On Newton’s Cannonball 2.2.4

Abstract Something that has gone unexplained in every physical theory to date is the principle of inertia, i.e. the tendency of things to remain at rest, or to continue moving uniformly in a straight line, unless acted on by a force. In this note, we offer an explanation of this principle.


Newton imagined firing a canon from the top of a tall mountain (1). 

From the first law of motion (2), the canon ball will travel in a straight line at a constant speed forever. But gravity pulls the ball downward. If its speed is low, the cannonball hits the ground near the mountain. The higher the speed, the further away the ball lands. If its speed is high enough, the cannonball will travel all the way around the earth and settle into orbit. The orbit of the cannonball around the earth is a balancing act between the cannonball’s tendency to fly off in a straight line and the gravitational force of the earth pulling the ball towards its center. With still more speed, the cannonball will break free of the earth altogether…

In Newtonian physics, there are basically two forces – a centrifugal (center-avoiding) force arising from the natural tendency of things to travel in a straight line, and an centripetal (center-seeking) force arising from the attraction of gravity (1, 2). In Einsteinian physics, the centrifugal force is due to the principle of inertia, and the centripetal force to curvature of space time caused by the mass of objects occupying regions of space (3, 4). 

The earth for example orbits the sun because, like the cannonball ball in Newton’s thought experiment, it is guided, on the one hand, by the centrifugal force due to the principle of inertia and, on the other hand, by the centripetal force due to the curvature of space time. Newton wrote in 1662 that

That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it. (5)

and Einstein was able to remove this perplexing aspect of Newton’s theory of gravity. But the Einsteinian re-conception is nonetheless associated with a number of undesirable things, including the infinities at the centers of black holes (6), the incompatibility of gravity with the other forces and with quantum mechanics (7), and with the flat rotation curves of distant galaxies (8, 9) (these have lead to the ad hoc and ugly idea of dark matter). Whenever something of central importance goes unexplained by a theory (everything must be explained), the theory is incomplete -a more comprehensive theory is required- and a sure sign of the problems there are with these theories is that the role of the principle of inertia goes entirely unexplained by their authors. Richard Feynman captures the ignorance surrounding the principle of inertia when he recollects:

One day I was playing with an “express wagon,” a little wagon with a railing around it. It had a ball in it, and when I pulled the wagon I noticed something about the way the ball moved. I went to my father and said, “Say, Pop, I noticed something. When I pull the wagon, the ball rolls to the back of the wagon. And when I’m pulling it along and I suddenly stop, the ball rolls to the front of the wagon. Why is that?
“That, nobody knows,” he said. “The general principle is that things which are moving tend to keep on moving, and things which are standing still tend to stand still, unless you push them hard. This tendency is called ‘inertia,’ but nobody knows why it’s true.” (10)

To get at the explanation of the principle of inertia, consider that the great defect of Einsteins’s beautiful and practical theory of gravity is that the curvature of space is due to mass. This notion readily takes us back to an initial condition of the universe such that all the mass of the universe is compressed to a point, and this same infinite compression of mass is, by the terms of the theory, also to found at the centers of black holes. But the singular nature of the initial condition of the universe represents the beginning of the time, while the singularities at the centers of black holes in some sense represent the end of time, and these forms of curvature should therefore be quite distinct. More particularly, it should not be the case that both are attributable to the infinite action of the force of gravity. There is talk of the big bang versus the big crunch, but since both these states are associated with infinite gravity, General Relativity paints a picture of the universe that begins and ends in an identical state when, very clearly, there is an arrow of time leading from an energetic contracted state to an exhausted expanded one. If we give up this idea that curvature is due to mass (which is a combination of light and space), and employ instead the idea that curvature is due to imbalances of light and space (where the classical world is balanced, the atomic is unbalanced in the direction of light, and black holes are unbalanced in the direction of space), we will solve this and other problems, and also be able to give an account of the principle of inertia.

Mathematically, we capture what it is to be balanced, and what it is to depart from balance, thereby producing curvature, by re-expressing the tradition equation for a circle of area 1 (\pi \sqrt{\frac{1}{\pi }}^2=1) as

\lim_{x\to \infty } \left(e^{2 \gamma } \sqrt{\frac{1}{e^{2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}}\right){}^2=1

Where the traditional equation fails by implying that an energy source located at the center of this area unit-circle is undiminished from center to circumference (it has either a zero or an infinite radius), the second provides us with a potentially infinite hierarchy of energy levels that are necessarily non-infinite and non-zero. Given that gamma is a spacial case of \zeta (s)-\frac{1}{s-1} for s = 1, we can go from \lim_{x\to \infty } \left(e^{2 \gamma } \sqrt{\frac{1}{e^{2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}}\right){}^2=1 to the more general

\lim_{x\to \infty } \left(e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{\exp \left((s+1) \left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)}\right){}^{\frac{1}{s+1}}\right){}^{s+1}\right)=1

Let s = 12, and let

\left(\frac{e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2}{\hbar =e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^7 \frac{1}{n^s}-\int_1^7 \frac{1}{n^s} \, dn\right)}\right){}^2}\right){}^{1/s} = the critical line/radius on the right hand side of which curvature in the direction of light is non-classical.

If we look to the non-classical side of the critical line, we see the following series of quantized jumps goes on forever within bounds related to the Gibbs constant (which limits the size of the overshoots of Fourier sums at jump discontinuities) (17).

Since it is the differences between the partial sums/integrals and the limit that protect against the degeneration of a cosmic spiral into a circle with no radius (a point), or the degeneration of this spiral into a circle with infinite radius (a line), we know that these differences have minimum and maximum sizes. More particularly, we know that if and only if s = 1 and

e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)}}\right)^{\frac{1}{s+1}}\right)^{s+1}=e^{2 \gamma } \sqrt{\frac{1}{e^{2 \gamma }}}^2=1

then the progression associated to

\left(\frac{e^{2 \gamma } \left(e^{\frac{1}{s-1}-\zeta (s)}\right)^2}{e^{2 \gamma } \left(e^{\frac{1}{s-1}-\zeta (s)}\right)^2-e^{2 \gamma } \left(\exp \left(-\left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)\right){}^2}\right){}^{1/s}

is potentially infinite. In the case of s = 1, the difference between the partial sum/integral and the limit is always larger than the critical line, but if s is a positive real number greater or less than 1, then the difference is sometimes not less than the critical line where s > 1, and in both cases the progression is strictly finite. These two directions leading away from s = 1 give us two distinct notions of imbalance, and of curvature. Given that the fundamental state of the universe is infinite light, it follows that if s > 1, then the imbalance is in favour of light, and if s < 1, the imbalance is in favour of space. Let s = 0.9, and let

\left(\frac{e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2}{\sigma }\right)^{1/s} = the critical line on the right hand side of which curvature in the direction of space- gravitational curvature- is non-classical:

Polar plots of

e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2-e^{2 \gamma } \left(\exp \left(-\left(\sum _{n=1}^{p_x} \frac{1}{n^{12}}-\int_1^{p_x} \frac{1}{n^{12}} \, dn\right)\right)\right){}^2


e^{2 \gamma } \left(e^{-\left(\zeta (0.9)-\frac{1}{0.9\, -1}\right)}\right)^2-e^{2 \gamma } \left(\exp \left(-\left(\sum _{n=1}^{p_x} \frac{1}{n^{0.9}}-\int_1^{p_x} \frac{1}{n^{0.9}} \, dn\right)\right)\right){}^2

on the non classical side of the critical line:


Since the initial condition of the universe is, in this model, massless -since it involves no space and time, and an infinite concentration of light- it follows that absolutely speaking light has no speed, and that the apparent speed of light is due to the expansion of space. The principle that

  • light is propagated in straight lines at the velocity c regardless of the state of motion of the emitting body

is in a suitably balanced classical region equivalent to the principle

  • space expands in straight lines at the velocity c regardless of the state of motion of the body in space

It follows from the switch from the usual space-centric perspective to the light-centric perspective, that any departure in the speed of an object from the speed of the uniform expansion of space -c- requires the action of a force. So when an object is at rest or moving uniformly (when it is not departing from the speed of the uniform expansion of space), it will for obvious reasons remain that way unless acted upon by a force. This gives us an objective notion of uniformity and explains the principle of inertia. Recalling that balanced classical regions are governed by the equation \lim_{x\to \infty } \left(e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}\right){}^2\right)=1 – and therefore by the Generalized Riemann Hypothesis (18) – we can deduce all of Newton’s laws of motion:

  1. Every object persists in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.
  2. Force is equal to the change in momentum (mV) per change in time. For a constant mass force = mass times acceleration.
  3. For every action there is an equal and opposite reaction. (2)

Gravity we deduce as a departure from uniformity in the direction of accelerated expansion, and c can be tied to the fundamental physical constants using the language of the new model by reference to the fine structure constant – \alpha. The recommended CODATA value of \alpha is

\frac{7.29735}{10^3}=\frac{1}{137.036}=\left(e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^1 \frac{1.00009}{1^{12}}-\int_1^1 \frac{1.00009}{1^{12}} \, dn\right)}\right){}^2\right){}^2


\alpha = the fine structure constant

e = the elementary charge

\epsilon _0 = the electric constant

\hbar = the reduced Planck constant

k_e = the Coloumb constant

\mu _0 = the magnetic constant

R_K = the von Klitzing constant

Z_0 = vacuum impedance

and it follows that

\alpha =\frac{e^2}{\left((4 \pi ) \epsilon _0\right) (c \hbar )}=\frac{\left(e^2 c\right) \mu _0}{(4 \pi ) \hbar }=\frac{e^2 k_e}{c \hbar }=\frac{c \mu _0}{2 R_K}=\frac{e^2 Z_0}{(4 \pi ) \hbar }=\left(e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^1 \frac{1.00009}{1^{12}}-\int_1^1 \frac{1.00009}{1^{12}} \, dn\right)}\right){}^2\right){}^2

There is a difference of 0.0000127295 between the recommended value of \alpha and \left(e^{2 \gamma } \left(e^{-\left(\zeta (12)-\frac{1}{12-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^1 \frac{1}{1^{12}}-\int_1^1 \frac{1}{1^{12}} \, dn\right)}\right){}^2\right){}^2, but this value has changed over the decades, it is dependent on many assumptions, and positive and negative variations in \alpha of one part in 100, 000 are suggested by data on quasar absorption lines (20).

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(1) Newton, I (1728), A Treatise of the System of the World

(2) Newton, I (1687), The Principia: Mathematical Principles of Natural Philosophy

(3) Cajori, F (1934), Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World

(4) Einstein A. (1916), Relativity: The Special and General Theory

(5) O’Connor, J (1996), General relativity

(6) Wald, R (1997), Gravitational Collapse and Cosmic Censorship

(7) Wald, R (1984), General Relativity

(8) Rubin, V et al (1980), Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R = 4kpc) to UGC 2885 (R = 122kpc)

(9) de Swart, J. et al (2017), How dark matter came to matter

(10) Feynman, R (1988), What Do You Care What Other People Think?: Further Adventure of a Curious Character

(11) Zhou, Z (2002), An observation on the relation between the fine structure constant and the Gibbs phenomenon in Fourier analysis

(12) Davenport, H (2000), Multiplicative number theory

(13) Webb, J et al (2010), Evidence for the spatial variation of the fine-structure constant