I am reading Landau and Lifgarbagez's Classical Theory of Fields, 4th edition. In the beginning of page 18, the completely antisymmetric unit tensor is said to be a pseudotensor, because none of it components changes sign when we change the sign of one or three of the coordinates. Then, in the 2nd paragraph, the product [tex]e^{iklm}e^{prst}[/tex] is a tensor of rank 8 and it is a true tensor! Why? We know that [tex]e^{iklm}[/tex] does not change sign when one of the coordinates changes its sign. Either does [tex]e^{prst}[/tex]. Then the product does change its sign either. How could it be possible that the product is a true tensor? I totally cannot understand. I need your help, your hints. Thank you!
A pseudotensor has the determinant of the LT included in its transformation. This gives a minus sign compare to the transformation of a true tensor. If a pseudotensor is combined with another pseudotensor, the determinant is squared and always gives +1.