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

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  • #1
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Hi,
I am having trouble understanding why Tij can be non-zero for i≠j. Tij is the flux of the i-th component of momentum across a surface of constant xj. Isn't the i-th component of momentum tangent to the surface of constant xj and therefore its flux across that surface zero? What am I missing here?
 

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  • #2
pervect
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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.
 
  • #3
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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...
 
  • #4
Bill_K
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Tij is the flux of the i-th component of momentum across a surface of constant xj.
No, you have it slightly backwards! Tij is the i-th component of the flux of the momentum across the surface of constant xj. The momentum flux across the surface is a vector that can point in any direction. Tij is its i-th component, and there is no reason for it to be zero.
 
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  • #5
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OMG! I get it now. Thank you very much!!!
 

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