Why shear stress components of the Stress Energy tensor not zero?

[epovo]

Hi,
I am having trouble understanding why T^ij can be non-zero for i≠j. T^ij is the flux of the i-th component of momentum across a surface of constant x^j. Isn't the i-th component of momentum tangent to the surface of constant x^j and therefore its flux across that surface zero? What am I missing here?

 

[pervect]

The principle axis theorem says that there are some basis vectors for which $T^{ij}$ is diagonal and hence $T^{ij}$ zero for i $\neq$ j, but the basis vectors are not necessarily aligned with the principle axes and thus in general the off-diagonal entries can be nonzero. Consider the analogous case of the 3x3 Newtonian moment of inertia tensor, for instance.

 

[epovo]

I don't understand your answer, pervect. Why would we express the components of four-momentum in a basis vectors not aligned with the coordinates that define the surfaces of constant xⁱ? I have not been able to find a word of warning to that effect in any text...

 

[Bill_K]

T^ij is the flux of the i-th component of momentum across a surface of constant x^j.

No, you have it slightly backwards! T^ij is the i-th component of the flux of the momentum across the surface of constant x^j. The momentum flux across the surface is a vector that can point in any direction. T^ij is its i-th component, and there is no reason for it to be zero.

 

[epovo]

OMG! I get it now. Thank you very much!