Two kinds of reciprocity

The Taijitu, the emblem of the action of the Pole with respect to the world, and of the reciprocal action with respect to the Pole, inevitably reminds us of the most universal figure in physics; we are naturally referring to the ellipse —or rather, it should be said, to the idea of the generation of an ellipse with its two foci, since here there is no eccentricity. The ellipse appears in the orbits of the planets no less than in the atomic orbits of the electrons, and in the study of the refractive properties of light it gives rise to a whole field of analysis, ellipsometry. Kepler’s old problem has scale invariance, and plays a determining role in all our knowledge of physics from the Planck constant to the furthest galaxies.

In physics and mechanics, the principle of reciprocity par excellence is Newton’s third principle of action and reaction, which is at the base of all our ideas about energy conservation and allows us to “interrogate” forces when we are obliged to assume the constancy or proportionality of other quantities. The third principle does not speak of two different forces but of two different sides of the same force.

Now, the story of the third principle is curious, because we are forced to think that Newton established it as the keystone of his system to tie up the loose ends of celestial mechanics —particularly in Kepler’s problem- rather than for down-to-earth mechanics based on direct contact between bodies. The third principle allows us to define a closed system, and closed systems have been the given for all fundamental physics since then —however, it is precisely in celestial orbits, such as that of the Earth around the Sun, that this principle can be least verified, since the central body is not in the center, but in one of the foci. The force designated by the vectors would have to act on the void, where there is no matter.

Since the very first moment it was argued in the continent that Newton’s theory was more an exercise in geometry than in physics, although the truth is that, if physics and vectors were good for something, the first thing that failed was geometry. That is, if we assume that forces act from and on centers of mass, instead of on mere mathematical points. But, despite what intuition tells us —that an asymmetric ellipse can only result from a variable force, or from a simultaneous generation from the two foci-, the desire to expand the domain of calculus prevailed over anything else.

In fact the issue has remained so ambiguous that attempts have always been made to rationalize it with different arguments, either the system’s barycenter, or the variation in orbital velocity, or the initial conditions of the system. But none of them separately, nor the combination of the three, allows to solve the issue satisfactorily.

Since no one wants to think that the vectors are subjected to quantitative easing, and they lengthen and shorten at convenience, or that the planet accelerates and brakes on its own as a self-propelled rocket, in order to keep the orbit closed, physicists finally came to accept the combination in one quantity of the variable orbital speed and innate motion. But what happens is that if the centripetal force counteracts the orbital velocity, and this orbital velocity is variable despite the fact that the innate motion is invariable, the orbital velocity is in fact already a result of the interaction between the centripetal and the innate force, and then the centripetal force is also acting on itself. Therefore, the other options being ruled out, what we have is a case of feedback or self-interaction of the whole system.

So it must be said that the claim that Newton’s theory explains the shape of the ellipses is at best a pedagogical resource. However, this swift pedagogy has made us forget that our so called laws do not determine or “predict” the phenomena we observe, but try to fit them at most. Understanding the difference would help us to find our place in the overall picture.

The reciprocity of Newton’s third principle is simply a change of sign: the centrifugal force must be matched by an opposing force of equal magnitude. But the most elementary reciprocity of physics and calculus is that of the inverse product, as already expressed by the formula of velocity, (v = d/t), which is the distance divided by time. In this very basic sense, those who have pointed out that velocity is the primary fact and phenomenon of physics, from which time and space are derived, are absolutely right.

The first attempt to derive the laws of dynamics from the primary fact of velocity is due to Gauss, around 1835, when he proposed a law of electric force based not only on distance but also on relative velocities. The argument was that laws such as Newton’s or Coulomb’s were laws of statics rather than of dynamics. His disciple Weber refined the formula between 1846 and 1848 by including relative accelerations and a definition of potential —a retarded potential, in fact.

Weber’s electrodynamic force is the first case of a complete dynamic formula in which all quantities are strictly homogeneous or proportional [8]. Such formulas seemed to be exclusive to Archimedes’ statics, or Hooke’s elasticity law in its original form. In fact, although it is an specific formula for electric charges and not a field equation, it allows to derive Maxwell’s equations and the electromagnetic fields as a particular case, simply integrating over volume.

The logic of Weber’s law could be applied equally to gravity, and in fact Gerber used it to calculate the precession of Mercury’s orbit in 1898, seventeen years before the calculations of General Relativity. As is well-known, General Relativity aspired to include the so-called “Mach principle”, although in the end it did not succeed; but Weber’s law was entirely compatible with that principle in addition to explicitly using homogeneous quantities, well before Mach wrote about these issues.

It has been said that Gerber’s argument and equation was “merely empirical”, but in any other era not having to create ad hoc postulates would have been seen as the best virtue. In any case, if the new proportional law was used to calculate a tiny secular divergence, and not for the generic ellipse, it was for the simple reason that in a single orbital cycle there was nothing to calculate for either the old or the new theory.

Weber’s purely relational formula cannot “explain” the ellipse either, since force and potential are simply derived from motion —but at least there is nothing unphysical in the situation, and the fulfillment of the third principle is guaranteed while permitting a deeper meaning.

Ironically, as this new law changed the prevailing idea of central forces, understood with a string attached, Helmholtz and Maxwell blamed Weber’s law for not complying with energy conservation, although finally in 1871 Weber showed that it did so on the condition that the motion was cyclical —which in this issue was already the basic requirement for Newtonian or Lagrangian mechanics too. Conservation is a global property, not a local one, but the same was true for the orbits described in the Principia, not less than those of Lagrange. Strictly speaking there is no local conservation of forces that can make physical sense. Newton himself used the analogy of a slingshot, following Descartes’ example, when he spoke of the centrifugal motion, but nowhere in his definitions is there any talk that the central forces should be understood as if connected by a string. However, posterity took the simile at face value.

Why claim that there is in any case feedback, self-interaction? Because all gauge fields, characterized by the invariance of the Lagrangian under transformations, are equivalent to a non-trivial feedback between force and potential —the eternal “information problem”, namely how does the Moon know where the Sun is and how does it “know” its mass to behave as it does.

Indeed, if the Lagrangian of a system —the difference between kinetic and potential energy – has a certain value and is not equal to zero, this is equivalent to say that action-reaction is never immediately fulfilled. However, we use to assume that Newton’s third principle is immediate and takes place automatically and simultaneously, without mediation of any time sequence, and the same simultaneity is assumed in General Relativity. The presence of a retarded potential indicates at least the existence of a sequence or mechanism, even if we can not say anything else about it.

This shows us that additive and multiplicative reciprocity are notoriously different; and the one shown by the continuous proportion in the diagram of the Pole includes the second kind. The first is purely external and the second is internal to the order considered.

All the misunderstandings about what is mechanics come from here. And the essential difference between a mechanical system in the trivial sense and an ordered or self-organized system lies precisely at this point.

At the time it was believed that Hertz’s experiments confirmed Maxwell’s equations and disproved Weber’s, but that is another misunderstanding because if Weber’s law —which was the first to introduce the factor of the speed of light- did not predict electromagnetic waves, it did not exclude them either. It simply ignored them. On the other hand, some perceptive observers have noted that the only thing Hertz demonstrated was the reality of the action at a distance, not of waves, but that is another story.

As a counterpoint, it is worth remembering another fact that shows, among other things, that Weber had not fallen behind his time. Between the 1850s and 1870s he developed a stable model of the atom with elliptical orbits —many decades before Bohr proposed his model of the circular atom, without the need to postulate special forces for the nucleus.

Weber’s relational dynamics shows another aspect that may seem exotic in the light of the present theories: according to its equations, when two positive charges approach a critical distance, they produce a net attractive force, rather than a repulsive one. But is not the very idea of an elementary charge exotic in the first place, or should we just say a mere convention? In any case, this fits very well with the Taijitu diagram, in which a polarized force can potentially become its opposite. Without this spontaneous reversal, hardly we could speak of truly alive forces and potentials.