I guess it should be an easy question and I'm just missing smth, but I already spent on it much time and didn't get the answer. Here's a little quote from Landau's book "The Classical Theory of Fields" (2nd volume of 'Theoretical Physics' series)(adsbygoogle = window.adsbygoogle || []).push({});

"With respect to rotations of the coordinate system, the quantities e_iklm (iklm in the superscript) behave like the components of a tensor; but if we change the sign of one or three of the coordinates the components of that tensor, being defined as the same in all coordinate systems, do not change, whereas some of the components of a tensor should change sign. Thus e_iklm (iklm again in superscript)" is strictly speaking not a tensor, but rather a pseudotensor. Pseudotensors of any rank, in particular pseudoscalars, behave like tensors under all coordinate transformations except those that cannot be reduced to rotations, i.e. reflections, which are changes in sign of the coordinates that are not reducible to a rotation."

Now when I'm trying to consider the reflection of only one coordinte axis, which means the transformation matrix is A_ik=(1,1,1,-1) on main diagonal and all the rest 0s, trying to see how Levi-Chevita tensor (pseudotensor) and Kroneker delta (real tensor) transform, i.e. A_ip*A_kr*A_ls*A_mt*e_prst (again prst in the superscript, sorry I don't know how to put it there), I come to the exact opposite, some elements of the pseudotensors do change their signs, tensors do not!

Hope someone could help me, I'm close to despair :) Any help will be much appreciated

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# Tensor transformation under reflections

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