I'm trying to understand how the various EM tensors work in General Relativity. The only source I've found is https://en.wikipedia.org/wiki/Maxwell's_equations_in_curved_spacetime, but there are two things I don't get.(adsbygoogle = window.adsbygoogle || []).push({});

Why do they use ordinary partial derivatives instead of covariant ones? It's clear for the definition of F_{μν}because here the correction terms (Christoffel symbols) cancel, but I don't see how that should work for the definition of J^{μ}. And further down, the continuity equation is given as ##\partial_μ J^μ =0 ##. Isn't this equation coordinate system dependent? How can this be a law?

The second (possibly related) issue is their use of tensor densities for D and J. This means the time component of J has a dimension of charge per unit cube, whereas using a vector would mean charge per lengh cubed; is that right? Could that somehow fix the problem with the partial derivative? The maths is beyond me, I'm afraid.

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# I Electromagnetism in General Relativity

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