Tensors, defined mathematically, are simply arrays of numbers, or functions, that transform
according to certain rules under a change of coordinates. In physics, tensors characterize the
properties of a physical system, as is best illustrated by giving some examples (below).
A tensor may be defined at a single point or collection of isolated points of space (or space-time),
or it may be defined over a continuum of points. In the latter case, the elements of the tensor
are functions of position and the tensor forms what is called a tensor field. This just means that
the tensor is defined at every point within a region of space (or space-time), rather than just at a
point, or collection of isolated points.
A tensor may consist of a single number, in which case it is referred to as a tensor of order zero,
or simply a scalar. For reasons which will become apparent, a scalar may be thought of as an array
of dimension zero (same as the order of the tensor).
An example of a scalar would be the mass of a particle or object. An example of a scalar field
would be the density of a fluid as a function of position. A second example of a scalar field
would be the value of the gravitational potential energy as a function of position. Note that both
of these are single numbers (functions) that vary continuously from point-to-point, thereby
defining a scalar field.
The next most complicated tensor is the tensor of order one, otherwise known as a vector. Just as
tensors of any order, it may be defined at a point, or points, or it may vary continuously from
point-to-point, thereby defining a vector field. In ordinary three dimensional space, a vector has
three components (contains three numbers, or three functions of position). In four dimensional
space-time, a vector has four components. And, generally, in an n-dimensional space, a vector
(tensor of order one) has n components. A vector may be thought of as an array of dimension one.
This is because the components of a vector can be visualized as being written in a column or along
a line, which is one dimensional.
An example of a vector field is provided by the description of an electric field in space. The
electric field at any point requires more than one number to characterize because it has both a
magnitude (strength) and it acts along a definite direction, something not shared with a scalar,
such as mass. Generally, both the magnitude and the direction of the field vary from
point-to-point.
As might be suspected, tensors can be defined to all orders. Next above a vector are tensors of
order 2, which are often referred to as matrices. As might also be guessed, the components of a
second order tensor can be written as a two dimensional array.. Just as vectors represent physical
properties more complex than scalars, so too matrices represent physical properties yet more
complex than can be handled by vectors.
An example of a second order tensor is the so-called inertia matrix (or tensor) of an object. For
three dimensional objects, it is a 3 x 3 = 9 element array that characterizes the behavior of a
rotating body. As is well known to anyone who has played with a toy gyroscope, the response of a
gyroscope to a force along a particular direction (described by a vector), is generally
re-orientation along some other direction different from that of the applied force or torque.
Thus, rotation must be characterized by a mathematical entity more complex than either a scalar or
a vector; namely, a tensor of order two.
There are yet more complex phenomena that require tensors of even higher order. For example, in
Einstein's General Theory of Relativity, the curvature of space-time, which gives rise to gravity,
is described by the so-called Riemann curvature tensor, which is a tensor of order four. Since it
is defined in space-time, which is four dimensional, the Riemann curvature tensor can be represented
as a four dimensional array (because the order of the tensor is four), with four components
(because space-time is four dimensional) along each edge. That is, in this case, the Riemann
curvature tensor has 4 x 4 x 4 x 4 = 256 components! [Fortunately, it turns out that only 20 of
these components are mathematically independent of each other, vastly simplifying the solution of
Einstein's equations].
Finally, to return to the comment that tensors transform according to certain rules under a change
of coordinates, it should be remarked that other mathematical entities occur in physics that, like
tensors, generally consist of multi-dimensional arrays of numbers, or functions, but that are NOT
tensors. Most noteworthy are objects called spinors. Spinors differ from tensors in how the
values of their elements change under coordinate transformations. For example, the values of the
components of all tensors, regardless of order, return to their original values under a 360-degree
rotation of the coordinate system in which the components are described. By contrast, the
components of spinors change sign under a 360-degree rotation, and do not return to their original
values until the describing coordinate system has been rotated through two full rotations =
720-degrees!
Answered by: Warren Davis, Ph.D., President, Davis Associates, Inc., Newton, MA USA
'As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.'