Levi-Civita Symbol: Understanding Rank 8 Tensor Properties

In summary, in the book Classical Theory of Fields, 4th edition, it is noted that the completely antisymmetric unit tensor is considered a pseudotensor due to its transformation properties. However, the product of this pseudotensor with another pseudotensor results in a true tensor, despite the pseudotensor having a determinant that includes a minus sign in its transformation. This is because the determinant is squared when combining pseudotensors, always resulting in a positive sign. The abbreviation LT stands for Lorentz Transformation, not Leprous Tyrannosaurus as jokingly suggested.
  • #1
qinglong.1397
108
1
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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  • #2
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
clem said:
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!
 
  • #4
Lorentz Transformation, or Leprous Tyrannosaurus. I think here the first one is ment.
 
  • #5
haushofer said:
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?
 
  • #6
qinglong.1397 said:
Should be the first one. Is the second one an English name?
He was joking. A leprous tyrannosaurus would be a tyrannosaurus with leprosy.
 
  • #7
Fredrik said:
He was joking. A leprous tyrannosaurus would be a tyrannosaurus with leprosy.

Ha ha~~ That disease is scary...
 

1. What is the Levi-Civita symbol and how is it used in tensor analysis?

The Levi-Civita symbol, also known as the permutation symbol, is a mathematical tool used in tensor analysis to identify the orientation of a coordinate system. It is a rank 3 tensor with a value of 1, -1, or 0 depending on the order of the indices. This symbol is used to simplify the expression of tensor equations and to determine the properties of tensors.

2. How does the Levi-Civita symbol relate to the properties of rank 8 tensors?

The Levi-Civita symbol is a rank 3 tensor, meaning it has three indices. In the case of rank 8 tensors, the Levi-Civita symbol is raised to the power of 8, resulting in a rank 24 tensor. This tensor is used to determine the properties of rank 8 tensors, such as symmetry, tracelessness, and isotropy.

3. Can the Levi-Civita symbol be extended to higher rank tensors?

Yes, the Levi-Civita symbol can be extended to higher rank tensors. The number of indices in the symbol will depend on the rank of the tensor it is being applied to. For example, a rank 10 tensor would require a Levi-Civita symbol with 10 indices.

4. How is the Levi-Civita symbol used in physics and engineering?

The Levi-Civita symbol is used in various areas of physics and engineering, including electromagnetism, fluid mechanics, and general relativity. It allows for the concise expression of tensor equations and helps to determine the properties of physical systems, such as the magnetic field or stress tensor.

5. Are there any alternative symbols or methods for understanding rank 8 tensor properties?

Yes, there are alternative symbols and methods for understanding rank 8 tensor properties. The Kronecker delta symbol, for example, is commonly used in conjunction with the Levi-Civita symbol to simplify tensor equations. Additionally, there are other mathematical tools and techniques, such as matrix representations and transformation laws, that can be used to understand the properties of tensors.

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