All the laws in physics are local in nature and that's why their formulation follows differential equations. My doubt is whether the locality is a proven theorem or it is a postulate?
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1$\begingroup$ Theorems are mathematics, not physics. Mathematics deals with the properties of objects that exist only in the human imagination. Sometimes, those properties map well onto real objects in the universe, as verified by experiment. Then, theorems are useful, but one must never completely trust them as proven in physics. $\endgroup$– John DotyCommented Jun 6, 2022 at 23:14
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$\begingroup$ Quantum physics are non-local, as counterxample to your first assumption. $\endgroup$– JoakoCommented Jun 7, 2022 at 1:52
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In a thin tunnel (compared with the radius of the sphere) across a massive sphere, the acceleration of gravity in a point at a distance $r$ from the center is:$$\mathbf a = -\frac{4}{3}\pi G \rho \mathbf r$$ where $\rho$ is the density of the sphere, supposed uniform.
The divergence of $\mathbf a$ can be shown to be: $$\nabla \boldsymbol{ .a} = -4\pi G\rho$$.
This is a differential equation, and its solution refers not to the local density inside the tunnel (that is zero), but to $\rho$, the density of the sphere.
So, the symmetry of the situation matters. We can not say that only the value of the variables in a local neighborhood has to be taken in account.
answered Jun 6, 2022 at 22:54
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$\begingroup$ True, But a true test of the locality will be on how you measure the effect. And in that sense, any change in the density of the matter will not directly affect the gradient at a point. The gradient at a point is related only to its neighborhood, which is to their and so on till the density. The solution to that differential equation is sum of the local effects. $\endgroup$– rknCommented Jun 12, 2022 at 19:23
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$\begingroup$ If the density of the sphere as a whole changes, the potential and its gradient changes inside the tunnel. And the density there was zero and remains zero. $\endgroup$ Commented Jun 12, 2022 at 20:31