Assume we have D-dimensional space-time in spherical coordinates $(t, r, \theta_1, ..., \theta_k)$.

We have Maxwell equations for $k$-form $F^{\alpha_1, ..., \alpha_k}$

$$\partial_{\mu} F^{\mu \alpha_2 ... \alpha_k} = 0$$

Then we want to obtain magnetic monopole solution with charge $P$. How to show that our $k$-form $F$ should be the form on the transversal space $(\theta_1, ..., \theta_k)$?

P.S. An interesting case is in the presence of gravity:

$$ \partial_{\mu} (\sqrt{-g} F^{\mu \alpha_2 ... \alpha_k}) = 0$$

How to find magnetic solution in this case?

  • 1
    $\begingroup$ What exactly do you mean by "magnetic monopole solution" for a $k$-form, and how is this supposed to be electromagnetism? Electromagnetism has always a 2-form as its curvature, and you get magnetic monopole "solutions" for pure electromagnetsm only if you allow the solution to be singular/not-defined at the point where the monopole sits, and accept that you get additional singularities as coordinate artifacts like the Dirac string. $\endgroup$ – ACuriousMind Mar 18 '16 at 12:41
  • $\begingroup$ As far as I know, it's impossible to define magnetic monopole solution in arbitrary dimension $D > 4$ with electromagnetic 2-form, and one needs to introduce new $D-2$ dual form. $\endgroup$ – newt Mar 18 '16 at 12:46
  • $\begingroup$ Comments to the post (v3): Consider adding context and references in order to receive focused and useful answers. $\endgroup$ – Qmechanic Mar 20 '16 at 15:04
  • $\begingroup$ Related question by OP: physics.stackexchange.com/q/244471/2451 $\endgroup$ – Qmechanic May 20 '16 at 18:56

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