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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, $$ F_\mu= \frac{q}{c} E_{\mu \nu} U^\ \nu$$$$ F_\mu= \frac{q}{c} E_{\mu \nu} U^\nu$$ 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 $$F\mu U^\mu = 0$$$$F_\mu U^\mu = 0$$. So we need $$E_{\mu \nu} U^\mu U^\nu = 0$$ for all $ U^\mu$ , and hence the antisymmetry of the field tensor $$E_{\mu \nu}= −E_{\nu \mu}$$\

. . . 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 e 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.

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 $$E_{\mu \nu}= −E_{\nu \mu}$$\

. . . 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 e 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.

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 $$E_{\mu \nu}= −E_{\nu \mu}$$\

. . . 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.

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Генивалдо
Генивалдо
  • 1.3k
  • 7
  • 17

Demonstration of Electromagnetic Tensor antisymmetry

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 $$E_{\mu \nu}= −E_{\nu \mu}$$\

. . . 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 e 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.