There's a derivation of this at http://en.wikipedia.org/wiki/Stress_Tensor#Equilibrium_equations_and_symmetry_of_the_stress_tensor, followed by:[tex]
e_{ijk}\sigma_{jk}=0
[/tex]
However, in the presence of couple-stresses, i.e. moments per unit volume, the stress tensor is non-symmetric. This also is the case when the Knudsen number is close to one, K_{n} -> 1, e.g. Non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers.
Symmetry under spacetime translations implies (by Noether theorem) that the canonical energy-momentum (or stress) tensorCan someone explain to me why the stress tensor is symmetrical. I understand that Sij=Sji , but can someone give me the assumption or the physical reason why this is true.
[tex]
T_{ab} = \frac{\partial L}{\partial \partial_{a} \phi} \partial_{b} \phi - \eta_{ab}L
[/tex]
is conserved;
[tex] \partial^{a} T_{ab} = 0[/tex]
But it is not, in general, symmetric! Well, it is not unique either, for you could define a new tensor
[tex]
\Theta_{ab} = T_{ab} + \partial^{c} X_{cab}
[/tex]
which is also conserved, [itex]\partial^{a}\Theta_{ab} = 0[/itex], provided that
[tex]X_{cab} = - X_{acb}[/tex]
In a Lorentz invariant theories, we may choose [itex]X_{cab}[/itex] to make [ the new stress tensor] [itex]\Theta_{ab}[/itex] symmetric.
So, your question should have been: Why do we want the stress tensor to be symmetric?
There are two reasons for this:
1) In general relativity, the matter fields couple to gravity via the stress tensor and this is given by the Einstein equations
[tex]R_{ab} - \frac{1}{2} g_{ab} R = - k \Theta_{ab}[/tex]
Since the (geometrical) Ricci tensor [itex]R_{ab}[/itex] and the metric tensor [itex]g_{ab}[/itex] are both symmetric, so [itex]\Theta_{ab}[/itex] must be also.
2) The second reason for requiring a symmetric stress tensor comes from Lorentz symmetry:
Lorentz invariance implies that the ungular momentum tensor;
[tex]\mathcal{M}_{cab} = \Theta_{ca} x_{b} - \Theta_{cb} x_{a}[/tex]
is conserved! But
[tex]\partial^{c} \mathcal{M}_{cab} = \Theta_{ab} - \Theta_{ba}[/tex]
Thus, conservation of ungular momentum requires the stress tensor to be symmetric;
[tex]\Theta_{ab} = \Theta_{ba}[/tex]
regards
sam