Gravity vs. EM: action at a distance

Question

Countless texts point to Newton's theory $\nabla^2\phi = 4\pi G\rho$, and remark that the problem here is that a distribution of mass determines the potential instantaneously everywhere, which is obviously incompatible with SR.

Yet, no texts on electromagnetism make the same comment about the electrostatic potential $\nabla^2\phi_e = 4\pi\rho_e$. Of course, we see later with Maxwell's equations that when a charge is moved, the change in the field (at the source) detaches and radiates with speed $c$. The static potential $\phi_e$ does indeed affect every charge in the universe, assuming that the field has existed statically since $t=-\infty$.

Yet, no one ever seems to reason the same way with Newton's $\nabla^2\phi = 4\pi G\rho$. The comment that this implies "action at a distance" is repeated constantly (see Geroch's 1972 Lecture Notes). Why is it never asserted, by analogy with Coulomb's law, that the gravitational potential is a statement about statics only (assuming the mass distribution has existed, unchanged, since $t=-\infty$), and that if a mass is moved (accelerated) then the field will detach and radiate exactly the same way as the electric field?

Answer 1

Yet, no texts on electromagnetism make the same comment about the electrostatic potential $\nabla^2\phi_e = 4\pi\rho_e$.

I disagree with this claim. In fact, most EM textbooks do make this statement. Usually in the context of discussing the Coulomb gauge. For example, see the Coulomb gauge section of the Wikipedia article on gauge fixing.

https://en.m.wikipedia.org/wiki/Gauge_fixing

Why is it never asserted, by analogy with Coulomb's law, that the gravitational potential is a statement about statics only (assuming the mass distribution has existed, unchanged, since t=−∞), and that if a mass is moved (accelerated) then the field will detach and radiate exactly the same way as the electric field?

In Newtonian gravity there is no gravitational radiation. The analogy fails.

The issue at hand is that the potentials are not detectable, only the fields are. So you must look at the relationship between the potentials and the fields to see if there is an issue.

In the case of EM we have$$\vec E = -\nabla \phi -\frac{\partial}{\partial t}\vec A$$So although the EM $\phi$ in the Coulomb gauge violates causality, the Coulomb gauge forces a more complicated $\vec A$ which ensures that $\vec E$ does not violate causality.

In the case of Newtonian gravity we have $$\vec g = -\nabla \phi$$So there is no compensating vector potential. A non-causal (Newtonian) gravitational potential directly leads to a non-causal gravitational field. There is no part of Newtonian gravity that ensures that the field does not violate causality.

Therefore, a new theory of gravity is needed. That new theory is general relativity (GR). There was indeed some work on modifying Newtonian gravity in analogy with electromagnetism, called gravitoelectromagnetism, and in fact it is a specific limit of GR. So the analogy you make is reasonable in a limiting case. GR introduces the other terms that are necessary to restore relativistic causality to the theory of gravity.