Tensors for Physics

34 3 Symmetry of Second Rank Tensors, Cross Product

5.1 Isotropic and Symmetric Traceless Parts

The symmetric part of a second rank tensorAcan be decomposed further into an
isotropic partproportional to the product of the isotropic tensorδand thetrace

trA=Aλλand a symmetric traceless partAdefined by

Aμν =

1

2

(

Aμν+Aνμ

)

−

1

3

Aλλδμν. (3.3)

Thus the tensorAis decomposed into its isotropic, antisymmetric and symmetric
traceless parts according to

Aμν=

1

3

Aλλδμν+Aasyμν+Aμν. (3.4)

This decomposition is invariant under a rotation of the coordinate system.
The symbol...used to indicate the symmetric traceless part of a tensor, was
introduced by Ludwig Waldmann around 1960. Compared with the double arrow
←→, which also occurs in printing, the...has the advantage that it can be drawn
in one stroke. For second rank tensors,...first appeared in print in [20], and in
[21], it was applied for irreducible tensors of any rank. Alternative notations used in
the literature for symmetric traceless tensors are mentioned in Sect.3.1.7.

3.1.3 Trace of a Tensor

The isotropic part involves thetrace of the tensor

tr(A)=Aλλ=A 11 +A 22 +A 33. (3.5)

It is a scalar (tensor of rank=0), i.e., it is invariant under a rotation of the
coordinate system. The proof is: the tensor propertyA′μν=UμκUνλAκλimplies
A′μμ =UμκUμλAκλ, and due to the orthogonality (2.31) of the transformation
matrix, one hasA′μμ=δκλAκλ=Aκκ.
The termisotropicis used since the unit tensorδμνhas no directional properties,
it is not affected by a rotation of the coordinate system. Here the other orthogonality
(2.36) is used for the proof:δ′μν=UμκUνλδκλ=UμλUνλ=δμν.
Notice: the antisymmetric part of the tensor does not contribute to the trace:

tr(A)=Aλλ=A

sym

λλ =tr(A

sym). (3.6)

The trace of a second rank tensor is also given by the total contraction of this tensor
with the unit tensor:

tr(A)=δμνAμν=δμνAνμ=Aνν. (3.7)