My point is that you are tacitly augmenting the laws of Nature here. Local physics gives differential equations. By privileging a particular solution to those differential equations, you are going beyond local physics. And you must, too, to speak about radiation intelligibly.
A purely local formulation of Newtonian gravity theory, then, leads naturally and inevitably to the Equivalence Principle. In such a theory, the notion of gravitational "force" is meaningless. This notion can only be recovered by adding a non-local "law" to the theory.
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The same situation obtains in Maxwell theory. It is a purely local theory, whose laws take the form of partial differential equations. In this theory, no meaningful statements can be made about the "radiation" emitted by a source, until and unless boundary conditions are imposed.
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Take a charge. Not a theoretical one, take a real one. Now accelerate it. It will emit radiation that can be measured. And what happens at a distance L (e.g. the walls of my lab) only influence it after a time of order L/c. The walls of my lab matter, but not immediately.
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But I have nothing against adding an homogeneous solution to the inhomogeneous one. We do it all the time! For Maxwell eqs the homogeneous solution just represent any field that is not generated by the charges in your model.
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