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The wikipedia article for Kaluza-Klein theory states: 'In physics, Kaluza–Klein theory (KK theory) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism."

I am no expert in KK-theory (obviously), but I was wondering what's the use for these types of theories?

Einstein's special relativity is fully compatible with E&M since Einstein especially formulated his theory to keep Maxwell's equation the same but modified Newton's equations. General Relativity is an extension of special relativity to non-inertial reference frames.

Is it not logical, then, that GR is fully compatible with E&M (we just take the covariant form of Maxwell's equations and take partial derivatives to covariant derivatives)? Why do we need some other theory to "unify" gravity and E&M then? Why can't we just use GR?