I read yesterday the Norton Dome's paper, which shows that some Newtonian systems can be non-causal, based on specific solutions of Newton's laws. The author justifies the solutions in very nice, logically consistent ways, that made me unable to falsify his conclusions.

In brief words, the thought experiment is: If a sphere is on the apex (top) of a dome that can be geometrically described by the equation $h=\frac{2}{3g}r^{\frac{3}{2}}$ (see the Fig. 1a below), we can show with Newton's laws that this sphere can start moving with absolutely no cause (not even probabilistic one). If you find this very bizarre (like I did when I first heard about it), please take a look at the paper before attacking my post.

enter image description here

Getting closer to my question: The author even makes this sound more reasonable by saying that this can be made clearer by considering the reversibility of the system. Consider a sphere at the rim of the dome, and you give it a kick with some initial velocity to reach the apex (see Fig. 1b below). If the force you use is very small, the sphere will not reach the apex. If the force you use is very high, the sphere will go over the apex. If the force is just right, the sphere will exactly stop at the apex. This shows that this system is reversible, because exactly the same way the sphere rested at the apex by a force that reached the apex, if we reverse time, it'll take the same trajectory to go down (ignoring the radial symmetry of the apex).

enter image description here

My question: Following this logic, can't we say that every Newtonian system that reaches a steady state is non-causal, because otherwise it would be non-reversible, time-wise?

Note: Please don't involve Standard Model's CP/T symmetry related topics. I know that this world is CP-violating (and hence T-violating) due to weak interactions. My question is merely about classical mechanics.

  • $\begingroup$ More on Norton's dome: physics.stackexchange.com/q/39632/2451 and links therein. $\endgroup$ – Qmechanic Oct 13 '16 at 12:06
  • $\begingroup$ It's going to take infinite time to stop moving, so when you reverse time after an infinite amount of time, it's going to take infinite time to start moving. $\endgroup$ – immibis Oct 13 '16 at 21:23
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    $\begingroup$ @immibis Actually that's not true. The paper discusses this point. This would be true if the surface is a hemisphere, which is why this kind of special dome is chosen. I recommend that you read the paper, it's so much fun. $\endgroup$ – The Quantum Physicist Oct 13 '16 at 21:29
  • $\begingroup$ @TheQuantumPhysicist I agree with your comment. Still, I wonder how an object that has 0 (zero!) speed, acceleration and jerk can actually start moving at all. The first non-zero derivative of a $t^4$ trajectory is the time-derivative of the jerk, but where should that come from? (This is close to the whole point of the paper, but maybe it's more practical: what physical cause could a change of jerk correspond to?) $\endgroup$ – bers Oct 14 '16 at 15:36
  • $\begingroup$ @bers The paper shows that this doesn't violate Newton's laws at $t < T$, $t = T$ and $t > T$. That's one of the very interesting points! The particle moves, but still without a force. That's what the equations show, and I can't honestly explain it in an intuitive way. Apparently we have to go back to the classical Quantum Mechanics philosophy: "shut-up and calculate" :-) $\endgroup$ – The Quantum Physicist Oct 14 '16 at 15:40

You're right as far as it goes -- if you can come up with a Newtonian system that reaches a stationary state from a non-stationary one, then the system must be non-deterministic.

The point (to the extent there is a point here) is that this is not as easy as you seem to assume it is. The vast majority of nice smooth Newtonian systems cannot reach any stationary state, save for having been in it forever.

The value of Norton's Dome as a thought experiment is to provide a proof that there are Newtonian systems that can reach a stationary state at all. If you can define another system with this property, it will be just as good as the dome for making whichever point you would otherwise use the dome to make.

The (slight) controversy that appears to exist around Norton's Dome is not whether the conclusion the thought experiment reaches is correct, but whether it is an interesting conclusion to arrive at at all.

The pragmatic counterargument goes something like: Yes, yes, a precisely defined Newtonian system is not necessarily deterministic, but why should we care about that? We know our world doesn't function precisely along Newtonian lines anyway, so a shortcoming of the mathematical formulation of Newtonian mechanics which requires infinite precision and therefore -- even before we consider quantum effects -- is impossible to manufacture in practice ought not to keep us up at night. It would be much nicer to know, for example, whether your favorite quantum field theory is mathematically consistent!

As a counterpoint to this, Norton's Dome serves as a relatively simple didactic counterexample to the popular conception that "classical mechanics was nice and deterministic, but with quantum theory we suddenly have to grapple philosophically with nondeterminism. Oh, woe is us!" The dome example shows that the Newtonian picture does not necessarily give us determinism -- and actually it can give a kind of nondeterminism that is far worse than what quantum theory does, in that it doesn't even provide us with any principled way to assign probabilities to when a resting mass at the apex will start to slide down the dome.


We're dealing all the time with systems that are (possibly) reversible but effectively nondeterministic: chaotic systems. In a chaotic system, an arbitrarily small pertubation of the initial state can blow up to a large discrepancy after sufficiently long time. Or, phrased the other way around, to arrive at completely different final states you only need small pertubations at the initial state. How small is determined by the Lyapunov exponent. Crucially, the required amount of initial pertubation goes down exponentially as we increase the time span. So, if we do a sequence of experiments that are supposed to show coherent behaviour over an ever longer time span, we would need to make the initial preparation exponentially more accurate with each experiment. That's infeasible (for much the same reasons it's infeasible to employ monkeys for writing a novel); in this sense the experiment is nondeterministic / noncausal.

As an example a dome is excellent, but it doesn't even need to be Norton's dome: a normal hemispherical dome / inverted pendulum is already unpredictable in the sense that a ball bearing you place “exactly” on top will, as we know from experience, quickly roll down at some side, and you couldn't predict on which.
All what Norton's dome changes about this is that it reduces the required amount of pertubation from “physically zero” (i.e. zero within the uncertainty bounds) to “mathematically zero”. It essentially makes the Lyapunov exponent infinite at a single point – turns that this can arise from a system which has to the eye no obvious singularities, but only gets discontinuous after two derivatives.

Now, systems that macroscopically reach a steady state, i.e. thermal equilibrium, essentially do so also for the reason that the (microscopic) behaviour is chaotic. This causes an ensemble of given volume in phase space – which due to Liouville's theorem can't actually change – to eventually get smeared out to an indistinguishable blob of much larger volume, thereby increasing entropy.

In this sense, yes, the mathematical properties which cause noncausal behaviour of Newtonian dynamics in setups like Norton's dome are also the ones that allow macroscopic systems to reach a steady state despite the reversibility of the microscopic dynamics.


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