Proving Antisymmetry of Electromagnetic Field Tensor with 4-Force

[Little Gravity]

TL;DR

Trying to proove the antisymmetry of Electromagnetic Field Tensor via the orthogonality property of 4-Force with respect the 4-velocity of some particle

I've already made a post about this topic here, but I realized that I didn't understand the explanation on that post. in Chapter 7 of Rindler's book on relativity, in section about electromagnetic field tensor, he states that

_and introducing a factor 1/c for later convenience, we can ‘guess’ the tensor equation_, $$ F_\mu= \frac{q}{c} E_{\mu \nu} U^\nu$$
_thereby introducing the electromagnetic field tensor_$$E_{\mu \nu}$$
_We would surely want the
force $F\mu$ to be rest-mass preserving, which, according to (6.44) and (7.15), requires_
$$F_\mu U^\mu = 0$$. _So we need_
$$E_{\mu \nu} U^\mu U^\nu = 0$$
_for all $ U^\mu$ , and hence the antisymmetry of the field tensor_
$$E_{\mu \nu}= −E_{\nu \mu}$$\\

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I'm really confused about the correct way to show that the equation $$E_{\mu \nu} U^\mu U^\nu = 0$$ implies the fact that $$E_{\mu\nu}$$ is antisymmetric tensor. What is the correct demonstration of this implication?

OBS: I've saw some posts answering this kind of question with bilinear maps notation, instead of component notation. If possible, please make some demonstration using the index notation as in the post.

Another OBS: I'm NOT trying do proove that antisymmetry implies the null equation. I'm trying to proove that the null equation implies the antisymmetry of the field tensor.

 

[PeterDonis]

What is the correct demonstration of this implication?

The dyad $U^\mu U^\nu$ is obviously symmetric. So the only way for $E_{\mu \nu} U^\mu U^\nu = 0$ to always hold is for $E_{\mu \nu}$ to be antisymmetric.

 

[anuttarasammyak]

We can write
$$E_{\mu\nu}=S_{\mu\nu} + A_{\mu\nu}$$
where S is symmetric tensor and A is antisymmetric tensor.

$$E_{\mu\nu}U^\mu U^\nu=S_{\nu\mu}U^\nu U^\mu+A_{\nu\mu}U^\nu U^\mu$$
$$E_{\nu\mu}U^\nu U^\mu=S_{\mu\nu}U^\mu U^\nu+A_{\mu\nu}U^\mu U^\nu$$

Summing the both sides
$$E_{\mu\nu}U^\mu U^\nu+E_{\nu\mu}U^\nu U^\mu=2S_{\mu\nu}U^\mu U^\nu=0$$

$$S_{\mu\nu}=0$$
Only anti-symmetric component $A_{\mu\nu}$ survives.

EDIT: I found I have failed to delete my previous wrong post #2. Please disregard it.

 

[PeterDonis]

I found I have failed to delete my previous wrong post #2.

You can always ask a moderator to delete a post if it is outside your edit window. I have just deleted your post #2.

 

[PeterDonis]

Only anti-symmetric component $A_{\mu \nu}$ survives.

As long as you make use of the fact, which you have not explicitly stated, that $U^\mu U^\nu$ is symmetric. You have implicitly used that fact in your second, third, and fourth equations. And if you make use of that fact, you have a much simpler argument that gets you to the same conclusion--the argument I gave in my first post in this thread (which is now post #2).