# Levi-Civita symbol

1. Jan 15, 2012

### qinglong.1397

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 $$e^{iklm}e^{prst}$$ is a tensor of rank 8 and it is a true tensor! Why?

We know that $$e^{iklm}$$ does not change sign when one of the coordinates changes its sign. Either does $$e^{prst}$$. 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!

2. Jan 16, 2012

### clem

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.

3. Jan 16, 2012

### qinglong.1397

What does LT stand for? Thank you!

4. Jan 16, 2012

### haushofer

Lorentz Transformation, or Leprous Tyrannosaurus. I think here the first one is ment.

5. Jan 16, 2012

### qinglong.1397

Should be the first one. Is the second one an English name?

6. Jan 16, 2012

### Fredrik

Staff Emeritus
He was joking. A leprous tyrannosaurus would be a tyrannosaurus with leprosy.

7. Jan 16, 2012

### qinglong.1397

Ha ha~~ That disease is scary...