- #1
- 5,844
- 551
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
Last edited: