Antisymmetrization leads to an identically vanishing tensor

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In summary, Anderson's Principles of Relativity Physics states that for fifth- or higher-rank tensors, antisymmetrization will result in an identically vanishing tensor. This is because any tensor with repeated indices will be zero when antisymmetrized. Although this may not seem obvious at first, it is because tensors are defined as general geometrical objects and not specifically applied to the space-time manifold.
  • #1
jason12345
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This comes from Andersons's Principles of Relativity Physics:

"Of course, for fifth- or higher-rank tensors antisymmetrization leads to an
identically vanishing tensor"

But I don't understand why, even if it's "of course". So can someone show me why?
 
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  • #2
Hi jason12345! :smile:

Because any tensor Aabcde has elements which look like Axyztx (or worse), and if A is antisymmetric, then this must be zero. :wink:
 
  • #3


tiny-tim said:
Hi jason12345! :smile:

Because any tensor Aabcde has elements which look like Axyztx (or worse), and if A is antisymmetric, then this must be zero. :wink:

Antisymmetrising Aabcde means writing it as 1/5!(Aabcde + ... - Abacde -.. ) so that the indices abcde are merely permutated, whereas you have Axyztx which has indices x repeated. Why did you change the labelling from abcde to xyz and then t?

Thanks for your interest.
 
  • #4
Because the only candidates for the indices for the elements of the matrix are x y z and t (or 1 2 3 and 4, or whatever the four basis elements are) …

each element of the matrix has to have each of a b c d and e equal to x y z or t. :smile:
 
  • #5


tiny-tim said:
Because the only candidates for the indices for the elements of the matrix are x y z and t (or 1 2 3 and 4, or whatever the four basis elements are) …

each element of the matrix has to have each of a b c d and e equal to x y z or t. :smile:

Thanks, I understand what you're saying now :), although I still say it isn't obvious since Anderson was defining Tensors as general geometrical objects upto this point, it seems, rather than applying them to the space-time manifold.
 

What is antisymmetrization?

Antisymmetrization is a mathematical operation that rearranges the terms in a tensor, such that it is invariant under permutation of its indices.

What does it mean for a tensor to be identically vanishing?

A tensor is identically vanishing if all of its components are equal to zero. This means that the tensor has no physical significance and does not contribute to any equations or calculations.

Why does antisymmetrization lead to an identically vanishing tensor?

Antisymmetrization involves rearranging the terms in a tensor, which can result in some of the terms cancelling out due to the properties of the tensor. This ultimately leads to all components of the tensor being equal to zero.

What is the importance of understanding antisymmetrization and identically vanishing tensors?

Understanding these concepts is important in various fields of science, including mathematics, physics, and engineering. It allows for more efficient and accurate calculations and helps in identifying and simplifying complex equations.

Are there any real-world applications of antisymmetrization and identically vanishing tensors?

Yes, these concepts are used in various areas such as quantum mechanics, electromagnetism, and fluid dynamics. They also have applications in computer science, particularly in the study of algorithms and data structures.

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