Raising Index of Electromagnetic Energy Momentum Tensor

In summary, the General Relativity text being discussed presents two forms of the Electromagnetic Energy Momentum Tensor, one of which is incorrect according to the conversation. The correct form is derived in another source. Additionally, there is a discrepancy in the notation between different sections of the book.
  • #1
jstrunk
55
2
The General Relativity text I am using gives two forms of the Electromagnetic Energy Momentum Tensor:
[tex]{\rm{ }}\mu _0 S_{ij} = F_{ik} F_{jk} - \frac{1}{4}\delta _{ij} F_{kl} F_{kl} \\[/tex]
[tex]{\rm{ }}\mu _0 S_j^i = F^{ik} F_{jk} - \frac{1}{4}\delta _{ij} F^{kl} F_{kl} \\[/tex]

I don't see how these are equivalent. Raising one index on the left entitles you to raise one index on
each term on the right, but instead they raised two indexes on the right. Can anyone explain this?
 
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  • #2
jstrunk said:
The General Relativity text I am using gives two forms of the Electromagnetic Energy Momentum Tensor:
[tex]{\rm{ }}\mu _0 S_{ij} = F_{ik} F_{jk} - \frac{1}{4}\delta _{ij} F_{kl} F_{kl} \\[/tex]
[tex]{\rm{ }}\mu _0 S_j^i = F^{ik} F_{jk} - \frac{1}{4}\delta _{ij} F^{kl} F_{kl} \\[/tex]

I don't see how these are equivalent. Raising one index on the left entitles you to raise one index on
each term on the right, but instead they raised two indexes on the right. Can anyone explain this?
The first one has two covariant indexes called k in the rhs, also index l is down twice. That is not right.

The second equation is better , every term has the correct i, j indexes, except the ##\delta_{ij}## should be ##\delta^i_j##
 
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  • #3
The first equation which you say is wrong is the one that is derived in the book from the vector form of Maxwell's Equations.
The second equation is just stated 75 pages later as if it was an obvious consequence of the first.
I assumed the error was in the second equation but maybe I will have to check the derivation of the first equation again.
 
  • #4
jstrunk said:
The General Relativity text I am using

Which text?
 
  • #5
I am using Introduction to Tensor Calculus, Relativity and Cosmology by Lawden. He never actually derives the erroneous first equation but derives other things from it. I found a correct derivation that I can use but naturally every possible convention differs from Lawden so it was a bit of work to convert it.
 
  • #6
At this point in Lawden, he restricts to inertial frames in special relativity, and he uses the the ict convention, so, numerically (but not conceptually), there is no distinction between up and down indices.
 
  • #7
Lawden Section 28 says the Electromagnetic 4 - force density is [tex]D_i = F_{ij} J_j = \frac{1}{{\mu _0 }}F_{ij} F_{jk,k}[/tex].
In Section 56 it says [tex]D_i = F_{ij} J^i[/tex] which implies for flat space [tex]D_i = F_{ij} J^j = \frac{1}{{\mu _0 }}F_{ij} F_{,k}^{jk}[/tex].
Can someone verify that the statement from Section 28 is another mistake in the book and that [tex]D_i = F_{ij} J^j = \frac{1}{{\mu _0 }}F_{ij} F_{,k}^{jk}[/tex] is correct?
 
  • #8
jstrunk said:
Lawden Section 28 says the Electromagnetic 4 - force density is
..
..
[tex]D_i = F_{ij} J^j = \frac{1}{{\mu _0 }}F_{ij}{ F^{jk}}_{,k}[/tex] is correct?

That looks OK. The indexes are certainly well formed.
 

Related to Raising Index of Electromagnetic Energy Momentum Tensor

What is the "Raising Index of Electromagnetic Energy Momentum Tensor"?

The "Raising Index of Electromagnetic Energy Momentum Tensor" is a mathematical concept used in the study of electromagnetism. It refers to the process of converting a tensor with a lowered index into one with a raised index.

Why is the "Raising Index of Electromagnetic Energy Momentum Tensor" important?

This concept is important because it allows us to properly describe and analyze the behavior of electromagnetic fields and their interactions with matter. It also helps us understand the conservation of energy and momentum in these interactions.

How is the "Raising Index of Electromagnetic Energy Momentum Tensor" calculated?

The calculation involves using the metric tensor and the Levi-Civita symbol to raise the index of the electromagnetic energy momentum tensor. This is done through a specific mathematical equation that takes into account the properties of these tensors.

What are some practical applications of the "Raising Index of Electromagnetic Energy Momentum Tensor"?

This concept has applications in various fields such as astrophysics, particle physics, and engineering. It is used in the study of gravitational waves, black holes, and high-energy particle interactions. It also plays a role in the development of technologies such as MRI machines and particle accelerators.

Are there any limitations or criticisms of the "Raising Index of Electromagnetic Energy Momentum Tensor"?

While this concept is widely accepted and used in the scientific community, there are ongoing debates and discussions about its application in certain scenarios. Some critics argue that it may not accurately describe certain phenomena, such as the behavior of high-energy particles in extreme conditions. However, it remains a valuable tool in understanding the fundamental laws of electromagnetism.

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