I've already asked this question on Physics.SE, but it got no response; its not a conventional physics question, but really on how to interpret physical equations and physics.

Newtons law of gravity for two particles is proportional to the product of their masses divided by the square of their displacement.

Supposing that the particles are point particles then gravitional attraction will bring them closer together, and in fact infinitesimally closer together. Now in Newtons time there was no theory, as far as I am aware of inter-atomic forces that would have kept these two particles apart, so the gravitional attraction is asymptotically infinite. This is nonsensical, and either one can say that point particles cannot arbitrarily approach one another, or that particles can never be point particles and must have extension - this in fact includes the previous solution, as the notional point positions of the centre of mass of a particles with extension cannot obviously approach one another.

In Classical Mechanics, in retrospect this could have counted as evidence of either particles cannot be point masses; or of some then unknown repulsive force that acts at very small distances - are there any other alternatives? Why would physicists ignore then such a simple observation? What does this tell us about the process of theory-formation in Physics?

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    Classical point particle mechanics and Newtonian gravity do admit one (and only one) possible way to explain the stability of matter without resorting to additional inter-atomic forces: conservation of momentum and mechanical energy. As the gravitational potential energy becomes asymptotically infinite, so do the relative velocities. Since point particles have zero cross sectional area, the probability of collision is zero, and so particles that fall towards one another due to gravity get closer and closer until their distance equals zero, and then proceed to fall away from each other.
    – David H
    Commented Jun 23, 2014 at 16:41
  • "... and in fact infinitesimally closer together ..." Care to elaborate? If two masses are a distance d > 0 apart, at what time will they be "infinitesimally" close together, and what does that even mean? You know better, right?
    – user4894
    Commented Jun 23, 2014 at 17:45
  • Not sure I understand the question or that it has an answer. It could just mean that having d=0 was somehow impossible some other way. Commented Jun 23, 2014 at 18:02
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    Before even using the phrase "the centre of mass of a particles" (and talking about mass in general) you need to be precise about what you mean by "mass". This is such a difficult question (though to laymen and physicists it doesn't seem like a question at all) that there's even a book devoted to it: Concepts of Mass in Classical and Modern Physics by Max Jammer. I highly reccomend reading it and then coming back :)
    – user132181
    Commented Jun 23, 2014 at 18:20
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    Please consider taking these sorts of discussions to chat!! I will probably be coming through and cleaning these up... :(
    – Joseph Weissman
    Commented Jun 23, 2014 at 22:34

2 Answers 2


Theories of gravity were formed to describe interactions between very large objects: moons, planets, and starts. Certainly no scientist ever actually confused any of these objects with "point particles".

The use of point particles is an approximation when performing calculations. When the actual size/shape of an object becomes important, then scientists refine the approximations with further more complicated calculations. For example: in gravity, this led to understanding of tidal forces.

We now know that atoms, nuclei, and protons and neutrons are not point particles, because we have measured interactions that depend on their sizes. We still approximate leptons and quarks as point particles because we have not yet encountered situations where the sizes or shapes of those particles has become apparent. If/when we do, then we will approximate them differently.

  • I like your answer, but still - how do you really know elementary particles of today really do occupy some space, and are not "isomorphic" to mathematical points? If elementary particles "are" mathematical points, then @Mozibur Ullah's point (no pun intended) makes perfect sense.
    – user132181
    Commented Jun 24, 2014 at 21:42
  • Electrons and quarks are as good at point-like; at least for energies achievable in the lab. physics.stackexchange.com/questions/24001/…
    – Dave
    Commented Jun 24, 2014 at 23:55
  • How do we know that bowling balls occupy space? Because we measure them. Likewise for protons, neutrons, and atomic nuclei. We bounce photons (or other particles) off them and measure how large they are. Protons and neutrons are no more point-like than are bowling balls. Commented Jun 25, 2014 at 13:31
  • @user132181: they bounce of because of the strong force; one suppose the harder they are hurled the further in they will go; Commented Jun 25, 2014 at 16:57
  • @user3294068 neutrons and protons are composite, therefore not elementary (and thus do have a sensible size). As far as we can tell, fundamental particles (quarks, leptons) are point-like; i.e. we haven't been able to measure any deviations that contradict the model that they are point-like.
    – Dave
    Commented Jun 25, 2014 at 18:17

"Why would physicists ignore then such a simple observation? What does this tell us about the process of theory-formation in Physics?"

So far as I understand it, Einstein did a similar thing when proposing that space is "bent". One of his thought experiments was about establishing that a train travelling at the speed of light will have the tops of its wheels travelling at twice the speed of light. Impossible. The speed of light is constant, so it follows that if ..

Speed = distance / time, and we know the time element is a given, and the speed of light is constant, then the only other possibility is that space is able to distort to account for the 'extra' speed.

When this was explained to me (in a book) my initial reaction was "how did everyone miss that?!" but that's the great thing about having such notions pruned and cleansed so that even the likes of me can just about get my head around it. It also gives much credit to the brilliant abilities of Prof Brian Cox in explaining such things to the masses.

I think that's the nub of it though : It seems like a simple observation given how what was once ground-breaking and mind-stretching is now taught routinely.

So I guess what I'm saying is perhaps you (and I) are victims of the illusion that when reading a theory, we'll see things which seem obvious but absolutely were not obvious around the time the theory was being constructed.

  • Sure; but its also true that the scientific documentary record will show a more complex record - as you it is 'pruned' for the masses - I doubt that I'm the first to think of such a natural observation. Commented Jun 25, 2014 at 16:54
  • Perhaps the scientific community isn't immune to what a lot of people might feel: that "Surely someone else has thought of this", even if reading the full-pelt scientific document ? Commented Jun 25, 2014 at 17:19
  • Sure, plus the fact that a lot of detail gets missed out in the public presentation of most subjects; for example, to go with the example you've given of relativity, Hilbert came up with the equations of General Relativity pretty much at the same time that Einstein did - and this is because they were scientific colleagues; theres a similar story with Newton & Leibniz over the calculus and Newton & Hooke for gravity; I'm sure these stories can be multiplied indefinitely Commented Jun 25, 2014 at 17:48
  • BTW I've always wondered at that gedanken-experiment you meantioned - travelling with light; good to see it made clear. Commented Jun 25, 2014 at 17:50

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