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My question relates to Noether's theorem, which I've recently been reading about, and I couldn't find any good answers on the internet relating a conservation law to the symmetry between Newton's law of gravity and Coulomb's. Would it be something like the conservation of charge, or the conservation of angular momentum? Or both?

Also, is this an area of physics that's received much attention by researchers? I ask this because I can't seem to find any published papers on the subject and would really like to know the answer.

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    $\begingroup$ The laws look the same, formally, due to principles of conservation of mass and conservation of charge, combined with assumptions about spherical symmetry of the fields. If there is something deeper, it's beyond the standard model physics. $\endgroup$ – Jerry Schirmer Jul 11 '17 at 18:05
  • $\begingroup$ What do you mean by "the symmetry between Newton's law of gravitation and Coulomb's law"? If you refer to both resulting in a force scaling like $1/r^2$, that follows from charges/mass being sources of the corresponding fields, the Divergence theorem and our universe having 3 macroscopic space dimensions. $\endgroup$ – CodesInChaos Jul 11 '17 at 18:16
  • $\begingroup$ I mean that the two laws are an example of symmetry, Feynman says they are so I'll go along with it $\endgroup$ – Sam Cottle Jul 11 '17 at 18:20
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    $\begingroup$ Symmetry doesn't mean what you think it means. It doesn't mean that two equations look "the same": it means that the Lagrangian of the theory is invariant under some (one-parameter continuous) group of transformations. $\endgroup$ – gented Jul 11 '17 at 18:44
  • $\begingroup$ Related: physics.stackexchange.com/q/47084/2451 , physics.stackexchange.com/q/54942/2451 and links therein. $\endgroup$ – Qmechanic Jul 11 '17 at 19:25
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Noether's theorem says that given a system, a continuous symmetry of its action implies into a conserved quantity along the dynamical evolution of that system. For example, if the action of a system of particles is invariant under rotation, then the system's angular momentum is conserved. It does not apply to symmetries between physical laws and therefore it does not say anything about the relation between Coulomb law and Universal Gravitation law.

Strictly speaking a symmetry in physics is a map relating different states belonging to the same theory. In your example, you are relating two distinct theories and it would be more appropriate to talk about a possible duality. A duality maps two distinct theories such that the model of one of them (the original theory) can be used to study physical properties of the other (the dual theory). In your example, there is a quasi-duality between Coulomb law and Universal Gravitation law given by the fact that under a map $$q\leftrightarrow m,\quad G\leftrightarrow 1/4\pi\epsilon_0,$$ we almost can use Coulomb law to describe gravity and vice-versa. If the full duality was true, every property we obtain in electrostatic would also show up in gravity. Of course this is not a complete duality because electrostatic interaction can be attractive or repulsive whereas gravity is always attractive.

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  • $\begingroup$ Do you think a better question could be 'is there some relation between Newton's law of gravity and Coulomb's?'? $\endgroup$ – Sam Cottle Jul 11 '17 at 18:11
  • $\begingroup$ That question would make much more sense. $\endgroup$ – Diracology Jul 11 '17 at 18:12
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    $\begingroup$ Yeah, and it's probably been asked before. $\endgroup$ – Sam Cottle Jul 11 '17 at 18:13
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Hopefully, my intuitive explanation will not be considered too simple for this forum.

Gravity is a field force, and all points that are equidistant from the center of gravity of a massive object experience the same gravitational force. This means that in a sense, mass is a "point source" of the gravitational field.

The electric field is a field force, and all points that are equidistant from the "center" of a single charged object, experience the same electric field strength.

This means that the generator of each field is a "point source". Every point that is equidistant from a point source describes a sphere in space, with that point source at its center. The area of that sphere is $A = 4\pi r^2$. If you double your distance from either point source, the new locus of points that is equidistant from that source covers 4 times the area that it previously covered, so the field strength in both cases decreases as $1/r^2$, because the geometry involved is identical in both cases.

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