Astrophysical example of non-uniqueness of Newtonian mechanics via the Dome Paradox

v. 6 n. 44

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Actually, this article has already been written. [1] But in order to appreciate this astrophysical example of the Dome Paradox, it is necessary to appreciate the "Dome Paradox." [2][3]

In this apparently trivial but rather sophisticated thought experiment, a ball is placed at the apex of a dome and subject to a uniform gravitational field. The slope is zero where the ball is placed so that it is at rest. An infinitesimal distance from this zero slope is a non-zero slope that would permit the ball to start rolling down the dome.

Conventionally, there would be only one solution in this situation because of Newton's First Law of Motion: A body at rest remains at rest. Simple, even trivial apparently. The contention with this paradox is that there could be more than one solution to the behavior of the ball, apparently breaking Newtonian determinism, along with much of practical physics; possibilities include,

1. The ball remains indefinitely. Unique solution from Newton's First Law of Motion.
2. The ball is placed on a two-dimensional situation, and may have three solutions, stay put, roll left, roll right.
3. The ball is placed on a three-dimensional dome apex. It may stay put or roll down the dome in one of an infinite number of directions. Infinite number of solutions, suggesting that Newtonian mechanics is ideally of no value.

Recall, these are ideal surfaces and situations, thought experiments; no fair waiting for any table to shake supporting the dome because of any slight earth tremor or any quantum effects.

The seemingly inescapable logic of the paradox hinges on the lack of an initiating cause. There is no "first instant in which the mass moves." Rather, "It is the last instant at which the mass does not move." A real head-scratcher. This is carefully illustrated in mathematical terms (at about 19 minutes into the rather comprehensive video). It's like one of those epsilon-delta proofs that leads to college freshman mathematics student hair pulling. [3]

Newton's "Law of Universal Gravitation" is only strictly accurate below the Solar system scale, as evidenced by its apparent error in comparison with general relativity regarding the perihelion shift of Mercury and bending of light near the Sun experiments. However, there was shown to be another form of this Newtonian equation (without altering the classical form during the derivation) that provides different solutions when gravity is taken to be fundamentally repulsive, and only phenomenologically attractive at smaller scales. [1]

Therefore, any non-uniqueness of Newtonian mechanics may indicate broader application instead of being questionable regarding this attempt to degrade the determinism of Newtonian mechanics with this non-uniqueness paradox. There seems to be two useful ways to interpret Newton's gravity -- when gravity is assumed attractive, or when gravity is assumed repulsive (depending on scale). With the latter assumption, it is shown that Newton's gravity can be scale-invariant with an implicit cosmological constant, seemingly more general than the original interpretation. The "Loophole" in the title of the video below, then, may actually be a doorway to opening Newton's mechanics -- and possibly other theories -- to greater effect. A theory commonly employed, may be more encompassing in another setting. Trying to prove non-determinism, this thought experiment can suggest that the object of criticism could be freer and of greater utility elsewhere.

[1] Newton (and general relativity) or MOND in wide binary stars? | LinkedIn

[2] Original paper: The Dome: An Unexpectedly Simple Failure of Determinism | Philosophy of Science | Cambridge Core

[3] Well organized and presented video: (115) The Dome Paradox: A Loophole in Newton's Laws - YouTube

Cover image: https://sites.pitt.edu/~jdnorton/Goodies/Dome/