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Levi-Civita symbol

by qinglong.1397
Tags: levicivita, symbol
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qinglong.1397
#1
Jan15-12, 09:46 PM
P: 107
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!
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clem
#2
Jan16-12, 09:29 AM
Sci Advisor
P: 1,256
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.
qinglong.1397
#3
Jan16-12, 09:37 AM
P: 107
Quote Quote by clem View Post
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.
What does LT stand for? Thank you!

haushofer
#4
Jan16-12, 11:03 AM
Sci Advisor
P: 887
Levi-Civita symbol

Lorentz Transformation, or Leprous Tyrannosaurus. I think here the first one is ment.
qinglong.1397
#5
Jan16-12, 07:35 PM
P: 107
Quote Quote by haushofer View Post
Lorentz Transformation, or Leprous Tyrannosaurus. I think here the first one is ment.
Should be the first one. Is the second one an English name?
Fredrik
#6
Jan16-12, 08:18 PM
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Quote Quote by qinglong.1397 View Post
Should be the first one. Is the second one an English name?
He was joking. A leprous tyrannosaurus would be a tyrannosaurus with leprosy.
qinglong.1397
#7
Jan16-12, 08:57 PM
P: 107
Quote Quote by Fredrik View Post
He was joking. A leprous tyrannosaurus would be a tyrannosaurus with leprosy.
Ha ha~~ That disease is scary...


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