Page 84 - Intro to Tensor Calculus

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Associated Tensors

Associated tensors can be constructed by taking the inner product of known tensors with either the
metric or conjugate metric tensor.

Definition: (Associated tensor) Any tensor constructed by multiplying (inner
product) a given tensor with the metric or conjugate metric tensor is called an
associated tensor.

Associated tensors are different ways of representing a tensor. The multiplication of a tensor by the
metric or conjugate metric tensor has the effect of lowering or raising indices. For example the covariant
and contravariant components of a vector are different representations of the same vector in different forms.
These forms are associated with one another by way of the metric and conjugate metric tensor and

j
ij
g A i = A j g ij A = A i .
EXAMPLE 1.3-7. The following are some examples of associated tensors.

j ij i
A = g A i A j = g ij A
A m = g mi A ijk A i.k = g mj A ijk
m
.jk
.nm mk nj i
A i.. = g g A ijk A mjk = g im A .jk
Sometimes ‘dots’are used as indices in order to represent the location of the index that was raised or lowered.
If a tensor is symmetric, the position of the index is immaterial and so a dot is not needed. For example, if
A mn is a symmetric tensor, then it is easy to show that A n and A .n are equal and therefore can be written
.m
m
as A n without confusion.
m
Higher order tensors are similarly related. For example, if we find a fourth order covariant tensor T ijkm
we can then construct the fourth order contravariant tensor T pqrs from the relation

pi qj rk sm
T pqrs = g g g g T ijkm .

This fourth order tensor can also be expressed as a mixed tensor. Some mixed tensors associated with
the given fourth order covariant tensor are:

qj
pi
T p = g T ijkm , T pq = g T p .
.jkm ..km .jkm