Electromagnetism versus gravity I am wondering why classical electromagnetism predicts (classical, wavelike) photons , but classical Newtonian gravity does not predict analogous wave-like solutions (as far as I am aware)?
Stated differently, classical electromagnetism predicts its corresponding force-mediating particle - the photon (albeit as a classical wave) - but classical newtonian gravity does not predict its corresponding force mediating particle (as a classical wave)?
Im guessing because in Maxwell theory, the electric and magnetic fields are subject to Maxwell's equations. Whereas in Newtonian gravity, gravity is not treated as a field. The gravitational potential acts everywhere instantaneously, so does not propagate in any sense.
 A: A 2008 review of gravitomagnetism is available in this PDF but basically you are asking a question about history and that is harder to answer.
So it wasn’t until the 1830s that we had a preliminary unification just beginning to unfold, between electricity and magnetism. It took another generation to get James Clerk Maxwell’s 1860s observations that the laws as he had them were only consistent if charge did not accumulate at any point, but that if he viewed all of space as made out of little electromagnetic vortices then another term would enter the equations—what we now call a “displacement current”—and as a result these vortices could carry waves which would travel at the speed of light. Even then it took another generation, the 1890s, before we had the Lorentz invention of “local time” and the ether-based tensions brewing between Newton’s theory and Maxwell’s theory. Some have mused that the chief virtue of scientific revolution is that the old generation of scientists die out, allowing new generations to have radical new ideas: the timeline of electromagnetism certainly lends some support to this idea.
It is hard to believe that nobody had the idea of gravitomagnetism before Einstein. For example JJ Thompson’s Recent Researches in Electricity and Magnetism starts out with a discussion about how “Faraday tubes” (we’d call them electric field lines now) are instrumental to understanding electricity and magnetism, and that magnetism in fact gives us a fundamental idea that there might be a Faraday tube which does not begin and end on a charge but rather loops around on itself, having neither beginning nor end. It then immediately observes that there is a connection between the Faraday tubes and Le Sage’s theory of gravitation (which is that maybe space is filled with particles flying in all different directions and maybe matter absorbs a little bit of them, so that any given mass casts a shadow of these particles in all directions, leading to an attractive $1/r^2$ force between two masses as they sit in each others’ shadow). It is perhaps not a tremendous leap to imagine that perhaps there are analogously gravitational Faraday tubes and perhaps some of them do not terminate on masses but rather just loop around. But if Thomson made this leap then I do not know about it; he probably would have immediately objected that the force directions are opposite (like attracts like for gravitation, repels like for electromagnetism) preventing the mechanical analogy that Faraday was chasing.
So the first inklings I can find of gravitomagnetism are maybe in Einstein’s 1912 (four years before general relativity!) paper “On the theory of the static gravitational field, and note added in proof”, which was itself a correction-type paper to a different paper (just flip back several pages) also being published in 1912, itself a response to a third paper in 1912 by Max Abraham, with some history in a paper on arXiv. But even here I am not seeing a prediction of classical gravitational waves! Einstein just thinks that light might move slower in a gravitational field, because he is chasing this idea that gravitation should be the same as being in a uniformly accelerated field.
I am surprised that he did not go further immediately; you should surely want a Lorentz-covariant theory of gravitation plus the observation that the Coulomb law looked like the Newton law in having a $1/r^2$ force law corresponding to a straightforward $\operatorname {div} E \propto \rho$ equation. If you want to make this relativistic then you have to consider that moving masses should generate a “magnetic field” and that together these two should have an underlying wave theory.
Right after the above paper you of course have general relativity and then more explicitly the Lense-Thirring papers (paywall warning) which apparently treated gravitomagnetism even more seriously. The reason that gravitomagnetism is not a final theory and you need to “graduate” to general relativity eventually, has to do with the fact that your source (which in electromagnetism is the 4-vector $(\rho, \mathbf J/c)$) is no longer Lorentz-covariant when you write out these equations.
But yeah, the basic reason why classical Newtonian gravity does not predict waves is because before Einstein I can’t find any work into predicting that there would be a magnetic analogue to the “purely electrical” theory of Newtonian gravitation. If you formulate such a theory and then hook up the electric and magnetic fields the way Maxwell did, to get gravitational fields propagating at speed $c$ and gravitational waves when those gravitational fields oscillate suitably: then you have your prediction. But as far as I can tell, nobody had the electromagnetic model to perform this intriguing unification until Maxwell derived that light was just electromagnetic waves: and it looks like before the development of even special relativity folks were just more likely to say “Newton is right, Maxwell is wrong” than to say “Hm, maybe we can steal these ideas from Maxwell to say something interesting about Newton, too.” But with special relativity, Einstein’s hand was kind of forced, he was committing very strongly to “Maxwell is righter than he could ever have known” and this meant that Newton’s idea of instantaneous gravitational-force at a distance could no longer be sustained, since once you have special relativity, instantaneous forces (if you can choose multiple different reference frames for different forces to be instantaneous in) can transmit information backwards in time.
A: Between 1820 and 1830, after the invention of batteries, it was possible to get continuous current in wires. Ampére (I think he was the first to note, but not sure) realized that conducting wires were atracted or repelled according to its relative position, and the intensity of the current.
That is: moving charges generate forces, that are different from static charges, described by Coulomb law. And the effect is strong enough to be detected in a lab since those days. Eventually the concept of fields prevails over force at a distance, leading finally to EM waves.
If we compare the experimental challenge for determine the G value by Cavendish, imagine measuring the effect of moving masses on the force of gravity.
It is not a surprise that gravity remained "static" until GR.
A: I'm not sure that the comparison is really optimal ? Comparing Newton's law to EMG, it would be more fair to compare Newton's law to Coulomb's law. You cannot infer from Coulomb law that there is an electromagnetic field which propagates at finite speed etc. There isn't a speed in Newton's law. In that sense, the analog of EMG for gravity is GR, which predicts waves that propagates at c, a graviton etc.
A: The main reason for the lack of gravitstional waves in Newtonian gravity is that changes in gravity propagate to the entire universe instantaneously. As the Earth orbits around the Sun, the gravitational force vector is always pointed at the location where the Sun is right now. If the Sun suddenly moves to one side, the gravitational force immediately adjusts to the new position.
As an analogy, imagine you toss a rock into a lake. The reason the rock creates ripples (surface waves) is because not all of the lake knows about the impact of the rock. Only the water hit by the rock responds, which causes a response in the water next to that water, followed by the water next to to that, in an ever expanding wave. If the entire lake could react all at once to the thrown rock, then the only observable effect of the rock would be to raise the surface level of the lake. There would be no splash or ripples because the entire lack acts all at once.
The existence of waves implies there is a time delay between a change in a source and an effect at a distance. Newton's equations for gravity have no such delay, so there are no waves. One could be added so that the gravitational force points towards where the other mass used to be, but there's no natural choice for how long the delay should be (at least in Newton's time).
