Between stress and pressure

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We live between stress and pressure, this is the framework that defines our lives most immediately. But what is between stress and pressure? This Modern Prometheus’ X has a clear characterization in current Science and engineering, but oddly enough, to our scientists, so eager to make their concepts understandable to the average person, the potential of such elementary and universal connection has gone unnoticed. Between pressure and stress, only superficial opposites, we get strain or deformation, and both materials science and the constitutive aspect of electromagnetism just deal with that. This way we can establish a formal and rigorous analogy between a quantitative aspect of science and the qualitative aspects of traditional pairs of opposites that are even prior to perception. This key connection could be critical for the future of technology, our perception of nature and our self-understanding.

We all talk about blood pressure and hypertension, but do we know where the difference is? Blood pressure refers to the pressure that blood exerts on the vessels, which is lower when the blood circulates faster, and higher the slower the circulation is. Tension, on the other hand, refers to the stress to which the arterial wall is subjected, stress that can have different responses depending on whether this wall is more or less rigid, more or less elastic. And this is how we place ourselves, even without thinking: by pressure we mean the action of external forces on ourselves, and by stress or tension we mean the presence of forces in our inner constitution, which may yield or not yield at all, giving or not giving place to a deformation.

Materials science characterizes materials following this same pattern: they are the subject of strain and stress, and their resulting three-dimensional response is studied. The basic idea is that matter, which has the attribute of a relative impenetrability, can withstand stresses without deformation —thanks to its rigidity-, but can not exhibit deformations without the presence of a force, either stress or pressure, and these being mere expressions of opposite sign of the force itself.

Based on the mechanics of continuous media, the constitutive law of materials is a universal mathematical tool that applies to all types of particular subjects. And since the original formulation of electromagnetic fields by Maxwell is precisely a continuum one, it has the typical attributes of the Elasticity Theory and Continuum Mechanics, only with particular properties that are not found in ordinary matter. The symmetry of Maxwell’s equations has been widely celebrated, but nothing captures its essence better than this verbal formula we owe to Nicolae Mazilu: “The form of the stresses that are not accompanied by strains is what characterizes the case of a classic electric field classic electric, while the form of the strains that are not accompanied by stresses is what characterizes a classical magnetic field, and vice versa.”

The Continuum of the old electromagnetic theory was not only the medium between bodies, but what is both inside and outside of matter. In this constitutive framework, and in tune with the above, would be “that capable of withstanding stresses without strain when it is in the matter, and of exhibiting strains without being under stress when it is in free space“. A combination of both must be considered and this reveals “a mathematical structure describing electromagnetic radiation, and light, therefore, within them.”

In other words, electromagnetic waves themselves already support those apparently contradictory aspects attributed to the medium, so that, whether we are talking about fields or Ether, no theory can really circumvent them. Maxwell’s equations are composed of a constitutive part, relative to the material sources, and another part that corresponds to the fields emanating from them. And it is because science relies preferentially on the field laws, better suited for prediction, that Mazilu’s constitutive interpretation goes unnoticed.

In fact, the electromagnetic waves, whose two field components are not even measured with the same units or dimensions, can not really be that mutually transverse waves that geometric idealization displays in our mind. The perpendicularity of electric and magnetic fields “is not a geometrical property but a straight statistical one.”

So we have an average. But average of what? It is clear that we can not directly detect waves in vacuum, because for this we will always need some material means. And hence the only thing we can get is an average of the waves in free space and the waves in matter.

We can then consider the apparent repulsions and attractions between electric charges in terms of stress and pressure; in both cases it is a trivial relationship, since it only involves the change of sign. This is the field perspective and we don’t need to say it’s like looking from the outside. But the same does not happen with the constitutive or internal aspect: a material can be deformed or not depending on the force exerted, or else, a rigid material like a metal will have a lot of resistance but once deformed it will not recover, while an elastic material yields easily but also recover in the same measure. In this case there are margins, not everything measurable is controllable, there are reversible reactions and also irreversible responses. In short, it is not a perfectly symmetrical domain, but it is much more like life and the properties commonly observed in it —and yet, it is still obeying universal quantitative laws with which it can be measured and calculated.

The physicist will often say that the most fundamental aspect is not the constitutive one, but the one related to the field; and that the fields in turn obey the particle dynamics… but the particle is just the source or constituent part of the field. And then, if we ask what an electron is made of, the answer is usually that it is made of … electromagnetic fields, what else could it be. At most, this electron field, in order to avoid explosion, would be cohesive with a kind of tension, like the Poincaré stress, which even can be related to gravity or the cosmological constant of the Space-Time Continuum of General Relativity. And so we would shift from the Old Continuum to its modern version through the narrow tunnel of an electron. In Maxwell’s continuum there were no more limitations than those of the metric; but we only want to show that under the constitutive guise one can get something so fundamental as one could wish.

What is the Continuum? That for which it does not make sense to say “inside” and “outside”, that which transcends its categorical separation. Stress and pressure are internal and external expressions of the forces, deformations or strains can have both internal and external meaning. But if this is the case, and we oppose strains and stresses, this means that stresses can also be external or internal, and the same goes for pressures.

Our interest here is not the theory but our ordinary perception of the material reality. And it is clear that we can not aspire to a more tangible and earthly definition than this one, since the most immediate properties that we sensorially attribute to matter, its hardness, resistance or impenetrability, as well as its relative capacity to yield and deform under external pressure, are precisely those included in this definition. That is, no matter how hard we try, we will never find anything more material than this. But it is not something that depends only on our touch as an external sense: it belongs to our innermost primary sense as self-perception, as bodies in the middle of the interplay of forces.

The question arises: it is a commonplace to say there is no possible continuity between the quantitative descriptions of Physics and the qualitative aspect of our perceptions; as there can not be continuity between the analytical scientific method and the dialectical method that underlies the old world views of all peoples, those that spoke of the generation of the world, of course, by genders: the extensive and the intensive, the empty and the full, space and matter, Heaven and Earth, water and fire, mercury and sulfur, expansion and contraction, soft and hard, Yin and Yang.

But we can see that at the constitutive level the electromagnetic continuum is the most literal quantitative translation possible of the relationship of these pairs of primordial opposites, and that this relation can not be confused with the much more trivial one of the charges —action and reaction don’t follow automatically and we must consider the response capacity of the material. Of course, factors that are measurable but not controllable can occur here —which draws the limits of mathematical analysis. The mathematical relation of two 3 x 3 strain and stress matrices gives us the basic case. The continuum is partially or totally uncontrollable, and it is very appropriate to affirm, as Mazilu does, that Maxwell’s electromagnetic theory of light is an intellectual reaction to its uncontrollability.

We don’t need to convince ourselves, it is enough if we can ask about the appropriateness of this connection. Do we have here the quantitative translation of our innate sense of the pairs of opposites? I am more interested in rigorous analogy than algebra, in connection than substitution, in continuity than in discrete operations. There seems to be enough of the latter in this world, but a “sustainable development” of the former is missed.

I think that the analogy is perfectly justified in the qualitative, and that in the quantitative its limits are those of the equations of this type; but with limitations and all the connection is of the deepest interest. And electromagnetism in the language of external differential forms teaches us that it is always possible to leave down the constitutive part, with its particular metric, and to cleanly distill a field part that possesses natural invariance and greater universality. Is not this an excellent mathematical form of Alchemy? Thus seen, Maxwell’s equations are a highly universal structure applicable to thermodynamics, hydrodynamics, or many other disciplines. But it’s doubtful that the implications of this can be extracted without the proper physical motivation.

Actually all the physical fields and the ideas we can make of space are generalizations of a particular metric applied to matter, prolonging the oldest sense of the word “Geometry”. But the space of the physical continuum is not the space of geometries; a completely homogeneous medium has no interior or exterior, it is not full and therefore can’t be void either.

Today we use the fashionable language of Relativity and Quantum Mechanics, but we must not forget that both have arisen, by segregation, from the workshop of electromagnetism. Phenomena that not long ago surprised the most trained theoretical physicists, who did not even know how to fit into standard quantum mechanics such as the Aharonov-Bohm effect and the geometric phase that generalizes it, also appear in the classical electromagnetic field, and even in elementary hydrodynamic analogies on the surface of the water. There is nothing that can not be derived from the physical continuum if it is sought with due consideration —for neither quantum mechanics nor any other theory has been able to substitute the continuum for some definite type of discrete entity.

The same goes for the attempts to explain consciousness by relying on the nonlocal aspects of quantum mechanics. Is it not much easier and natural to say that Consciousness and Continuum are Identity itself, since they are the undifferentiated, and that it is Self-consciousness that introduces difference and the discrete operations of thought by virtue of this same background? The apprehension of physical reality can only be done from Reality as such, not the other way around.

Maxwell’s contemporary, the mathematician William Clifford, speculated with a pregeometry of feelings, beyond matter and geometry, in terms of contiguity and succession. But all this is already involved in Maxwell’s physical continuum, although we do not see it, and it can be appreciated perhaps in the topological aspects, independent of form and scale, that can be derived from electromagnetism or thermodynamics. Things like quantum foam seems more suitable for the surface than for the bottom.

To a large extent, and if the continuum is already the base assumption and the universal law, it is from this “pregeometry of sensations” that we are talking about. This physical continuum has a representable part as a game of forces, which is for which the equations have solutions, and has an uncontrollable but no less real part, since it can be even measured. Here we have not only a crossroads for physics, but also a touchstone for what technology can achieve with certain means. And this would be a good place to talk about possible applications of something as recent as continuous quantum measurement.

Indeed, we can explore our own internal environment with technologies no less than space can be explored: in both cases we have both the immediate impression that everything is possible, and that everything is terribly limited —as it should be, being the means always limited by its very nature.

We attempt to transcend with machine, measurement and mathematics the obvious limitations of our five senses, which, we are told, are basically reduced to small variations of the electromagnetic fields of our constituent molecules. But in reality, just as physical space can not be limited because it can not even be void, the small village of our senses are five huts built on an indeterminate ground, but that doesn’t exclude perception, and that reminds us of the old sensorium communis of the ancient philosphers. And how could this be an abstraction, I wonder, if it is the pillow of darkness over which our impressions vanish every night when we fall asleep.

Could this have some practical application? We are speaking here of the match of a physical-mathematical language, not only with the most tangible, sensorial properties of matter, but with our own self-perception and the exercising of our internal physical space too, in which the body itself acts as an interface between the objective and subjective. Therefore, almost by definition, to develop this line of reasoning would affect, rather than one or another technology, the same Gordian knot, the man/machine interface that defines their mutual relationship in both directions.

More than anything, it seems a matter of perseverance and method, of the creation of an appropriate language too. On the one hand, we see that information technologies accumulate layers and layers of increasing complexity precisely to facilitate the user experience; on the other hand, the same users demand certain resistance and naturalness in order than this same experience can have depth and “a touch of reality”. For arbitrary complexity there are no limits, but for simplicity it is the continuum what set the rules.

If we talk about the whole complex world of man/machine interfaces, it’s clear that on the side of the machine the main weight will fall in the electromechanical issues, that is, the relations between electromagnetism and mechanics. Both aspects are already integrated in the aforementioned constitutive view; but it results that for the human being as an organism those aspects are equally inherent in his internal environment, biomechanics and external activity. Therefore the deep development of this connection should have huge implications, since it gravitates towards the optimal point between immediacy and the interface depth: what we call an appropriate and appropriable technology.

And as is often the case, as the relationship deepens, the back of both extremes recede in the depth.

In fact, if the inadequacy between man and machine is not greater, it is because these factors, being constitutive or inherent, are already considered and integrated implicitly in the complex “equation” that defines their coupling —in a range of factors that goes from the resistance of the materials to the subjective experience of the user. They range from the most basic level to very subtle aspects, and the question is whether they can find a common unifying language, which would transform not only our technological culture but culture in general.

The electro-biomechanical interfaces are increasingly diversified and in them the digital took command over the analogical to an overwhelming degree, but this doesn’t belittle the least the interest of the line here pointed out. Today many wonder which, among the many possible, will be the decisive level for the direct connection between hardware and wetware, computers and neurons: this seems to be the critical line of the digital vanishing point. And yet Nature seems determined to put a brake on our impatience, or rather we should say she is totally indifferent. Although the firing of neurons is a discrete act of all-or-nothing type, the building of potentials has little to do with binary logics.

Everything suggests that these binary logics are last minute by-products, and that if we want to place ourselves at Nature’s level we have to attend the lower level functions and thence, follow their guidelines. And I think that the lowest level is neither chemical nor electrical, but constitutive or electromechanical in the sense already mentioned, as it is also the most plastic in allowing analogies or quantitative extensions —thermodynamic, hydrodynamic, and so on. The lower the level, more present this feature must be; and I, for one, cannot conceive a more basic frame with relevance for macro and microstates, and, what is more important, for subjective and objective states.

If Physics seems to play such a subordinate role in this area, it is mainly because the customary field standards and definitions are used, operating from the local to the global instead of the opposite as Maxwell’s original theory —plainly organic and integral- did; and because in these descriptions issues of orientation that can be very relevant are largely ignored. A vision that included them could perhaps shift more fluidly to the questions posed by information theory, semiotics or semantics, just as one can pass from exterior differential forms to geometric algebras (Clifford), or to Grassmann algebras that are their common ancestor.

There would be a completely unexpected twist in all this, no matter how much we would expecting for it. We are fumbling the socket to Nature in the dark, and to ourselves in passing, but in the process, if we want to match the issue, what has to change, and to a great extent, is the shape of our plug.

Physics is worthy of interest only when we remember that things are not reducible to extension and motion. In fact, when we pretend that material particles are punctual, their properties are attributed to completely intensive phenomena. Unlike in mathematics, local and global in Physics do not mean “small” and “large”. The dualities that cross modern physics, and that have their origin in the electromagnetic duality, are diverse expressions of this, too: eminent pledges of non-duality, rather than of some pretended Cartesian dualism that has never taken place except in our heads.

The same thing happened in the old hermetic version of the artists of Fire: the male or sulfur, the dissipative, the intensive, has been caught in matter; the female mercury, space or spirit is nevertheless the matrix of everything we consider conservative or mechanical, fully reducible to movement. When this is understood, the slackening arch of Manifestation recover again its Tone, and we see a Male Nature where we thought we were seeing only a Female Nature, and vice versa, and the commerce between “high” and “low” abandon a routine that couldn’t have another place than in our heads either.

Digital/analog and analog/digital converters, sensors and medicine are three of the areas that first come to mind when it comes to test these ideas at the most basic level. An ever-expanding field that specifically puts to the test the connection between mechanics, electromagnetism and constitutive laws of materials is the design of metamaterials. We will dedicate a separate article to medicine, health, aging and evolution.

References
Mazilu, N, Mechanical problem of Ether, Apeiron, Vol. 15, No. 1, January 2008
Post, E J, A History of Physics as an exercise in Philosophy