- #1
- 5,844
- 550
Homework Statement
I have to take the curved space - time homogenous and inhomogeneous maxwell equations, [itex]\triangledown ^{a}F_{ab} = -4\pi j_{b}[/itex] and [itex]\triangledown _{[a}F_{bc]} = 0[/itex], and show they can be put in terms of differential forms as [itex]dF = 0[/itex] and [itex] d*F = 4\pi *j[/itex] (here [itex]*[/itex] is the hodge dual defined for any p - form [itex]\alpha [/itex] as [itex](*\alpha )_{b_1...b_{n-p}} = \frac{1}{p!}\alpha ^{a_1...a_p}\epsilon _{a_1...a_pb_1...b_{n-p}}[/itex] where [itex]n[/itex] is the dimension of the manifold and [itex]\epsilon [/itex] is the natural volume element for the manifold i.e. a totally anti - symmetric nowhere vanishing continuous tensor field).
The Attempt at a Solution
Since the exterior derivative [itex]d[/itex] is independent of the choice of derivative operator, I chose for convenience the unique metric compatible derivative operator [itex]\triangledown _{a}[/itex] because [itex]\triangledown _{c}\epsilon _{a_1...a_p} = 0[/itex] identically and this simplifies the calculation. The homogenous ones are trivial since [itex](dF)_{ba_1a_2} = 3\triangledown_{[b}F_{a_1a_2]} = 0[/itex]. The inhomogeneous ones are really starting to annoy me . This is 4 dimensional space - time so we have that [itex]*F_{b_1b_2} = \frac{1}{2}F^{a_1a_2}\epsilon _{a_1a_2b_1b_2}[/itex] hence [itex](d*F)_{cb_1b_2} = 3\triangledown _{[c}*F_{b_1b_2]} = \triangledown _{c}*F_{b_1b_2} - \triangledown _{b_1}*F_{cb_2} + \triangledown _{b_2}*F_{cb_1}[/itex] (where I have assumed the anti - symmetry of the hodge dual of the EM field strength tensor to combine terms). Plugging in the equation for the hodge dual gives us [itex](d*F)_{cb_1b_2} = \frac{1}{2}(\epsilon _{a_1a_2b_1b_2}\triangledown _{c}F^{a_1a_2} - \epsilon _{a_1a_2cb_2}\triangledown _{b_1}F^{a_1a_2} + \epsilon _{a_1a_2cb_1}\triangledown _{b_2}F^{a_1a_2})[/itex]. On the other hand, [itex]4\pi (*j)_{b_1b_2b_3} = 4\pi j^{a_1}\epsilon _{a_1b_1b_2b_3} = \triangledown _{r}F^{a_1r}\epsilon _{a_1b_1b_2b_3}[/itex]. Has my work been correct so far? This is where I'm stuck because the two different equations (the one for the hodge dual of the 4 - current density and the one for the exterior derivative of the hodge dual of the EM tensor) are being summed over totally different indices, as you can see, so I cannot begin to even see how to resolve the two so they can be equal.
Last edited: