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Newton's explanation of gravity as an attractive force seems to have been superseded by Einstein's explanation of gravity as warping of space-time. Was there any advances in math and science that was not known in Newton's time, that would have laid the foundation for Einstein to give a more accurate description of gravity in General Relativity?

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  • $\begingroup$ physics.stackexchange.com/q/178417/37364 $\endgroup$ – mmesser314 Oct 23 at 5:33
  • $\begingroup$ Non-linear (non-Euclidean) geometry was conceived and developed first late 1700 and early 1800s by Lobachewski and Bolyai and probably more guys. Actual warping of space would be difficult to express without this. $\endgroup$ – mathreadler Oct 25 at 8:48
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  • A well-developed idea of a field theory. Newton thought of the force of gravitation to be operating with an action-at-a-distance mechanism. While this bothered him, it remained an unresolved question to him. However, by the time of Einstein, the idea of thinking of the force of gravitation in terms of a field theory had been developed.
  • Lorentz invariance. While the shift from thinking of the theory of the force of gravitation in terms of a field theory was an important conceptual shift, nothing really changed in terms of the mathematical description of the force of gravitation. But, with the development of special relativity, Einstein had realized that the laws of physics should be Lorentz invariant, unlike the Newtonian law of gravitation which was Galilean invariant.
  • Mass-energy equivalence. This is another aspect of the development of special relativity which was relevant to going beyond the Newtonian law of gravitation. Einstein had realized through special relativity that mass and energy are not distinct properties but are rather unified in a profound way. This led him to believe that if mass plays a role in causing gravitational attraction then so should energy. However, as I said, this is closely related to my previous point: Lorentz invariance.
  • Riemannian geometry. Putting together all the physical axioms that Einstein had developed crucially required the use of Riemannian geometry. In fact, learning the tools of Riemannian geometry was the hardest part for Einstein in his journey of developing his theory of gravity.

Finally, I would like to mention that two crucial elements that went into the development of general relativity (perhaps, the most crucial two elements) were already present at the time of Newton. One of them was the equality of the inertial and the gravitational mass (something that Newton also found curious) and the other was the question of what determines which frame is an inertial frame (to which, Einstein ultimately found the answer: the freely falling frame is the inertial frame). This is not to say that Newton should've developed general relativity had he been clever enough. Lorentz invariance and non-Euclidean geometry were absolutely indispensable in the development of general relativity and they were too way ahead in the future to be discovered at the time of Newton.

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    $\begingroup$ how about Mach ideas about mechanics? They were also crucial for Einstein to free his mind from idea of fixed background. As far as i know in Newtons time no one considered background to be able to be dynamic. Note that this idea could be imagined even before Riemann geometry, Netwon himself is famous for inventing math he needed. $\endgroup$ – Umaxo Oct 23 at 4:24
  • $\begingroup$ @Umaxo Thanks for pointing it out. I know that Einstein was heavily inspired and influenced by Mach's work but due to my lack of knowledge about Mach's ideas, I am not educated enough in asserting how much of an actual scientific role they played in the development of GR. Moreover, the final results of GR stand in contradiction with the Machian principles so I am of the understanding that, in principle, GR could've been developed without the existence of Machian ideas while realizing that they heavily influenced Einstein. $\endgroup$ – Dvij Mankad Oct 23 at 4:50
  • $\begingroup$ Me neither, but in my university it was common to say that GR are fullfiling Mach's ideas and we were starting the GR lecture from those. But as far as i know they were never well formulated (which makes sense, since in that case the whole theory would be born), so i guess the question wheter GR contradicts Mach's principles might be controversial (and what does it mean contradicts? as far as i know, there are Machian solutions in GR). But i think he was the first that introduced the idea of (kind of) dynamic background. This idea is of course essential for GR. $\endgroup$ – Umaxo Oct 23 at 5:13
  • $\begingroup$ @Umaxo Again, I only have a very superficial understanding of Mach's ideas (and maybe there were vague enough to be completely debunked?) but a central idea was that distant matter decides which frame is an inertial frame. This is not true in GR since the local metric decides the local inertial frames. Sean Carroll once tweeted about this: twitter.com/seanmcarroll/status/954459178143133696?s=20 $\endgroup$ – Dvij Mankad Oct 23 at 5:36
  • $\begingroup$ I know there are solutions called Wheeler-Mach-Einstein spacetimes but I am uneducated as to how exactly they satisfy the Machian principle. In particular, the metric is still local even if the energy-momentum tensor on a Cauchy surface determines the local inertial frame everywhere in the spacetime. $\endgroup$ – Dvij Mankad Oct 23 at 5:40
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Riemannian geometry, the mathematical basis for General Relativity, was unknown in Newton’s day. The only geometry available to Newton was Euclidean geometry.

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    $\begingroup$ Calculus, the mathematical basis for Newtonian mechanics, was also unknown in Newton's day... $\endgroup$ – leftaroundabout Oct 24 at 15:13
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    $\begingroup$ Newton and Leibnitz invented calculus. $\endgroup$ – G. Smith Oct 24 at 15:33
  • $\begingroup$ Of course they did. That's my point: it has happened a couple of times that, if some particular mathematical tool didn't exist at the time, physicists would just invent it if that's needed for some physical theory. So just the absence of Riemannian geometry wouldn't necessarily be a show-stopper for GR. More critically, both calculus (“shoulders of giants”) and Riemannian geometry have lots of other maths behind them, without which it would be unlikely to do that final step. $\endgroup$ – leftaroundabout Oct 24 at 15:44
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    $\begingroup$ And my point was that Einstein did not invent Riemannian geometry, and probably would have been incapable of doing so as he turned to others for help with mathematics. $\endgroup$ – G. Smith Oct 24 at 15:47
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    $\begingroup$ If Einstein had not discovered GR, Hilbert would have. (He was basically a co-discoverer.) I have no idea how long it would have taken without Riemann, but probably not very long because others had been investigating non-Euclidean geometry. $\endgroup$ – G. Smith Oct 24 at 15:54
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In addition to all the answers listing the improved mathematical tools, I think it's important to mention the enormous progress made in astronomy, thanks to both the vastly improved manufacturing techniques that enabled telescopes far beyond anything possible in Newton's time (remember that Newton himself laid an important foundation in the then-new field by inventing the reflector telescope), and, well, the widespread use of Newtonian mechanics in developing celestial mechanics. The progress of astronomy gave an extremely important source of insight: the known problems that were encountered since Newton developed his theories. It's the kind of input that's only possible once you have your theory widely used and tested.

The perihelion shift of Mercury's orbit in particular was an important indication of success of the general relativity, being a well-known problem that both showed that classical gravity had shortcomings, and that the general relativity was on the right track in explaining it.

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An understanding of electromagnetism was required for the development of Special Relativity, which then motivated General Relativity. Specifically, the construction of Maxwell's equations was needed. The second and third sentences of Einstein's 1905 paper "ON THE ELECTRODYNAMICS OF MOVING BODIES" are:

Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion.

As stated by Einstein himself, the incompatibility of Maxwell's equations with Newtonian mechanics was the motivation for Special Relativity, which was, in turn, the motivation for General Relativity.

It's interesting that we had to understand electromagnetism to understand gravity, but it's pretty clear that's what happened.

Newton died in 1727. The development of electromagnetism required an enormous amount of experimental and theoretical work that hadn't been done in the time of Newton. Furthermore, some improvements in Newton's and Leibniz's calculus were needed to represent Maxwell's equations. Here are some major developments in electrical theory:

The required experimental work also required a lot of industrial development. The availability of inexpensive interchangeable parts and cheap metal wire were undoubtedly major contributors in the growth of electrical technology and the study of electrical phenomena:

Quoting Wikipedia:

The metal screw did not become a common fastener until machine tools for their mass production were developed toward the end of the 18th century. This development blossomed in the 1760s and 1770s.

Furthermore, both Special Relativity and developments in mathematics were needed for General Relativity.

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Special Relativity was the strongest input for Einstein. Spacetime and its 4D metric was needed. There was no notion of Lorentz transformations and invariance in Newton's time. An invariant light velocity would have been absurd for Newton!

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