3.1 Symmetry 35
Again, notice that summation indices can have different names, as long as no index
appears more than twice.
The trace of the unit tensor is equal to the dimension D, here D=3. Thus
δνν= 3. (3.8)This is the reason why the fraction^13 occurs in (3.3) and (3.4).
3.1.4 Multiplication and Total Contraction of Tensors, Norm
The multiplication of a tensorAμνwith a tensorBλκyields a fourth rank tensor. The
contraction withν=λ, corresponding to a “dot-product”A·B, gives a second rank
tensor. The total contraction or “double dot-product”
A:B=AμνBνμ (3.9)is a scalar. The order of the indices is such that it corresponds to the trace of the
matrix product ofAwithB.
In such a total contraction, the symmetry of one tensor is imposed on the other
one. This means, e.g. whenAis symmetric, the symmetric part ofBonly contributes
in the productAμνBνμ. Likewise, whenAis antisymmetric, the antisymmetric part
ofBonly contributes in the productAμνBνμ. Furthermore, whenAis isotropic, i.e.
proportional to the unit tensor, then the trace ofBonly contributes to the product.
WhenAis symmetric traceless, then only the symmetric traceless part ofBgives a
contribution. With both tensors decomposed according to (3.4), one obtains
AμνBνμ=1
3
AλλBκκ+AasyμνBνμasy+Aμν Bνμ. (3.10)Notice that one hasAμνBνμ=AμνBμνonly when at least one of the two tensors is
symmetric.
The square of the norm or magnitude|A|of a second rank tensor is determined
by the total contraction ofAwith its transposedAT, viz.:
|A|^2 =AμνATνμ=AμνAμν. (3.11)Of course, the order of the subscript does not matter whenAis symmetric.