Why does Newton's Third Law work for fields?
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third law of motion. Formally stated, Newton's third law is: For every action, there is an equal and opposite reaction.
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In Newtonian physics, the field is not really something physical that has an independent existence. Particularly in Newtonian gravity, the gravitational force is really action at a distance with nothing mediating the force in between. The field gg defined here is simply for mathematical convenience and is not the usual field that you talk about in a fully relativistic classical field theory.
So as long as you're not doing any relativistic calculations and asking questions like "Does the force act instantaneously? In what frame?", it's perfectly fine to just treat the particles as obeying the Third Law, with equal and opposite forces between them that act at a distance, and use the field gg only as a convenience.
Of course, if you want a relativistic theory, you have to introduce a real physical field that can carry momentum and energy at every point in space. This is done in classical electrodynamics and general relativity, for example. You will see there that particles do not obey a simple Newton's Third Law since you have to take into account the dynamics of the field too. (There is no dynamics for the field itself in Newtonian physics, because like I said, the field is not something physical in Newtonian physics) This is what gives rise to electromagnetic and gravitational waves in those two field theories, respectively.
Summary: Don't take the "field" gg in a physical sense.
So as long as you're not doing any relativistic calculations and asking questions like "Does the force act instantaneously? In what frame?", it's perfectly fine to just treat the particles as obeying the Third Law, with equal and opposite forces between them that act at a distance, and use the field gg only as a convenience.
Of course, if you want a relativistic theory, you have to introduce a real physical field that can carry momentum and energy at every point in space. This is done in classical electrodynamics and general relativity, for example. You will see there that particles do not obey a simple Newton's Third Law since you have to take into account the dynamics of the field too. (There is no dynamics for the field itself in Newtonian physics, because like I said, the field is not something physical in Newtonian physics) This is what gives rise to electromagnetic and gravitational waves in those two field theories, respectively.
Summary: Don't take the "field" gg in a physical sense.
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