Geometric Formulation
The geometric view of the electromagnetic field is that it is the curvature 2-form of a principal U(1)-bundle, and acts on charged matter by holonomy. In this view, one of Maxwell's two equations, d F= 0, is a mathematical identity known as the Bianchi identity. This equation implies, by the Poincaré lemma, that there exists (at least locally) a 1-form A satisfying F = d A. The other Maxwell equation is
where the curvature 2-form F is known as the Faraday 2-form in this context, J is the current 3-form, the asterisk * denotes the Hodge star operator, and d is the exterior derivative operator. The dependence of Maxwell's equation (there is only one with any physical content in this language) on the metric of spacetime lies in the Hodge star operator. Written this way, Maxwell's equation is the same in any spacetime.
Read more about this topic: Maxwell's Equations In Curved Spacetime
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