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Newton's explanation of gravity as an attractive force seems to have been superseded by Einstein's explanation of gravity as warping of space-time.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?

Qmechanic
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  • https://physics.stackexchange.com/q/178417/37364 – mmesser314 Oct 23 '19 at 05:33
  • 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. – mathreadler Oct 25 '19 at 08:48

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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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    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. – Umaxo Oct 23 '19 at 04:24
  • @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. –  Oct 23 '19 at 04:50
  • 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. – Umaxo Oct 23 '19 at 05:13
  • @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: https://twitter.com/seanmcarroll/status/954459178143133696?s=20 –  Oct 23 '19 at 05:36
  • 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. –  Oct 23 '19 at 05:40
  • i don't get it. Is not local metric determined by distant matter? – Umaxo Oct 23 '19 at 05:43
  • @Umaxo No, because the Einstein field equations are local, right? I mean the $T_\mu\nu$ here determines $g_\mu\nu$ here (up to general coordinate transformations, of course). –  Oct 23 '19 at 05:45
  • Ah, this is a nice and definitive discussion on why Mach's principle is wrong. Can't believe the arguments slipped my mind. Anyway, have a look: https://physics.stackexchange.com/q/5483/20427 –  Oct 23 '19 at 05:48
  • Mach principle isn't necessarily wrong. Yes, the metric (and its first and second derivatives) depends on the local stress-energy, but Einstein's equation is still a set of some complicated non-linear differential equations that need some initial and boundary constraints to be fully solved. The boundary may have an indirect effect on the local geometry. – Cham Oct 23 '19 at 19:29
  • The key assault on Mach's principle is that there is a notion of an inertial and a non-inertial frame even in a purely empty spacetime so spacetime has an intrinsic geometrical structure--something that Mach couldn't approve. In case you haven't, see Lubosh's answer in the post linked in my previous comment. –  Oct 23 '19 at 19:31
  • @DvijMankad, I have read Lubosh's answer, but I don't find it very illuminating or satisfaying. There is about a dozen of interpretations of the vague principle formulated by Mach, some are wrong, some are interesting and some are compatible (or partially compatible) with full GR. "Mach principle" is still very controversial today. There is no clear consensus about it even today. – Cham Oct 23 '19 at 19:47
  • @DvijMankad, here are two papers (among hundreds of papers on Mach's principle) which are worth the read, I think. In the first one, the authors define 14 versions of Mach principle: https://arxiv.org/abs/gr-qc/9607009. The second paper is more in relation to the boundary constraint that need to be imposed while solving Einstein's equation: https://arxiv.org/abs/hep-th/0612117. Maybe this comment is in the wrong place and should be placed in the other thread you cited. – Cham Oct 23 '19 at 19:54
  • @Cham Yes, you might want to link these papers there as well. Thanks a lot for the links, looks like an interesting read. :) –  Oct 23 '19 at 19:55
  • Your answer is fascinating although I don't understand the background that you take for granted in answering the question. Please consider providing links to "Lorentz invariant", "Galilean invariant", etc. – CJ Dennis Oct 25 '19 at 05:34
  • @CJDennis Thank you for pointing it out. Now that I think about it, it's a bit awkward answer given that a person familiar with all these concepts would probably already understand that these are the things that enabled Einstein to formulate a better theory of gravity. I will provide links to the relevant sources. Thanks again! :) –  Oct 25 '19 at 05:42
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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.

G. Smith
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    Calculus, the mathematical basis for Newtonian mechanics, was also unknown in Newton's day... – leftaroundabout Oct 24 '19 at 15:13
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    Newton and Leibnitz invented calculus. – G. Smith Oct 24 '19 at 15:33
  • 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. – leftaroundabout Oct 24 '19 at 15:44
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    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. – G. Smith Oct 24 '19 at 15:47
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    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. – G. Smith Oct 24 '19 at 15:54
  • Exactly, others were doing related stuff – that's the crucial thing, not Riemannian geometry in particular. – leftaroundabout Oct 24 '19 at 16:02
  • @GSmith Regarding your comment about Hilbert, in my understanding, Hilbert wouldn't have worked on it if it hadn't been for Einstein's pioneering work on the problem between around 1908-1912 tho, right? I think it was Einstein who realized the link between geometry and gravity. From there, Hilbert would've done it all by himself, I agree. –  Oct 25 '19 at 05:57
  • @DvijMankad I’ve read that Hilbert was racing Einstein to find the correct form of the field equations, but I don’t know much about the early history. You’re probably right that the key insight that curved spacetime could explain gravity was Einstein’s, although even Gauss realized that physical space might be non-Euclidean. – G. Smith Oct 25 '19 at 19:03
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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.

mathrick
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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.

WaterMolecule
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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!

Cham
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