Can anyone recommend a good reference for classical electrodynamics that goes over electrodynamics in curved spacetime that doesn't assume much knowledge of GR -- that is it builds up the tensor calculus and GR principles itself?
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The Classical Theory of Fields by Landau and Lifshitz fits the bill reasonably well. It doesn't develop electrodynamics from scratch in the context of a curved spacetime, but it does have a three-page section covering the equations of electrodynamics in a curved spacetime, after spending seven chapters developing electrodynamics in the context of flat spacetime. Unlike Griffiths or Jackson, the development of electrodynamics maintains contact with special relativity from the start. Indeed, the book's first two chapters are a coverage of the basics of SR. The book still presents the material using the traditional 3D vector fields with div, grad and curl, but it alternates between using that pre-relativistic treatment of electrodynamics with a special relativistic treatment using four-vectors and tensors.
The book requires about as little knowledge of GR as possible prior to its coverage of electrodynamics in curved spacetime. In fact, in a sense the book hasn't really yet covered GR at that point, since it doesn't cover the Einstein field equations until later. The only material in between the special relativistic treatment of electrodynamics and the treatment of electrodynamics in curved spacetime is one chapter that extends the tools used in SR to curved spacetime.
You might find Gauge Fields, Knots, and Gravity by Baez & Muniain useful.
It doesn't specifically deal with electromagnetism in curved spacetime, however, the first chapter develops differential geometry with the goal of reformulating Maxwell's equations. It assumes no knowledge of GR. The third and final chapter develops some aspects of GR.