- #1
timeant
- 16
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From Maxwell's equations [itex] \partial_\nu F^{\mu\nu}=J^{\mu},[/itex] one can derive charge conservation. The derivation is
[tex] 0\equiv \partial_\mu \partial_\nu F^{\mu\nu}= \partial_\mu J^{\mu} { \Rightarrow}\partial_\mu J^{\mu}=0. [/tex]
However, a circular reasoning exists in it. For the sake of better understanding, we suppose [itex] F^{kl} [/itex] is an antisymmetric n-dimenstional (n > 2) tensor. We consider the following equation
[tex] \partial_l F^{kl}= J^{k}, \qquad n=3,4,5,\cdots \qquad (\star) [/tex]
Where [itex] J^{k} [/itex] is known source. If the source is chosen as [itex] \partial_k J^{k}\neq 0 (e.g. J^{k} \propto x^k) [/itex], then the above equation(*) has no solutions. Hence, [itex] \partial_k J^{k}= 0 [/itex] is one of preconditions of existence about solutions of the above equation (*). If [itex] \partial_k J^{k}= 0 [/itex] is considerd as a corollary of Eq.(*) ([itex] 0\equiv \partial_k \partial_l F^{kl}= \partial_k J^{k} { \Rightarrow}\partial_k J^{k}=0 [/itex]), and at the same time it is one of preconditions of existence about Eq.(*)'s solutions. It must involve circular reasoning. Therefore, [itex] \partial_k J^{k}= 0 [/itex] is NOT a corollary of Eq.(*) for any n. When n=4, Eq(*) is one of Maxwell equations.
Hence the charge conservation law can NOT be derived from Maxwell equations.
[tex] 0\equiv \partial_\mu \partial_\nu F^{\mu\nu}= \partial_\mu J^{\mu} { \Rightarrow}\partial_\mu J^{\mu}=0. [/tex]
However, a circular reasoning exists in it. For the sake of better understanding, we suppose [itex] F^{kl} [/itex] is an antisymmetric n-dimenstional (n > 2) tensor. We consider the following equation
[tex] \partial_l F^{kl}= J^{k}, \qquad n=3,4,5,\cdots \qquad (\star) [/tex]
Where [itex] J^{k} [/itex] is known source. If the source is chosen as [itex] \partial_k J^{k}\neq 0 (e.g. J^{k} \propto x^k) [/itex], then the above equation(*) has no solutions. Hence, [itex] \partial_k J^{k}= 0 [/itex] is one of preconditions of existence about solutions of the above equation (*). If [itex] \partial_k J^{k}= 0 [/itex] is considerd as a corollary of Eq.(*) ([itex] 0\equiv \partial_k \partial_l F^{kl}= \partial_k J^{k} { \Rightarrow}\partial_k J^{k}=0 [/itex]), and at the same time it is one of preconditions of existence about Eq.(*)'s solutions. It must involve circular reasoning. Therefore, [itex] \partial_k J^{k}= 0 [/itex] is NOT a corollary of Eq.(*) for any n. When n=4, Eq(*) is one of Maxwell equations.
Hence the charge conservation law can NOT be derived from Maxwell equations.
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