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For a spherically symmetric mass m of radius r and charge q, how does the electric field vary with distance d from the center where d > r.

Does it still vary as $\dfrac{1}{r^2}$ or is there a correction factor (like $\dfrac{1}{r^4}$ which is for gravitation field) as per General Relativity??

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The electromagnetic field tensor $F$ satisfies the two equations $dF=0$ and $\star d \star F=J$. The first equation is purely topological and does not change regardless of the metric. For the Schwarzschild metric, the determinant of the metric is 1 (in the usual coordinates) so that the hodge star is the same as flat spacetime. Hence, the second equation is also the same as flat spacetime. Thus, all Maxwell's equations are the same in a Shwarzschild metric as it is in flat spacetime, and so the solution is also the same.

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    $\begingroup$ - 1. This is wrong. It is meaningless to say “solution is the same” for two different spacetimes. $\endgroup$
    – A.V.S.
    Commented Jun 27 at 7:37
  • $\begingroup$ What I meant was if you use the usual spherically symmetric coordinates, the expression for the field tensor as functions of r would be the same. Although now thinking about it, r is not the proper distance, but if you change coordinates to proper distance, then you can figure out the corrections I suppose. $\endgroup$
    – Toyesh Jayaswal
    Commented Jun 27 at 13:49
  • $\begingroup$ Could you answer for someone not too familiar with GR...The answer could be something with expressions as given in the question $\endgroup$
    – Ajay
    Commented Jun 29 at 11:25

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