Raising Index of Electromagnetic Energy Momentum Tensor

[jstrunk]

The General Relativity text I am using gives two forms of the Electromagnetic Energy Momentum Tensor:
$${\rm{ }}\mu _0 S_{ij} = F_{ik} F_{jk} - \frac{1}{4}\delta _{ij} F_{kl} F_{kl} \\$$
$${\rm{ }}\mu _0 S_j^i = F^{ik} F_{jk} - \frac{1}{4}\delta _{ij} F^{kl} F_{kl} \\$$

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?

 

[Mentz114]

The General Relativity text I am using gives two forms of the Electromagnetic Energy Momentum Tensor:
$${\rm{ }}\mu _0 S_{ij} = F_{ik} F_{jk} - \frac{1}{4}\delta _{ij} F_{kl} F_{kl} \\$$
$${\rm{ }}\mu _0 S_j^i = F^{ik} F_{jk} - \frac{1}{4}\delta _{ij} F^{kl} F_{kl} \\$$

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 $\delta^i_j$

 

[jstrunk]

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.

 

[PeterDonis]

The General Relativity text I am using

Which text?

 

[jstrunk]

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.

 

[George Jones]

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.

 

[jstrunk]

Lawden Section 28 says the Electromagnetic 4 - force density is $$D_i = F_{ij} J_j = \frac{1}{{\mu _0 }}F_{ij} F_{jk,k}$$.
In Section 56 it says $$D_i = F_{ij} J^i$$ which implies for flat space $$D_i = F_{ij} J^j = \frac{1}{{\mu _0 }}F_{ij} F_{,k}^{jk}$$.
Can someone verify that the statement from Section 28 is another mistake in the book and that $$D_i = F_{ij} J^j = \frac{1}{{\mu _0 }}F_{ij} F_{,k}^{jk}$$ is correct?

 

[Mentz114]

Lawden Section 28 says the Electromagnetic 4 - force density is
..
..
$$D_i = F_{ij} J^j = \frac{1}{{\mu _0 }}F_{ij}{ F^{jk}}_{,k}$$ is correct?

That looks OK. The indexes are certainly well formed.