A Maxwell's equation in differential forms formalism

[cosmic dust]

Homework Statement

This is not actually a homework but a personal work. Here it is:
Using the differential forms:

$F=\tfrac{1}{2!}{{F}_{\mu \nu }}d{{x}^{\mu }}\wedge d{{x}^{\nu }}$ and $J=\tfrac{1}{3!}{{J}^{\mu }}{{\varepsilon }_{\mu \alpha \beta \gamma }}d{{x}^{\alpha }}\wedge d{{x}^{\beta }}\wedge d{{x}^{\gamma }}$

proof that the Maxwell equation: ${{F}^{\mu \nu }}_{,\mu }={{\mu }_{0}}{{J}^{\nu }}$ can be written as:

$d*F={{\mu }_{0}}J$

Homework Equations

$*F\equiv \tfrac{1}{4}{{F}^{\mu \nu }}{{\varepsilon }_{\mu \nu \rho \sigma }}d{{x}^{\rho }}\wedge d{{x}^{\sigma }}$

$d*F=\tfrac{1}{4}{{F}^{\mu \nu }}_{,a}{{\varepsilon }_{\mu \nu \rho \sigma }}d{{x}^{\alpha }}\wedge d{{x}^{\rho }}\wedge d{{x}^{\sigma }}$

The Attempt at a Solution

I have tried several ways to do this but they either fail (not giving the desired result) or get to a point where I cannot proceed any further.

Here are some attempts:
A) I multiplied both sides of the given equation with ${{\varepsilon }_{\nu \alpha \beta \gamma }}d{{x}^{\alpha }}\wedge d{{x}^{\beta }}\wedge d{{x}^{\gamma }}$ in order to make the current 3-form to appear in the RHS and then I tried to manipulate the LHS of the equation in order to make the $d*F$ to appear. This is the process:

${{F}^{\mu \nu }}_{,\mu }={{\mu }_{0}}{{J}^{\nu }}\Rightarrow {{F}^{\mu \nu }}_{,\mu }{{\varepsilon }_{\nu \alpha \beta \gamma }}d{{x}^{\alpha }}\wedge d{{x}^{\beta }}\wedge d{{x}^{\gamma }}=3!J$

As you can see, only one of the indices of Levi-Civita Symbol (LCS) contracts with the one indices of the field tensor (in contrast with the $d*F$ expression where two indices contract with the two indices of the field tensor) and also in the $d*F$ expression the derivative contracts with a basis 1-form while in the above equation the derivative contracts with one indice of the field tensor. In order to bypass the problem, I tried to decouple those disturbing contractions using Kronecker deltas:

${{F}^{\mu \nu }}_{,\mu }{{\varepsilon }_{\nu \alpha \beta \gamma }}d{{x}^{\alpha }}\wedge d{{x}^{\beta }}\wedge d{{x}^{\gamma }}={{\delta }_{\mu }}^{\rho }{{\delta }_{\alpha }}^{\sigma }{{F}^{\mu \nu }}_{,\rho }{{\varepsilon }_{\nu \sigma \beta \gamma }}d{{x}^{\alpha }}\wedge d{{x}^{\beta }}\wedge d{{x}^{\gamma }}$

I don’t know how to proceed further, in order to get the desired contractions. I tried to represent the deltas with LCSs, like this:

${{\varepsilon }_{\rho \sigma \mu \alpha }}{{\varepsilon }^{\rho \sigma \mu \nu }}=3!{{\delta }_{\alpha }}^{\nu }$

but then the equation gets more complex. I suppose that I have to use some property of the LCSs in order to manipulate the 5 LCS that appear.

B) I tried to connect $*d*F$ with $*J$ , a process that involves simpler manipulations, but I come up with an equation that has an opposite sign of the desired. Here is what I did:

$*d*F=\tfrac{1}{24}{{\varepsilon }^{\mu \nu \beta \gamma }}{{F}_{\mu \nu }}^{,\alpha }{{\varepsilon }_{\alpha \beta \gamma \delta }}d{{x}^{\delta }}=\tfrac{1}{6}{{F}_{\mu \nu }}^{,\mu }d{{x}^{\nu }}\Rightarrow 6*d*F={{F}_{\mu \nu }}^{,\mu }d{{x}^{\nu }}$

$*{{J}_{}}=\tfrac{1}{3!}\tfrac{1}{3!}{{J}_{\mu }}{{\varepsilon }^{\mu \alpha \beta \gamma }}{{\varepsilon }_{\alpha \beta \gamma \delta }}d{{x}^{\delta }}=-\tfrac{1}{3!}{{J}_{\nu }}d{{x}^{\nu }}\Rightarrow -6*{J}={{J}_{\nu }}d{{x}^{\nu }}$

(I used the identity ${{\varepsilon }_{\mu \nu \alpha \beta }}{{\varepsilon }^{\mu \nu \rho \sigma }}=2\left( {{\delta }_{\alpha }}^{\rho }{{\delta }_{\beta }}^{\sigma }-{{\delta }_{\beta }}^{\rho }{{\delta }_{\alpha }}^{\sigma } \right)$ in the first equation and ${{\varepsilon }_{\rho \sigma \mu \alpha }}{{\varepsilon }^{\rho \sigma \mu \nu }}=3!{{\delta }_{\alpha }}^{\nu }$ in the second)

So, according to the given equation, the above 1-forms are related like this:

${{F}_{\mu \nu }}^{,\mu }={{\mu }_{0}}{{J}_{\nu }}\Rightarrow *d*F=-{{\mu }_{0}}*{J}\Rightarrow d*F=-{{\mu }_{0}}{J}$

That minus sign shouldn’t be there!

Please check this last process for mistakes; it is the one that gets me closer to the result. If it correct, then apparently what I want to proof is the result with minus sign. But in the literature this equation has always a positive sign. I checked the definitions of the forms and the Hodge duality that are used in literature (which are the same as I those presented above), but the result they get has a positive sign (for reference check here http://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field , in the “Differential forms approach” section)

 

[andrien]

it is better to go with one form current.
j=j_adx^a
with the definition of F,we have
*F=1/4(ε_ab^cd F_cd dx^aΔdx^b)
we can employ riemann normal coordinates in case of curved spacetime,
g=η,zero christoffel symbols this all means dε=0
d*F=1/4 d(ε_ab^cd F_cd dx^aΔdx^b)=1/4 ε_[ab ^cd∂_e]F_cd dx^eΔdx^aΔdx^b
this is three form with components
(d*F)_eab=3/2 ε_[ab ^cd∂_e]F_cd
then component of *d*F are
(*d*F)_p=1/6 ε_p^eab(3/2 ε_[ab ^cd∂_e]F_cd)=1/4 ε^ab_p^e ε_ab^cd ∂_eF_cd=-δ^[c|_p g^e|d]∂_eF_cd
=-δ^c_pg^ed ∂_eF_cd=-∂^d F_pd=∂^d F_dp
which from maxwell eqn
∇_aF^ab=-j^b gives what is required.your minus sign is ok.they are just using different faraday tensor ie. the equation used is ∇_aF^ba=j^b .that's all.see wiki page on electromagnetic tensor.

 

[cosmic dust]

Thank's!