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I am not a physicist - just a curious person.

We know from observation that Newton's formulation of physics is incomplete. We could suppose that, on a planet of perpetual fog, no-one would have thought to question it. Prior to discovering flight, we wouldn't have had much in the way of cosmology to point us in the right direction.

Question

Without physical experimentation, are there any inconsistencies in Newton's laws that would hint at them being incorrect? I'm thinking of calculations that would require dividing by zero or similar.

For example, could we have discovered, say relativity, purely from mathematical inconsistencies in the Newtonian formulation?

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    $\begingroup$ What do you mean by Newtonian physics? Is electromagnetism (before special relativity) included? is it just a way to indicate "the physics before Poincare' and Einstein"? or are you talking about non-relativistic classical mechanics and gravitation of point particles and extended bodies? $\endgroup$
    – Quillo
    Jan 5, 2021 at 2:08
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    $\begingroup$ @Quillo - Tbh, my knowledge of physics does not extend far enough to give a precise cut-off point. If I provide one now, I may invalidate existing answers. My initial thought was classical mechanics I suppose but then, foggy planet or no, I guess electromagnetism would have been discovered sooner or later. $\endgroup$ Jan 5, 2021 at 8:34
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    $\begingroup$ Re "fog"/"flight": I don't think seeing into space was necessary for the beginnings of modern physics (given the stipulation that Newtonian physics would have been discovered). The discovery of electromagnetism (and hence special relativity), electrons, photons, and nuclei (and hence quantum mechanics) in the late 1800s and early 1900s all came from terrestrial experiments. (Special relativity, although likely delayed a bit because the Michelson-Morley experiment relied on knowing Earth's orbital speed, would also be obtained from motion of relativistic electron beams.) $\endgroup$
    – nanoman
    Jan 5, 2021 at 11:08
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    $\begingroup$ Greg Egan wrote a sci-fi book, Incandescence, which describes how an alien race derives general relativity from underground experiments without access to astronomical observations. $\endgroup$
    – JBentley
    Jan 7, 2021 at 4:57
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    $\begingroup$ Special relativity was first required to make sense of ElectroMagnetism. You can derive special relativity from Newtonian principles and Maxwell's equations (describing EM). If we consider "Newton's Universe" to include EM, then it's incomplete. If we consider "Newton's Universe" to exclude EM, then the question is open (but I would guess it's complete/self-consistent)! $\endgroup$
    – PhilMacKay
    Jan 7, 2021 at 19:20

8 Answers 8

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No. Newtonian physics is self consistent.

You do get into some non-trivial conceptual difficulties with point particles. But you can simply take those difficulties as an indication of the logical impossibility of classical point particles.

There are also some interesting edge-cases where either determinism or time reversal symmetry seem to be in conflict (Norton’s dome). But pure logic does not require that the Newtonian universe must be both deterministic and time-reversible in all cases.

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    $\begingroup$ For a finite density $\nabla^2 \phi = 4\pi G \rho$ is well behaved. The potential does not diverge nor does the force for arbitrary objects. $\endgroup$
    – Dale
    Jan 5, 2021 at 0:32
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    $\begingroup$ how about electromagnetism? it was the cause of the crisis of Newtonian physics. $\endgroup$
    – Quillo
    Jan 5, 2021 at 1:59
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    $\begingroup$ While it doesn't lead to any inconsistency, the issue of the (un)reality of absolute motion and Newton's, Liebniz 's, and later views on the matter (e.g. Newton's bucket argument in favor of absolute motion, tension in a rotating rod when there is nothing else in the universe, Mach's principle, etc.) does speak to some conceptual ambiguities. $\endgroup$
    – jwimberley
    Jan 5, 2021 at 4:00
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    $\begingroup$ There's also a 5-body setup involving infinite velocities. Also, "Newtonian Physics is self-consistent" needs a source - AFAIK it's not possible to prove that, thanks to Godel's Incompleteness Theorem. $\endgroup$ Jan 5, 2021 at 10:12
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    $\begingroup$ @hyportnex Olbers paradox would not be an issue of self consistency, but consistency with observation. @ BlueRaja the 5-body setup is based on point masses, so again I would say it demonstrates a problem with classical point particles rather than Newtonian mechanics $\endgroup$
    – Dale
    Jan 5, 2021 at 19:46
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It is a little bit hard to say whether there are inconsistencies or not. It depends on what you consider Newton's Universe to be.

For example, Newton invented calculus to help with calculations. Calculus involves infinitesimals. Infinity is filled with logical difficulties that are hard to put on a mathematically consistent footing. It wasn't done until around 1900. So maybe you could find inconsistencies in math as Newton did it. But it is possible to do that math right.

Newton's laws do not predict the world as it is. It is an approximation. So are the best theories we have today. So I understand that when Newtonian mechanics predicts that atoms are not stable, that isn't what you are talking about. But again, Newton didn't predict anything about atoms. The laws of electromagnetism came afterward.

So Newtonian laws are incomplete. They don't talk about E&M, strong, and weak forces. You can add them. If you do it naively, you get laws that don't describe the universe. If you fix the laws, you get away from Newtonian mechanics.

Newton did talk about point particles, forces and accelerations, and gravity. Also optics and other topics. If you start up the universe, mechanics will predict its future forever.

So you can talk about objects orbiting each other. But you have to invent what those objects are. You can't invent something realistic without relativity and quantum mechanics and future theories we haven't worked out yet.

So you invent something unreal. That might have inconsistencies, but that isn't Newton's fault. For example, massless frictionless pulleys are often used in high school physics lessons. What is the acceleration of such a pulley given a force of $0$? You get $a = F/m = 0/0$.

You can invent classical atoms where electrons spiral into the nucleus. That isn't real either. If there are logical inconsistencies with it, they are your own fault for inventing it. So infinitely dense point particles are not Newton's problem.

Gradients of fields are inconsistent at $r = 0$ for such particles. Fields were invented by Faraday, but you might consider that fixing up the math inherent in Newton. Sort of like fixing any problems with infinity. Or you might consider it a problem inherent in something that Newtonian mechanics can describe.

Some of the math that improved on Newton came from Leibniz. He also invented calculus, and not in terms of Newton's fluxions and such. More came from LaGrange and Hamilton. These rearranged Newton's laws into other useful forms.

The math of orbital mechanics was refined. When Ceres was discovered, only 3 observations were made before it disappeared behind the sun. It would come out in a few months, but nobody know exactly when or where. It was lost. Gauss took those few months to solve the equation. Along the way, he invented error analysis and the Gaussian distribution. Ceres was found within a degree of where he predicted.

All of this was both beyond what Newton did, and consequences of it. We haven't found any logical inconsistencies that were introduced.

One last point. At the end of the 1800's, physics was a solved problem except for a few loose ends. Relativity and quantum mechanics came as a surprise and overturned that. Likewise, Russel and Whitehead had put all of mathematics on a solid consistent footing. Godel's Incompleteness Theorem came out of the blue and showed that math is either inconsistent (unlikely) or incomplete (much more likely). We might find a logical inconsistency in Newtonian mechanics in the future. Like the correctness of physics theories, consistency can't be proven. Only disproven.

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Navier-Stokes

Proving there exists a smooth and globally defined solution to the Navier-Stokes equation for arbitrary inputs is an unsolved Clay Mathematics Institute Millennium Problem. This is probably stretching a bit, but I identify the fundamental issue as the fact that "Newton's Universe" has no hard limits. As far as I can tell, infinite distance, mass, energy, velocity, and density are all allowed. If you want to get to Alpha Centauri quickly, just hop into a "Newtonian rocket" at 1g, and you'll arrive in 2.78 years, well ahead of a light beam that you send at the same time. Oh, sure, the light will leave you in the dust at the starting line, but slow and steady wins the race, right? In this race, you'll beat the light beam by a good 1.2 years, because you passed $c$ well along the way...

Now, you excluded astronomy or experiment, so we are just left with the question: "Why does anyone care about the smoothness of Navier-Stokes?" A non-smooth solution is one that diverges, or "blows up". It might try to claim that a turbulent vortex contains an infinite angular velocity or some other "obviously" non-physical statement. But is such a claim truly "non-physical" in a "Newtonian universe"? I say "no". Physicists and mathematicians working in a purely Newtonian universe may not even pose this Millennium Problem, because it is no more a problem than a rocket that arrives at Alpha Centauri before a light beam.

Or, conversely, Newtonian physicists might consider this issue and say: "Maybe it is a problem that the theory allows infinite values here. A fluid system with finite energy inputs should not obtain infinite energy densities, even over very small regions." Note that Navier-Stokes itself is a very straightforward application of Newton's laws to fluid particles, and in that regard, quite apropos, I think.

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    $\begingroup$ Starting with Newtonian physics, we get NS as an approximation. Surely, the approximation might break down, but I don't see how it points to any inconsistency in the microscopic (Newtonian) dynamics. $\endgroup$ Jan 5, 2021 at 9:48
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    $\begingroup$ @YardenSheffer if you accept NS as not just an approximation, but a precisely derived component of "Newton's Laws", then a divergence in a solution implies a situation which is intuitively non-physical. Newton might not have had a good reason to believe the universe has a speed limit, or an energy density limit for massive objects like stars, but I would hope that every physicist would be surprised by a small, closed, "cold", finite system producing any infinities. $\endgroup$ Jan 5, 2021 at 9:56
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    $\begingroup$ Surely infinite energies are unphysical, but there is nothing in Newtonian mechanics that requires NS to be valid as an exact description of the dynamics. A world of interacting small spheres is newtonian and described approximately by NS, but with no actual divergences. I am not familiar with the literature, but I am pretty sure that you could also write consistent (with no divergences) continuum mechanics by adding a \nabla^4 term to NS. $\endgroup$ Jan 5, 2021 at 10:28
  • $\begingroup$ This answer is misleading to say the least. I guess the point you are trying to convey is that there may be some equations defined in classical mechanics whose solutions may not satisfy some certain conditions, in some regimes, that we assume must always hold true. This is however true for more or less all differential equations of classical mechanics (if you drop the a-priori constraints that remove their "singularities") - so starting the answer with "Navier-Stokes" in capital giant letters implies that those are the problems, whereas this really isn't it. $\endgroup$
    – gented
    Jan 6, 2021 at 16:45
  • $\begingroup$ @gented If you automatically remove solutions that you deem "non-physical", then you have defined away the OP's problem. That's a reasonable answer, which some have given; but I was attempting to answer the OP's question in the spirit in which it was asked: could "Newtonian physicists" infer the incompleteness of Newtonian mechanics on a purely mathematical basis alone? That requires such a physicist to take all solutions seriously, and ask what they say about physical reality, to determine whether they are "reasonable" or not. $\endgroup$ Jan 6, 2021 at 21:15
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Newton never actually gave a complete description of a newtonian universe, since his work on optics was never put in a clear mathematical form. (He misinterpreted the Newton's rings experiment as a disproof of the wave theory of light, but his particle explanation wasn't fully developed.)

As far as the mechanical portion of the universe, Newton has a problem with stability, which he never clearly realized.

At cosmological scales, he came up with a way to avoid gravitational collapse by assuming perfect symmetry, but this equilibrium was unstable.

He was a die-hard atomist, so he thought rocks and houses were made out of particles interacting through instantaneous action at a distance. Results like Earnshaw's theorem tend to make this sort of thing unstable. We could try to get into more detail on whether there are newtonian models that can produce stable matter, but then we again get into the problem that newton never really developed his theories to that degree of completeness and sophistication. E.g., what are the dissipative mechanisms that would allow things to settle down to equilibrium? This was centuries before thermodynamics was understood. To avoid collapse, you could for example imagine making the particles have a finite size, rather than being pointlike as we tend to imagine them today. Or, somewhat equivalently, you could make their interaction have a repulsive hard core. This raises questions about whether, for example, a system of interacting, infinitely hard billiard balls has equations of motion that give uniqueness and existence for solutions. If you had asked Newton questions like this, he probably would have answered in terms of his experiments with alchemy (which was his passion, more so than physics).

You can worry about stuff like Norton's dome, but the lack of stability of matter means you can't build a dome in the first place -- you can't build any solid, stable matter.

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Newtonian physics has been shown to be uncomputable. It has also been shown that this uncomputability in a Newtonian universe can be exploited to build a hypercomputer that can perform a countably infinite number of computations using a countably infinite amount of memory in a finite time. Such a device works by doubling its computing speed and doubling its working memory by writing information on smaller and smaller scales every clock cycle.

This means that mathematicians living in such a universe can find the answer to many mathematical problems that may be unsolvable to us. For example, they can establish whether or not the Riemann hypothesis is true by letting the hypercomputer check if all the countably infinite number of nontrivial zeros are on the critical line.

Suppose they find the expected answer that it is true. They can then use the hypercomputer to find the proof if it exists, by generating all mathematical statements one by one and using Hilbert's proof checker algorithm to see if it is a valid proof of the Riemann hypothesis. If a proof exists, the hypercomputer is guaranteed to find it. So, if the Riemann hypothesis is unprovable, they'll find that out.

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  • $\begingroup$ Do you have citations for this? $\endgroup$
    – DKNguyen
    Jan 6, 2021 at 2:30
  • $\begingroup$ @DKNguyen Yes, I'll look it up. There is an old website on the so-called "rapidly accelerating computer", which contains a citation to the peer reviewed literature about the way one can construct such a computer in a Newtonian universe. $\endgroup$ Jan 6, 2021 at 7:58
  • $\begingroup$ Not every true statement is provable within a given axiom system. This insight is at the core of Godel's incompleteness theorem. $\endgroup$
    – tparker
    Jan 8, 2021 at 3:00
  • $\begingroup$ @tparker Yes, but this can be circumvented in some cases using a hypercomputer. One can then some cases determine if a theorem is true or not, independently from whether or not it's provable. $\endgroup$ Jan 8, 2021 at 16:05
  • $\begingroup$ Good point and interesting result. But does it answer the OP’s request for an actual inconsistency? Seems to me that it just indicates that the Church-Turing thesis fails in a perfectly Newtonian universe. $\endgroup$
    – tparker
    Jan 8, 2021 at 16:17
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It seems that no answer focuses on the cosmology angle yet. I'm no historical expert, but I believe Newton believed the universe was infinite in both time and space, and static, meaning that the stars are neither flying away from each other nor falling towards each other.

However, this can't actually work. To be a static universe each star has to experience no net gravitational force on average from all the other stars. But as soon as one region of the universe becomes, by chance, more dense than the rest, it will exert a greater force on nearby stars and thus become denser, causing a process that would snowball. The situation is unstable, and the slightest perturbation would cause the uniformity to collapse.

This is not an inconsistency in Newtonian gravity, but only a wrong assumption about what the universe looks like at large scales. In principle you could have an "expanding" Newtonian universe, with its own big bang. (The difference between that and our universe is that the space wouldn't be expanding, but rather the objects within it would all just be moving away from each other.)

In fact I think Einstein also believed the universe to be static at first, and introduced the cosmological constant in order to make that possible, before Hubble discovered that the galaxies are in fact all moving away from each other.

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  • $\begingroup$ Although this just depends on how you define "Newtonian physics", I wouldn't count this assumption of a static universe as a necessary ingredient of "Newtonian physics". It was just an additional (incorrect) physical assumption that Newton (perhaps) happened to believe. $\endgroup$
    – tparker
    Jan 8, 2021 at 2:58
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Following my comment, Newtonian gravitation involves instantaneous action at an arbitrary distance. This is not a mathematical problem, but is kind of weird with the Newtonian idea that the universe works like a machine.

I cannot seem to track it down where he says so explicitly in the text, but other references seem to suggest he does so and was aware that this is a challenge. [ Anyone who can clarify, please go ahead and edit.]

Another way of putting this is that Newtonian gravity is non-local.

As I understand it, Einstein's problem definition had the specific goal of resolving this, as well as having Special Relativity incorporated within it.

Interesting consequences of the instantaneous action include:

  • Moving an object (for example your hand) immediately affects every other object in the universe (which we now know is much bigger than was known in Newton's time).
  • The earth and sun, as examples, are currently acted on by all the mass in the universe. What is the net effect of that and can we measure it?
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This is a long shot, but in the first law there is no device for distinguishing matter at rest from matter in constant motion so perhaps it is inconsistent in Identity? So you can observe the body at rest on a table and not be aware that the table itself may be or not be in motion. Einstein made this apparent in clear thought experiment but Lorenz had done so for him in the foggy conditions.

Maybe Lorenz was already on the non-fogged other side because of optical data at his disposal?

Edit: Turns out even Einstein considered the Michelson-Morely experiment as the turning point for Aether's demise in favour of Constant speed light. So Lorentz was in un-fogged territory, and the Identity symmetry is an abstract flaw that did not render the Mechanics inconsistent.

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