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However, in three-dimensional continuum mechanics, the stress tensor takes a normal vector as an input and gives a stress vector as an output. In three dimensions, a normal vector is an axial vector (even under parity), while a stress vector is a true vector (odd under parity). Therefore it seems that the stress 3-tensor must be odd under parity, which makes it not a real tensor.

Is this analysis correct? I'm used to thinking of the stress 3-tensor as a block of elements in the stress-energy tensor, when they're expressed in Minkowski coordinates. Doesn't that imply that they should have the same parity properties?