensor.

Although we cannot usefully divide tensors, one by another, an entity
like
in the equation
, where
and
are tensors, can be formally regarded as the quotient of
and
. This gives the name to a particularly useful rule for
recognizing tensors, the *quotient rule*. This rule states that
if a set of components, when combined by a given type of multiplication
with all tensors of a given type yields a tensor, then the set is
itself a tensor. In other words, if the product
transforms like a tensor for all tensors
then it follows that
is a tensor.

Let

(1674) |

Then if is a tensor, differentiation of the general tensor transformation (1671) yields

(1675) |

where , et cetera, are terms involving derivatives of the 's. Clearly, is not a tensor under a general coordinate transformation. However, under a linear coordinate transformation ( 's constant) behaves as a tensor of the type indicated by the suffixes, because the , et cetera, all vanish. Similarly, all higher partial derivatives,

(1676) |

et cetera, also behave as tensors under linear transformations. Each partial differentiation has the effect of adding a new covariant suffix.

So far, the space to which the coordinates
refer has been without
structure. We can impose a structure on it by defining the distance
between all pairs of neighboring points by means of a *metric*,

where the are functions of position. We can assume that without loss of generality. The previous metric is analogous to, but more general than, the metric of Euclidian -space, . A space whose structure is determined by a metric of the type (1679) is called

The elements of the inverse of the matrix are denoted by . These elements are uniquely defined by the equations

(1678) |

It is easily seen that the constitute the elements of a contravariant tensor. This tensor is said to be conjugate to . The conjugate metric tensor is symmetric (i.e., ) just like the metric tensor itself.

The tensors and allow us to introduce the important operations of raising and lowering suffixes. These operations consist of forming inner products of a given tensor with or . For example, given a contravariant vector , we define its covariant components by the equation

(1679) |

Conversely, given a covariant vector , we can define its contravariant components by the equation

(1680) |

More generally, we can raise or lower any or all of the free suffixes of any given tensor. Thus, if is a tensor we define by the equation

(1681) |

Note that once the operations of raising and lowering suffixes has been defined, the order of raised suffixes relative to lowered suffixes becomes significant.

By analogy with Euclidian space, we define the *squared magnitude*
of a vector
with respect to the metric
by the equation

(1682) |

A vector termed a

(1683) |

Finally, let us consider differentiation with respect to an invariant distance, . The vector is a contravariant tensor, because

The derivative of some tensor with respect to is not, in general, a tensor, because

(1685) |

and, as we have seen, the first factor on the right-hand side is not generally a tensor. However, under linear transformations it behaves as a tensor, so under linear transformations the derivative of a tensor with respect to an invariant distance behaves as a tensor of the same type.