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# Do the laws of nature scale?

edited January 2016
I would like to share with you some thoughts on the question of scale invariance of the laws of nature.

In the paper Newtonian mechanics: An implication of extended relativity (the paper is attached below for your convenience; if you have trouble retrieving the attachment, try here) I entertain a seemingly preposterous idea that Newtonian mechanics in its entirety can be inductively derived from a single principle which I call extended relativity. By employing the notion of inductive implication (as opposed to deductive one) I want to emphasize that my derivation relies on assumptions induced by common sense and everyday experience (like additivity of inertial mass, for example, or the simple observation that kinetic energy grows monotonically with both mass and velocity).

The essence of the paper can be summarized as follows:

(1) functional relation between kinetic energy, mass, and velocity (K ~ mv^2), as well as conservation of momentum (m1V1+ m2V2= 0) are both inductive implications of Galilean relativity. This effectively means that all three laws of Newton follow from a single principle – the principle of relativity of inertial motion. In mathematical terms, the derivation boils down to solving the following two functional equations:

2g(m) = g(2m),
f(–v) + f(v) = f(–v + w) + f(v + w) – 2f(w); and

(2) inverse square law is not merely a geometrical law from which it follows that the intensity of emanation must decay as inverse square of the distance from a point source - inverse square law is an inductive implication of relativity of scale.

When I came up with this idea of relativity of scale I was quite ecstatic - I thought I hit upon something new and of great importance. But shortly after, I had the opportunity to see the light of the Russian adage: Everything new is actually well-forgotten old. It turns out that the great Galileo has already pondered over the possibility of scale invariance of the laws of nature, but he rejected it due to its unacceptable, as he thought, implications. The following quote from Feynman gives an idea of Galileo’s reasoning that lead him to reject the principle of scale invariance:
The fact that the laws of physics are not unchanged under a change of scale was discovered by Galileo. He realized that the strengths of materials were not in exactly the right proportion to their sizes, and he illustrated this property that we were just discussing, about the cathedral of matchsticks, by drawing two bones, the bone of one dog, in the right proportion for holding up his weight, and the imaginary bone of a ‘super dog’ that would be, say, ten or a hundred times bigger – that bone was a big, solid thing with quite different proportions. We do not know whether he ever carried the argument quite to the conclusion that the laws of nature must have a definite scale, but he was so impressed with this discovery that he considered it to be as important as the discovery of the laws of motion, because he published them both in the same volume, called ‘On Two New Sciences’
One can easily see that Galileo's reasoning is not quite free of logical fallacies here: the scaling is not applied consistently at all levels of fractal structure of material universe. But let us take a look at Feynman’s own arguments against the principle of relativity of scale:
Suppose that we ask: ‘Are the physical laws symmetrical under a change of scale?’ Suppose we build a certain piece of apparatus, and then build another apparatus five times bigger in every part, will it work exactly the same way? The answer is, in this case, no! The wavelength of light emitted, for example, by the atoms inside one box of sodium atoms and the wavelength of light emitted by a gas of sodium atoms five times in volume is not five times longer, but is in fact exactly the same as the other. So the ratio of the wavelength to the size of the emitter will change... Today, of course, we understand the fact that phenomena depend on the scale on the grounds that matter is atomic in nature, and certainly if we built an apparatus that was so small there were only five atoms in it, it would clearly be something we could not scale up and down arbitrarily. The scale of an individual atom is not at all arbitrary – it is quite definite.
From the logical perspective, Feynman’s reasoning is not much better than Galileo’s. Indeed, how does Feynman know that “the wavelength of light emitted, for example, by the atoms inside one box of sodium atoms and the wavelength of light emitted by a gas of sodium atoms five times in volume is not five times longer, but is in fact exactly the same as the other”? Did he, or someone else for that matter, conduct an experiment with enlarged five times atoms? He says: “matter is atomic in nature” and “scale of an individual atom is not at all arbitrary – it is quite definite”. How so? All our experience suggests otherwise. First we have discovered molecules, then atoms, then nucleus, then quarks (not entirely sure about quarks though - this could be a bogus theoretical construct). On what grounds is based Feynman’s belief that there is an end to that process? Does not Feynman know that the hypothesis of ‘elementary’ particles that occupy mere mathematical points in space without extended structures of any kind invariably leads to all kinds of nonsense and irreconcilable contradictions? It seems logically more consistent to conjecture that there is no such thing as ‘elementary’ particle in nature at all, that the distribution of matter is inherently fractal both up and down.

Speaking of the fractal nature of matter structure, I recall an incredibly bold and daring picture A.N. Kolmogorov had painted in his 1973 lecture inaugurating us – the newly-fledged students of his mathematical school. He said something to the effect: There is nothing in physics, or in science in general, that could rebuff the conjecture that the entire visible universe is nothing but a smoke from a businessman’s cigar. He was speaking figuratively, of course, but the idea is clear: The distribution of matter in the universe is fractal in nature. A.N. Kolmogorov evidently had a clear understanding of the idea behind the notion of fractals long before B. Mandelbrot coined that word in 1975. To substantiate that claim, I resort to the words of Mandelbrot himself (see “Fractals and Scaling in Finance” by Benoit Mandelbrot, p. 115):
Scaling in turbulence. Taking a path-breaking intellectual step, Richardson 1922 adapted [Jonathan] Swift as follows

Big whorls have little whorls,
Which feed on their velocity
And little whorls have lesser whorls,
And so on to viscosity

(in the molecular sense).

The next step after Richardson was taken in Kolmogorov 1941. In a class only with Lévy, Kolmogorov (1903 – 1987) was the greatest probabilist of this century. I barely knew him personally, but greatly admired his extraordinary range of achievement. At the mathematical end of his range of interest in probability theory, Kolmogorov 1933 seemed to me too close to comfort to the work of the ultimate decorator who rearranges existing material. But Kolmogorov’s papers on turbulence were filled with novelty and daring.

• edited May 2014
Now, it is not enough to throw in some crazy idea and leave it at that. One has to deal with the multitude of logical consequences of the novel idea. While contemplating the idea of scaling invariance of the laws of nature, the inevitable and seemingly insurmountable objection is as follows. We are all accustomed to the paradigm that at small scales the nature is administered by the laws of quantum mechanics and at large scales the laws of Newton rule. But what does small scale mean? Small compared to what? And what does large scale mean? One obvious answer that pops up in mind almost automatically is this: If the object is discernible by unaided human perception then it is a large object, if not – then it is a small object. But this is an anthropological principle – a principle that we are trying very hard to get rid of in science! How come the laws of nature are tied so intimately with the size of human body and its abilities? Why are we so special?

If the laws of nature are the same at all scales, than an inescapable logical conclusion follows: Quantum effects should manifest themselves at the level of human scale, as well as at the level of Kolmogorov’s cigar-smoking businessman and, in fact, at any other scale. Evidently we don’t have much of empirical data to talk seriously about quantum effects in Kolmogorov’s unimaginably slow and big world. Such empirical data is simply impossible: long before Kolmogorov's imaginary businessman even contemplates making the next puff, the human kind will disappear without a trace as far as that businessman's perception goes. But why don’t we observe quantum effects in our own macro world, like the Solar system? My answer to that is: Maybe because we are not looking, or looking in the wrong places.

It is quite reasonable to think of the empirical Titius-Bode law as a manifestation of quantization of planetary orbits in the Solar system . Quantization is the final result of the drive to acquire more or less long lasting stable orbits via chaotic transition which is so short (relatively, of course) as to appear as a jump. The mechanism behind this quantum “jump” is the non-linear effects of orbital resonance that exist on the scale of atoms as well as on the scale of Solar system (http://en.wikipedia.org/wiki/Orbital_resonance).

One could object to this line of reasoning by saying that it is highly speculative. True, but so is the theoretical speculations about big bang, black holes, dark matter and other highly exotic and doubtful things. For example, what is the basis for expecting that the speed of gravitational waves is the same as that of electromagnetic waves? If anything, common sense suggests quite the opposite. Indeed, all of human experience tells us that the speed of wave propagation is a function of the properties of the medium in which it propagates. The speed of sound is not the same as the speed of waves in a liquid water, or in a bar of steel. Call it ether, or whatever you like, but electromagnetic wave also needs some medium to propagate in. So does the gravitational wave (assuming its existence to begin with). Taking into account enormous differences between gravitational and electromagnetic phenomena, it doesn't seem reasonable at all to believe that the speed of wave propagation in two such different mediums as gravitational and electromagnetic fields would be the same. In matters like this, it is preferable to rely on common sense judgments rather than on mathematical implications of highly speculative theories like the general relativity.

Here is what Laplace had to say about the speed of propagation of gravitational action:
We have no method of measuring the length of time in which gravity is propagated, because the action of the Sun having once attained the planets, it continues to act on them as if the attractive force was communicated instantaneously to the extremities of the system; we cannot therefore ascertain in how long a time it is transmitted to the Earth, no more than we could measure the velocity of light, were it not for the aberration and eclipses of Jupiter's satellites. But it it not the same with the small difference that may exist in the action of gravity upon bodies, according to the direction and quantity of their velocity. Analysis has shewn me, that there should result an acceleration in the mean motions of the planets round the Sun, and in the mean motions of the satellites about their planets. I had imagined this method of explaining the secular equation of the Moon, when I believed with other geometricians that it was inexplicable on the principle of universal gravitation. I found that if it arose from this cause, we must suppose in the Moon, in order to release it entirely from its gravity towards the Earth, a velocity in the centre of this planet, at least six million times greater than that of light; the true cause of this equation being now known, we are certain that the activity of gravity is much greater than this. This force therefore acts with a velocity which we may consider as infinite, and we may conclude that the action of the Sun is transmitted in an indivisible instant to the extremities of the planetary system.

The System of the World by P.S. Laplace translated by J. Pond, London, 1809. Vol II, p.233.
That’s what I call common sense.
• edited May 2014
Laplace saw also very clearly the intimate connection between the inverse square law and the principle of scale invariance. Here is how he expressed it:
We have seen in the First Chapter with what precision the almost absolute state of repose of the perihelia of the planetary orbits, indicate that the force of gravity varies according to the inverse square of the distance, and now that we know the cause of the motions of these perihelia, we may regard this law as rigorously exact. It is the same with all emanations which proceed from a center, such as light; it seems as if all forces whose action is perceived at sensible distances follow this law. It has lately been observed, that the attractions and repulsions of electricity and magnetism decrease in proportion to the squares of the distances. A remarkable property of this law is, that if the dimensions of all the bodies of the universe, their mutual distances and velocities, were to be augmented or diminished proportionally, they would describe curves entirely similar to those described at present, and their appearances would be entirely the same. For the forces which animate them, being the result of attractions, proportional to the masses divided by the squares of the distances, they would augment and diminish proportionally as the dimensions of this imaginary universe. It may be remarked at the same time that this property can only belong to the law of nature. Thus the appearances of the motions of the universe are independent of its absolute dimensions, as they are likewise of the absolute motion it may have in space, and we can only observe and recognize relative phenomena.

The System of the World by P.S. Laplace translated by J. Pond, London, 1809. Vol II, p.231-232.
• edited June 2014
... we have merely stripped off the hideous mask with which materialism has covered the face of nature to find underneath (what every one with faith in anything at all must expect to find) something of surpassing, but yet of inscrutable depth. For indeed we are entire believers in the infinite depth of nature, and hold that just as we must imagine space and duration to be infinite, so must we imagine the structural complexity of the universe to be infinite also. To our minds it appears no less false to pronounce eternal that aggregation we call the atom, than it would be to pronounce eternal that aggregation we call the Sun. All this follows from the principle of Continuity

The Unseen Universe by B. Stewart and P.G. Tait
The trite principle of continuity strikes me as ill defined. I would say instead: All this follows from the principle of scale invariance.
• Long ago it had been pointed out by Olbers and others, that if the stars be infinite in numbers, and be distributed with anything roughly approximating to an average density through infinite space, the sky ought, night and day, to be all over of a brightness of the same order as that of the sun. Is the number of stars, then, finite; or does the ether absorb their light?

The Unseen Universe by B. Stewart and P.G. Tait, pp. 113.
Authors fail to mention that there is a third hypothesis (which is the most credible, in my view) that dissolves the paradox: If the distribution of matter is fractal in the universe, then average density through infinite space is meaningless, i.e. the premise of the paradox is void.
• edited July 2014
The scale invariance of nature is fundamentally a human perception . For me The fractal paradigm came alive when I realised that logic and other cultural thought systems were merely the limiting man made rules for shaping the mind of the young. The older wiser mind comes to realise that it's perception rests on what I call the principle of acceptance. Change what one( specifically you as an individual ) accept and you change everything that depends on what you accept!

But what you accept shapes what and how,when and where you perceive. It shapes who you perceive, but says nothing about why you perceive. This is because why is a mystical hypnogogic trigger in every language system. . The question why is the last language control any person can impose on another or on oneself. "Why" in this regard becomes the way we can shift between paradigms or free ourselves from sets or systems of acceptance. "Why" enables us to choose or others to choose for us without us ever realising quite how!!

The child's question" who made god?" is fundamentally the key that reveals the scale invariance that we naturally assume. That natural thought process is driven out of some of us by social and religious pressure or adaptation, but in terms of evolution, it is absolutely vital that one can adapt to changes of scale implied in that mental process.

Where do we stop? The principle of exhaustion is the only acceptable answer. This gives each individual the choice. Most stop way too short. A few are burned out because they cannot stop! The way of all is freedom to know one can stop whenever one chooses. The consequence of stopping is the simple acceptance of where your limitations are and then the rigorous synthesis of a connection between your perceptions and the consensus of others around you( providing you accept the independent existence of others).
• Once again I lose a piece to the aether! Well never mind , it is not time to set it down here in this thread!
I will pass on its message when it comes again to be written. But almost self similarity is the key constraint Mandelbrot introduced into what is an ancient conception of scale invariance.
• edited September 2015
Behind it all is surely an idea so simple, so beautiful that when we grasp it – in a decade, a century, or a millennium – we will all say to each other, how could it have been otherwise? How could we have been so stupid for so long?

– John Archibald Wheeler
I can't help but think that the idea of the universe where scale has no fundamental role to play is simple yet powerful enough to answer John Wheeler's longing for revelation provided that this idea's rich and far-reaching implications are thoroughly grasped and fearlessly followed wherever they may lead without any reservations period.

All that is required for quantitative realization of this qualitative program is the extension of the Newtonian concept of material point (which is characterized fully by its mass and its translational mode of motion) by supplementing it with attributes that would properly characterize the spinning mode of motion of material point.
• edited January 2016
In Worlds in Collision by Immanuel Velikovsky we read:
... celestial mechanics does not conflict with cosmic catastrophism. I must admit, however, that in searching for the causes of the great upheavals of the past and in considering their effects, I became skeptical of the great theories concerning the celestial motions that were formulated when the historical facts described here were not known to science. The subject deserves to be discussed in detail and quantitatively. All that I would venture to say at this time and in this place is the following: The accepted celestial mechanics, notwithstanding the many calculations that have been carried out to many decimal places, or verified by celestial motions, stands only if the sun, the source of light, warmth, and other radiation produced by fusion and fission of atoms, is as a whole an electrically neutral body, and also if the planets, in their usual orbits, are neutral bodies.

Fundamental principles in celestial mechanics including the law of gravitation, must come into question if the sun possesses a charge sufficient to influence the planets in their orbits or the comets in theirs. In the Newtonian celestial mechanics, based on the theory of gravitation, electricity and magnetism play no role. When physicists came upon the idea that the atom is built like a solar system, the atoms of various chemical elements differing in the mass of their suns (nuclei) and the number of their planets (electrons), the notion was looked upon with much favor. But it was stressed that "an atom differs from the solar system by the fact that it is not gravitation that makes the electrons go round the nucleus, but electricity" (H. N. Russell).

Besides this, another difference was found: an electron in an atom, on absorbing the energy of a photon (light), jumps to another orbit, and again to another when it emits light and releases the energy of a photon. Because of this phenomenon, comparison with the solar system no longer seemed valid. "We do not read in the morning newspapers that Mars leaped to the orbit of Saturn, or Saturn to the orbit of Mars," wrote a critic. True, we do not read it in the morning papers; but in ancient records we have found similar events described in detail, and we have tried to reconstruct the facts by comparing many ancient records. The solar system is actually built like an atom; only, in keeping with the smallness of the atom, the jumping of electrons from one orbit to another, when hit by the energy of a photon, takes place many times a second, whereas in accord with the vastness of the solar system, a similar phenomenon occurs there once in hundreds or thousands of years. In the middle of the second millennium before the present era, the terrestrial globe experienced two displacements; and in the eighth or seventh century before the present era, it experienced three or four more. In the period between, Mars and Venus, and the moon also, shifted.
This is absolutely fascinating! And I'll tell you why. Stay tuned.

I too am curious about universal scalability and have a similar intuition that it must be infinite it both directions. Further, along these lines I'm interested in spin effects that are, apparently, perpetual at quantum and cosmic scale and how they may me harnessed.
• edited September 2015
I am pleased to report that the concept of scale invariance of the laws of nature - notwithstanding it is not phrased in those precise terms - is slowly but surely getting foothold: The Long Path to Understanding Gravity by Wallace Thornhill, a champion of Velikovsky views, at EU2015 conference Paths of Discovery.
• edited September 2015
Now let's get back to that fascinating quote from Velikovsky. So why "we do not read in the morning newspapers that Mars leaped to the orbit of Saturn, or Saturn to the orbit of Mars"? Velikovsky answers, and I quote:
True, we do not read it in the morning papers; but in ancient records we have found similar events described in detail, and we have tried to reconstruct the facts by comparing many ancient records. The solar system is actually built like an atom; only, in keeping with the smallness of the atom, the jumping of electrons from one orbit to another, when hit by the energy of a photon, takes place many times a second, whereas in accord with the vastness of the solar system, a similar phenomenon occurs there once in hundreds or thousands of years.
Following Velikovsky's suggestion ("The subject deserves to be discussed in detail and quantitatively"), let us run some detailed, albeit rough, estimates using the principle of scale invariance of the laws of nature.

How fast is moving the electron (counterpart of the Earth) on its orbit around the proton (counterpart of the Sun)? That's an easy question to answer. According to the Coulomb law and the second law of Newton we have:

ke^2/r^2 = mv^2/r

where
k ~ 10^10 (Nm^2/C^2) constant
e ~ 10^(-19) C (electric charge of electron and proton)
r ~ 10^(-10) m (radius of electron orbit)
m ~ 10^(-30) kg (mass of electron)

So, v (the orbital velocity of electron we are looking for) is given by

v = e√[k/(rm)] ~ 1000000 m/s, i.e. around one thousand kilometers per second.

Therefore the electron rotates around the proton with a frequency, v/(2πr), in the order of 10^15 per second.

Velikovsky says, and I quote:
... the jumping of electrons from one orbit to another, when hit by the energy of a photon, takes place many times a second.
How many is "many times"? It depends, of course, on what is happening around the atom - it can be anywhere from almost none (at temperatures close to absolute zero) to millions or, perhaps, even billions of jumps per second. Let it be in the order of billion jumps per second. Then the electron will, before it makes the next jump, circle the proton one million times. Now, in accord with the principle of scale invariance of the laws of nature, let us replace the word electron in the previous sentence with its counterpart Earth, and proton - with its counterpart Sun and see what we get:

Then the Earth will, before it makes the next jump, circle the Sun one million times.

Is it any wonder now that "we do not read in the morning newspapers that Mars leaped to the orbit of Saturn, or Saturn to the orbit of Mars". From the perspective of an "atom" in the scale of solar system, humans - and all other species on Earth for that matter - are creatures with unimaginably short lifespan. We are mayflies, at best, brothers and sisters. That's why it seems to us that our solar system is rock-solid stable structure - an illusion that goes back to Laplace.

Tell me about bogus, inherently acausal Copenhagen interpretation of quantum mechanics now. Absolutely fascinating!
• I would like to accept this analogy, or rather example of scale invariance but it has some flaws. In particular with respect to physical chemistry theory which allows for multiple electrons per valence shell and the theory of electron orbitals.

This is irreconcilable with patterns we see in our solar system. Perhaps there is more to understand in the way of universal fraciticality or atom and cosmic theory.

For example, we can apply this idea to us as creatures. We're made of organs, which are made of cells, which are made of molecules, etc. And in the opposite direction organs make people, which make families and companies, which make neighborhoods, communities, cities, states, etc. Any one of these levels represents a different order of magnitude with the same general pattern (albeit this is very crude). As in the atom/solar system analogy they are not quite identical in their physical manifestation from level to level (they perform very different functions for example, RNA transcription is very different from going to work and typing on a computer) but perhaps on some other level of abstraction these patterns are not all that different.

My point is perhaps it doesn't make sense to say that an atom is a solar system or vice verse, especially since we know they behave differently in the physical sense. But in an abstract way are similar. I believe these lines of thought will bring us close to a sound magnetic base.
• edited January 2018
I dream of a new science coming into view. A science in which there is the very real prospect of understanding the universal laws of conservation of order and disorder that manifest themselves in the abundance and dazzling variety of life forms – all around us – subjected to never ending transformations through the death-doorway to other, up and down, fractal dimensions of the universe.

I see life as organized and ordered form of energy, while death – as disorganized and disordered form of it. I imagine that there is a certain amount of both at every given moment, both being strictly conserved across the borders of the fractal layers of the universe.

Life must be eternal – just like the energy is. Life cannot be created and it cannot be destroyed – just like the energy is. And – just like the energy – life can only be transformed from one form to another, the transition taking place across different scales of the hierarchical structure of the universe.

Death, i.e. the disorganized and disordered form of energy, is nothing but a transient state of energy undergoing a transition from one form of life to another across the embedded fractal layers of the universe.
• edited January 2018
I cannot stress enough the importance of the so-called law of equipartition or, to be more precise, the importance of the violations of this law, which is everywhere we look. It is everywhere where there is an ordered, organized, harmonious interplay of matter.

The law of equipartition has no place in life forms of matter, which is almost everywhere. And by life forms I do not necessarily mean biological forms. The Earth taken as a whole – with its intricate interplay with the surrounding medium in the framework of the solar system – is clearly a life form.

The law of equipartition can hold true only in conditions of thermal equilibrium which is basically impossible in a structure which is fractal in nature, i.e. composed of imbedded layers of vastly different scales.
• Look at the life span of a fly, compared to the life span of a elephant. Which lives longer, a horse or a cat? Granted no predator or sickness. Humming bird compared to a eagle, which lives longer? How many heart beats a day? This number changes.the heart beat of a humming bird heart is high frequency, the beat of the elephant heart is low frequency. How many heart beats per life time? Average is close enough.