In 1919 the theory of general relativity was confirmed by observing and photographing stars seemingly changing their position during a full solar eclipse due to the effects of gravity on light.
How early in history the result of this kind of experiment could potentially been observed and recorded? The experiment doesn't need to be conducted to prove general relativity and doesn't need to be immediately followed by any coherent light-gravity theory, but it needs to be rigorous enough not to be dismissed as an instrument flaw or observation error.
The latter is a tall order! It's rather implausible that those conditions can be met under reasonable circumstances, since even with GTR predictions being highly anticipated, it was not completely unambiguous at the time that Eddington's 1919 observations supported GTR over Newton (though later analysis supported Eddington).
For that matter, what would be the implications of the observation that light can bend in if general relativity has not yet been introduced? Alternatively, could you think of some wrong explanation for that phenomena that could be adopted instead?
Unless it's measured very accurately, it'll be a puzzle but not an obviously critical one. It's consistent with Newtonian gravitation that light can bend in a gravitational field--famously, Laplace even predicted the possibility of a 'dark star', the Newtonian analogue of a black hole, in 1796. The puzzle would be how to reconcile it with the newer wave theory, since it was completely straightforward in the old corpuscular theory of light, but wave theory had much more critical puzzles of the same type anyway.
Carl Sagan supposedly theorized that James Clerk Maxwell could have developed the idea of relativity if he had lived longer. If the theory is developed in the 1880s (for example) it may be tested much earlier than OTL too.
Maxwell was smart, but he didn't have nearly the mathematical machinery for GTR. Here Einstein had the nigh-immesurable advantage of having been born later. But even for STR, Maxwell probably didn't have the mindset for it.
I also recall reading that had Maxwell lived longer (he died at the age of 48), he could have developed Special Relativity. He certainly had the creativity and sheer mathematical brainpower to do so, and his latest efforts before he died were leading in that direction already.
Are you sure about that? What are you thinking of?
I know he made a brief digression on gravity in his original 1864 paper on electromagnetism noting the failure of the stressed-mechanical-medium (i.e., aether) approch for gravity, despite his use of it for EM. In 1875 wrote for Britannica:
A theory of this kind is worked out in greater detail in Clerk Maxwell's Treatise on Electricity and Magnetism. It is there shown that, if we assume that the medium is in a state of stress, consisting of tension along the lines of force and pressure in all directions at right angles to the lines of force, the tension and the pressure being equal in numerical value and proportional to the square of the intensity of the field at the given point, the observed electrostatic and electromagnetic forces will be completely accounted for.
...
The force of gravitation ... differs from the electric and magnetic forces in this respect, that the bodies between which it acts cannot be divided into two opposite kinds, one positive and the other negative, but are in respect of gravitation all of the same kind, and that the force between them is in every case attractive. To account for such a force by means of stress in an intervening medium, on the plan adopted for electric and magnetic forces, we must assume a stress of an opposite kind from that already mentioned. We must suppose that there is a pressure in the direction of the lines of force, combined with a tension in all directions at right angles to the lines of force. Such a state of stress would, no doubt, account for the observed effects of gravitation. We have not, however, been able hitherto to imagine any physical cause for such a state of stress.
...
Another theory of the mechanism of gravitation, that of Le Sage, who attributes it to the impact of "ultramundane corpuscules," has been already discussed in the article ATOM, supra, p. 46.
He did die soon after (1878), but if his 1864 paper and 1875 article are representative of his actual views--and this might not be the case, but I've seen nothing to suggest they aren't--then Maxwell not only did not advance toward STR, but in some sense made negative progress, by having a strange sort of 'double aether' for gravity.
Special Relativity is really just the completion of what Maxwell originally created. This is one reason why SR is considered part of Classical Mechanics. Quantum Mechanics, on the other hand, was something entirely new.
Here's an exercise for and alternate history of QM: wave optics must reduce to geometrical (ray) optics, ray optics are essentially the ancient particle (corpuscular) theory of light. In phase space, Fermat's principle of least time for light rays plays the same role as Hamilton's principle of least action for particles, with the eikonal (light) corresponding to the abbreviated action (particles).
The de Broglie relations of QM fall out taking this analogy seriously; in an alternate timeline that someone had the
unnatural idea to try describing particles with waves because the Hamiltonian formalism works well for both, the de Broglie relations and the Schrödinger equation are derivable without knowing any lick of experimental support for QM. So, what? Is QM "really just the completion" of the Hamiltonian (or Lagrangian) formulations of classical mechanics? Well, not really (still need Born rule), but if we're making uncharitable enough comparisons, there is nothing new under the sun. Pretty much nothing ever has sprung up from an intellectual vacuum.
By the way, STR is not part of Classical Mechanics. It's a theory so general it might even be called a 'meta-theory', because it is a core constraint on every mainstream theory in the last eighty years or so.
As for General Relativity, it has become increasingly evident that it is merely a close approximation to physical reality. I don't know what form the "true" theory of gravity will take, although I do like Verlinde's Entropic Gravity. So far, it appears to explain in a very natural way many (and perhaps all) of the increasingly numerous anomalous observations at various scales, ...
Well, there's no accounting for taste. However, it is very unnatural and goes directly against the lessons from every other fundamental forces.
... that GR cannot explain without recourse to magical elves (AKA Dark Matter and Dark Energy).
Dark matter is a problem, but I wouldn't (and more importantly, most physicists on this problem wouldn't) consider it likely to be a problem with
gravity, but rather a problem for GUTs, which almost generically predict heavy DM particles to some extent anyway. From that point of view, it's not even a qualitatively new kind of thing relative to experimental data, since the standard model includes light DM particles.
Dark energy isn't classically a problem at all; it's been in GTR since 1917, being quite literally the simplest possible theory after the initial
1915 1916 version of GTR. Quantum-mechanically, it's again a quantitative rather than qualitative issue, since the quantum mechanics predicts energy in vacuum. So then it appears to be just an analogue of the old QM renormalization problems.