- #1
Zag
- 49
- 9
Hello everyone,
I have recently read a puzzling statement on my Electromagnetism (Chapter on Special Relativity) material regarding the Field Strength Tensor, [itex]F^{\mu\nu}[/itex], and its dual, [itex]\tilde{F}^{\mu\nu}[/itex]. Since I've been thinking about this for a while now, and still can't understand it, I was hoping to hear your thoughts about it.
I believe we all agree that a possible definition for the Dual Tensor in terms of the original Field Strength Tensor is the following:
[itex]\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\sigma\rho}F_{\sigma\rho}[/itex]
Having that in mind, the text states the following:
"It can be shown that this is the only way in which we can construct a Lorentz-invariant four-tensor involving the fields that is independent of the original field strength tensor."
I can't understand why should this be the case, nor can I come up with a proof for this statement. Any thoughts on this matter would be greatly appreciated!
Thank you very much,
Zag
I have recently read a puzzling statement on my Electromagnetism (Chapter on Special Relativity) material regarding the Field Strength Tensor, [itex]F^{\mu\nu}[/itex], and its dual, [itex]\tilde{F}^{\mu\nu}[/itex]. Since I've been thinking about this for a while now, and still can't understand it, I was hoping to hear your thoughts about it.
I believe we all agree that a possible definition for the Dual Tensor in terms of the original Field Strength Tensor is the following:
[itex]\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\sigma\rho}F_{\sigma\rho}[/itex]
Having that in mind, the text states the following:
"It can be shown that this is the only way in which we can construct a Lorentz-invariant four-tensor involving the fields that is independent of the original field strength tensor."
I can't understand why should this be the case, nor can I come up with a proof for this statement. Any thoughts on this matter would be greatly appreciated!
Thank you very much,
Zag