Wheeler, Mechanics

many objects besides scalars that are useful, including many that are invariant. For example, a cylindrically
symmetric system may be characterized by an invariant vector along the symmetry axis, but measurement
of that vector relies on forming scalars from that vector.
The most important class of non-scalars are the tensors. There are two principal reasons that tensors
are useful. First, their transformation properties are so simple that it is easy to construct scalars from them.
Second, the form of tensor equations is unchanged by transformation. Specifically, tensors are those objects
which transform linearly and homogeneously under the action of a group symmetry (Λ), or the under inverse
action of the group symmetry (Λ −1 ). This linear, homogeneous transformation property is called covariance.
If we write, schematically,
T ′ = ΛT
S ′ = SΛ −1
for some tensors of each type, then it is immediate that combining such a pair gives a scalar, or invariant
quantity,
S ′ T ′ = SΛ −1 ΛT = ST
It is also immediate that tensor equations are covariant. This means that the form of tensor equations does
not change when the system is transformed. Thus, if we arrange any tensor equation to have the form
T = 0
where T may be an arbitrarily complicated tensor expression, we immediately have the same equation after
transformation, since
T ′ = ΛT = 0
Knowing the symmetry and associated tensors of a physical system we can quickly go beyond the dynamical
law in making predictions by asking what other objects besides the dynamical law are preserved by
the transformations. Relations between these covariant objects express possible physical relationships, while
relationships among other, non-covariant quantities, will not.
4.1 Examples of tensors
Before proceeding to a formal treatment of tensors, we provide some concrete examples of scalars, of vector
transformations, and of some familiar second rank tensors.
4.1.1 Scalars and non-scalars
If we want to describe a rod, its length is a relevant feature because its length is independent of what
coordinate transformations we perform. However, it isn’t reasonable to associate the change, ∆x, in the x
coordinate between the ends of the rod with anything physical because as the rod moves around ∆x changes
arbitrarily. Tensors allow us to separate the properties like length,
√
L = (∆x) 2 + (∆y) 2 + (∆z) 2
from properties like ∆z; invariant quantities like L can be physical but coordinate dependent quantities like
∆z cannot.
There are many kinds of objects that contain physical information. You are probably most familiar with
numbers, functions and vectors. A function of position such as temperature has a certain value at each
point, regardless of how we label the point. Similarly, we can characterize vectors by their magnitude and
direction relative to other vectors, and this information is independent of coordinates. But there are other
objects that share these properties.
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