DETAILS ON ENCRYPTION AND DECRYPTION USING 2×2 MATRICES

There are a few issues related to the cryptography using the properties of the 2×2 families of matrices. As we perceived some public interest for the task at hand, it is perhaps appropriate to discuss them in a little detail.

The most important issue is obviously the possibility of a successful attack. As we can see the attack cannot be done but through Alice’s public key, or through Bob’s encryption part from the message to Alice. The intruder has the form of the function and one value of it corresponding to the value x. Most probably he will try to find Alice’s private key, for which he has at disposal the linear equation connecting the parts of the key. As the choice of n-part of the key is at random from among the positive real numbers, the chance to recover Alice’s private key, or Bob’s encryption key is given by the probability to find exactly such a value, which for all intents and purposes is very dim.

More to the point, the choice of m and n is not quite arbitrary, if we want the family of functions to behave in a certain way. For instance we want them to be monotonous and to have no discontinuity between the points a and b.  Assuming a > b, in order to assure proper continuity conditions for the encrypting homographies, we need to have m > n, in order to ensure increasing rational functions on the interval (b, a). In other words, by the simple condition m > n, we are able to secure an infinity of increasing rational function serving for encryption.

The choice of m and n should not be therefore quite arbitrary. The monotony and lack of discontinuity between the fixed points b and a can be both realized by the choice m = R(1+r); n = R(1-r), where R is an arbitrary positive number generated by a random procedure, and r is a random number between 0 and 1. The condition m > n is automatically satisfied, and the two levels of randomness make the encryption practically unbreakable.

The Physics Underlying the Differential Geometry of Color Space

There is no doubt anymore, today, that Wien’s displacement law is the crux of the methodology leading to quantum concept, and therefore to modern physics of light. Being a key point in physics, one would expect a good characterization and understanding of the law. Yet, it is quantitatively quite ambiguous: the location of the maximum of spectral density of the blackbody radiation depends on the way we represent this density as a function. This is a critical issue, especially when it comes to teaching physics purposes. However, from the very same point of view, we think that the emphasis on quantitative aspects of the displacement law is misplaced. Indeed, for teaching purposes, the logical understanding should come first, rather than the quantitative argument. From this perspective, the Wien’s displacement law, as well as its time-honored product, the Planck law of radiation, do have a strong natural-philosophical reason: they are an explicit expression of the human possibilities of performing experiments with light.

No wonder, then, that this natural-philosophical motivation emerged, by means of a special mathematical expression, into the modern theory of color vision. After all, from physical point of view the human eye serving to define the colors, is just an apparatus measuring the light, like any other one. Therefore the mathematics we are talking about can very well be taken as a theoretical tool helping in unifying the physics of light with the psychophysical theory of color vision.

The mathematical expression of the physical reasons leading to its explicit form can be historically traced back to Fresnel, the founder of modern physical optics. The case is best illustrated by the means of statistics underlying the Planck’s law of radiation, and finally it is reiterated, in a special form, to be found within a particular chapter of the modern mathematical theory of color vision – the differential geometry of color space. By this we actually build the physical foundations of that differential geometry.

Simply put, the laws of the physics we practice – specifically the physics of light – are the expression of an invariance principle deriving, on one hand, from man’s condition of being confined to the Earth and, on the other hand, from the urge to pass judgments on the environment. If, therefore, one can speak of an evolution at any level, and physics is brought to bear on the explanations, an invariance principle should be the main reason of that physics, any other one should be only consequential. This is the starting point and the conclusion of the work in the link below.

Guided by such ideas, we came to the conclusion that the insistence on a precise quantitative expressions of the Wien’s displacement law should not be critical. Rather, the mathematical form of the invariance leading to the modern physical theory of light is far more important. By the same token, the physical theory of light and the modern differential theory of colors should spring from the same general physical principle. We found a mathematical formulation of that principle in the form of a theory of measurement of the spectral energy density of light. Why the spectral energy of light? First of all because the light is the main agent carrying the information from places to Earth. Secondly, the eye is the main physical tool, so to speak, measuring the light as far as the color is concerned.

It turns out that modern physical theory of light, as well as the modern theory of color vision, are only different mathematical expressions of the very same theory of measurement, as we formulated here. This theory, in turn, can be discovered starting from the Wien’s displacement law, and uncovering its relation with the Fresnel theory of light and with Planck’s theory of quanta. This is the program we have followed in the work.

Besides building confidence in the case we have made of light and color, this program has some important marginal conclusions on the very physics in general, and the physics we use in the psychophysical theory of colors. First of all, this fact is an indication that, either all of the measurements involving the light are thermodynamic in character, or that the temperature has to be replaced with a more general equilibrium quantity, that reduces to temperature in some particular instances. This conclusion has far reaching theoretical consequences on every level of natural philosophy, all of these consequences being however intrinsically connected with each other.

Thus, on the biology and physiology level: any theory on the working principle of the eye, for instance, should start from the idea that it is a physical apparatus adapted, first and foremost, in receiving the light from different spaces, not to color. The color of the environment comes only as a consequence of this capability of the eye. One could rightfully say that the color is quite accidental. The color sensitivity could not, indeed, be an argument in a theory of evolutionary type. By the same token, one has to find the equivalent of color for the parts of light spectrum not perceived by the eye, a fact for which the transcendence of the principle formulated in the present work helps explicitly.

Secondly, the “thermodynamical character” has to be either more precisely defined, or adequately generalized. The thermodynamics itself should transcend the current concept of temperature, as indeed the modern theoretical speculations seem to indicate. The current physical theory of light accepts the temperature as classically defined, a parameter related to a sufficient statistic. It inherits this tract from the classical quintessential guiding model of thermodynamics – the ideal gas. There the temperature is related to a well defined sufficient statistic though, the only one of this kind we are aware of – the kinetic energy of molecules. It is easy to see that this is not the case with the light anymore. First of all, in the case of light, a second equilibrium parameter occurs: the frequency. This equilibrium is, however, not of the nature of thermal equilibrium, but of a more general type. For particular situations, a combined statistics of the two equilibrium parameters, temperature and frequency, seems to have worked thus far, building the modern quantum theory, but the latter developments of the theoretical physics, show its insufficiency as a concept. The fact that there is no unique relationship between color and temperature, or between color and frequency for that matter, just add heavily to this case.

the-physical-foundations-of-differential-geometry-of-colors.pdf

The Quarks and the Confinement of Matter

In a work soon to appear at Nova Publishers (Mazilu, Agop, 2012) the idea is promoted that there is no discontinuity between the classical Newtonian natural philosophy ang modern quantum theory of atoms. The book insists upon the hot topic of the modern theory of nuclear matter, placing it explicitly within classical Newtonian philosophy, through the modern theory of Skyrmions, in the geometrisation of Nicholas Manton (Manton, 1987). It suggests that, from the point of view of harmonic maps - one of the essential tools of the modern theoretical physics (Misner, 1978) - there should be no difference between the theory of gravitation in Einstein acceptance and the theory of nuclear matter in the Skyrme acceptance. While insisting upon a classical image of nucleus, the mentioned work has no attempt of touching the wider realm of elementary particle physics, other than suggesting a relationship between Newtonian view of forces and modern Regge-like theory of particles via SU(3) representation of some Regge families (Dothan, Gell-Mann, Ne’eman, 1965), and also a specific justification of the very Skyrme ansatz. Here it is shown, with the help of articles from the present pages, that the general theory of particles can be safely linked to the classical Newtonian natural philosophy, indicating also the way of this link.

The general philosophy starts with the observation that there is free space, containing nothing in it, and space containing matter. Any point from the space filled with matter has some constraints in the form of pressures. These are taken here as being both positive and negative. It is in this respect that the space containing matter is different from the free space. The concept of pressure is quite complex, even disregarding the important issue of the sign of pressure, but can be explained with the help of the idea of force, which is quite handy for man.

The force was always used rationally in order to create motion on a given direction. That motion is usually employed in conjunction with the inherent property of inertia of matter, otherwise one is not able to exploit the technological possibilities of force. It is this process that led further to the ideas of vector and of material point. Indeed, the force has unconditionally the anticipated result, i.e. it is a vector, only in case it acts on a point. One might say that it is for this reason that the concept of material point was created.

One can simply show, however, that it is not always the case of the anticipated result in the action of a force, if we squeeze a piece of wax or modeling clay between fingers. First of all, in such a case we are by no means entitled to use the idea of material point, simply because the piece of wax has a noticeable space expansion. It is due to this expansion that in spite of the fact that the force acts only along the direction of the squeezing fingers, the matter responds along a perpendicular direction too. We can imagine that this effect is universal – every piece of matter has it, more or less of course – or even independent of the force – the so-called phenomenon of creep of matter.

In cases where the matter can be considered as a continuum, the pressures in any point of the space containing it, can be algebraically described by a 3×3 matrix with real entries. This description is a consequence of the fact that the human experience shows that the pressures display, in every point and always, three extreme values along three directions in space. These three directions can thus be characterized as those directions in that point, along which, when a force acts upon matter, has the maximum anticipated effect. This is the essential difference between the force in free space and the force in the space filled with matter: the force in matter cannot be a vector. In any other direction in space the values of the pressures are not extremes because part of the force is dissipated for the displacement of matter on directions perpendicular to the direction of the action of force, just like in the case of clay.

Ideally, i.e. in the absolutely continuous matter – and only from mathematical point of view we should say – one can guess that the three extreme pressures are determined, in any point of matter’s host space, by some directional statistics. In building these statistics, one can start with the observation that in every plane through a point, the external applied force is ‘distributed’ along normal, as a normal component, and in plane, with components in all directions of that plane. In some special planes, however, these components are simply determined as statistics, only by the three extremal pressures in the point in question. These statistics are two in number: one of them is the mean pressure on the special plane, which can be calculated simply as the arithmetic mean of the three extremal pressures. The other is the mean shear on the special plane – a measure of the capability of isotropic continuous matter to deflect the effect of the force from the direction of its action. This last statistic, a vector according to its geometrical nature, is defined in an arbitrary direction from a special plane among those of which we are talking here. One can thus say that, conversely, the extremal pressures in a point of the host space of the ideally continuous matter, are uniquely determined up to that arbitrary direction, on which they depend in a specific way: there are three extremal pressures for any such direction.

It is now the time to say something about the special planes we are talking here. The scientists call them octahedral planes, for they are the planes of the faces of a regular octahedron in space. Therefore they are eight in number, each one with two statistics of the pressure on it, and with the extremal pressures depending on the direction in the plane. However, even such an ideal octahedron has an arbitrary orientation in a point in space.

It can be proved out though that the space has a word to say here. Namely if the host space of the matter has constant curvature, again an ideal situation related to the continuity of matter and to the physical possibility of electromagnetism, then the directions of constant curvature are the ones determining the orientation of the “local octahedron”, and therefore of the eight local special planes.

Any material structure humanly perceived, no matter how ephemeral, means necessarily discontinuity in the continuous matter: at least one of the properties of the continuum is affected. The human perception seems to be the basis of the old principle παντα ρει of Heraclit of Ephesus, for nothing can be perceived if it is not discontinuous: ideal continuity means immovability. It is in this way that the idea of Newton appeared, referring to the force between different pieces of matter. This idea extends the practical understanding of force, namely that of creating motion. For, if a force creates motion, nothing precludes us of thinking that a force is what moves the celestial bodies. This is the Newtonian force, acting through space at a distance, and it cannot be appreciated but mathematically, through the relative motions of the pieces of matter. Asking that the matter has a structure, means therefore implicitly that it is discontinuous and the principle of discontinuity extends even to the space filled with matter. It is this extension that led to the explanation of pressures by Newtonian forces, and here is the place where the eightfold way starts to… unfold, entering the realm of the fundamental forces of physics.

Phenomenologically the particles of the microscopic world have only very small lifetimes. More to the point, their existence is not accessible directly to our senses, at the usual scale of time, but only can be inferred from specific experiments, the particles having very limited times of existence by comparison with the times accessible to our senses. It is like these particles are simple accidents. A hint of their objectivity is however given throughout mathematics, which allowed a certain ‘census’ so to speak, by the way of some algebraical structures. This census stirred in turn one of the most fruitful theoretical ideas of modern physics, that of the existence of three basic constituents of the matter made of particles – the quarks. These fundamental pieces of matter have never been observed in a free state, but always ‘second-hand’ as it were, i.e. from experiments indicating only, their theoretically anticipated presence. The idea started then getting in, that these fundamental bricks of matter cannot even be observed in a free state, because they are actually ‘confined’ in matter by their very nature. Any speculation on their confinement (and deconfinement, naturally!), is always conducted through arguments reminding us almost explicitly of the experimental origin of the problem, aptly baptised by Gell-Mann ‘the eightfold way’. Its experimental roots and the main characteristic – the confinement – is easily explained in terms of the pressures in matter as presented above.

Going as far back as the times when Augustin Fresnel started building our theoretical image of light – the wave surface – from bits and pieces accessible to experiment, the idea started getting in, that the technology must, at any rate, enter the stage of human possibilities of describing the nature. The modern particle physics is but the highest technological expression of such philosophy. Indeed, we have no other possibility of experimentally revealing the properties of matter but through matter: there is no other way! The matter revealing the properties of particles has specific forms, expressions of some top technologies: particle counters, bubble chambers, photo-plates and any variations technologically advanced of these devices. The principle of revealing is quite simple: the particle penetrating the revealing matter in a certain experiment is visualized by a theoretically anticipated trajectory, and experimentally ‘dotted’ by local changes in the structure of the revealing matter. One can understand, and even explain rationally, the ‘dotting’ process by the fact that the external particle perturbs the local state of pressures in a point from the revealing matter. In any point of the host space exists such a state of pressures in matter. However, the external particle only perceives them through their two means described above, whose space support should be one of the eight local octahedral planes through that point. The particle perturbation can then be described by a ‘phase angle’ representing an arbitrary direction in that plane. The choice of octahedral plane in a point depends on the nature of the revealed particle. The octahedron itself is imaginary: it is hard to believe that the matter organizes geometrically so… Platonically! However it gets through in the theoretical census of the particles, in the form of an ‘eightfold way’. To round up the idea, every particle, to the extent to which it is ‘elementary’, chooses one single resultant of the eight possible resultants of the local forces in the revealing matter, and by this, one of the octahedral planes supporting the force that goes with that resultant. The octahedron itself is an expression of the curvature of the host space of matter. It is then quite natural that the idea of quark, generated according to this technological manner of visualizing the microscopic world, should come along with the essential property of the matter revealing the particles: the confinement. From this point of view the forces of confinement must be therefore described by the mathematics of pressures rather than by Newtonian vector forces, deriving for instance from a potential, as it is the current habit in the physics of elementary particles.

References

Dothan, Y., Gell-Mann, M, Ne’eman, Y. (1965): Series of Hadron Energy Levels as Representations of Non-Compact Groups, Physics Letters, Vol. 17 (31), pp. 148 – 151 

Manton, N. S. (1987): Geometry of Skyrmions, Communications in Mathematical Physics, Vol. 111, pp. 469–478

Mazilu, N., Agop, M. (2012): Skyrmions – a Great Finishing Touch to Classical Newtonian Philosophy, Nova Publishers, New York

Misner, C. W. (1978): Harmonic Maps as Models for Physical Theories, Physical Review D18, pp. 4510 – 4524

New Testament and the Science

Nicolae Mazilu 

We are plainly aware that the New Testament helps us in choosing the right way in life, at any level of manifestation of the human being: from the individual to social level. In virtue of this observation we claim here that this maxim is valid for science also. Moreover, it is valid only for science. For, in conducting a rational life, if the man doesn’t have the New Testament, then he has not at his disposal but only the science.

  • Perceived like this, the New Testament has a first lesson for us: if the science, like any other human enterprise, does not realize to what extent it comes from and is committed to sin, then it will have no law. In proper words:

    … For by the law is the knowledge of sin. (Romans, 3:20)

    The true science has therefore this sacred task of giving the man the sense of law and by this the sense of sin. If the science does not offer us this awareness, then it has really no laws. This seems to be indeed the reality today. Speaking specifically of physics, the basic laws are only conventional limitations; sometimes we cannot even say that they are axioms. One can recognize this by the fact that they always lead to paradoxes, especially when it comes to their mathematical consequences.

  • Can we pinpoint the sin that comes with the science by a law in the Scriptures? Certainly! It is the second commandment of the Old Testament

    Thou shalt not make unto thee any graven image, or any likeness of any thing that is in heaven above, or that is in the earth beneath, or that is in the water under the earth. Thou shalt not bow down thyself to them, nor serve them: for I the LORD thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me; And showing mercy unto thousands of them that love me, and keep my commandments. (Exodus, 20: 4, 5; our Italics)

    Clearly, on occasions (very often lately!) the science transgresses this second commandment of the Old Testament. And we cannot do anything, for this is the nature of science: to work with likenesses. The only escape here would be to work with acceptable likenesses, and the whole New Testament shows us what this likeness is: the human body.

  • The human body is indeed among the earthly things, however not a “likeness” among those prohibited by the second commandment. In fact, if it is ever considered a “likeness”, it should be taken as in “after Our likeness”, from the sixth day of Creation (Genesis, 1:26): it is indeed holy. Pushing this conclusion a little further, one can say that the real science springs from those moments where it was not a sin, of which we can mention only two, the ones that make explicit reference to human body. These are the Copernican model of the Universe and the Newtonian association of the force with the celestial harmony. This last moment of science gave us the very possibility of application of mathematics in scientific predictions, and shows that in this science the contradiction is obsolete - the concept of falsifiability is practically superfluous. Therefore it proves that the mathematics itself is actually carried over from the Edenic life of man.
  • It is mainly from this perspective - of the human body - that we need to consider the New Testament. Following this idea we can extract the principles of a correct natural philosophy, of which many signs are already at our disposal in the modern scientific observation of Nature. For, Jesus did not proceed, did not act or speak, but only respecting the Law, i.e. without sin. We shall evaluate later those signs from science, by limiting our analysis only to the very foundation of the modern science. For now, let’s just give our reader the New Testament criteria of selection of the essential facts of the science.
  • To begin with, if the human body is sanctified in the Old Testament, by the very act of Creation, there is nowhere a more harmonious description of it, of its role and its essential tasks, than in the New Testament. Here Jesus Christ makes a key point from giving us the understanding that the human body is the very temple - the place of Trinity - that cannot be actually destroyed:

    … Destroy this temple, and in three days I will raise it up (John, 2:19)

    The reaction of Jews was genuinely human, and certainly it can qualify as scientific, in the sense in which the science is usually accepted and understood today. It is actually the reaction that the man has without the awareness of sin - strictly speaking with the consciousness of the fact that he is right. Everyone of us, facing the facts of social life, can logically understand such a reaction. On the other hand, facing those very facts of the social life is what makes apparently hard to understand the acts and explanations of Jesus. For instance:

    Then said the Jews, Forty and six years was this temple in building, and wilt thou rear it up in three days? (John, 2:20)

    This reaction is clearly dictated by the lack of understanding of the actual role played by the temple, i.e. by locating the sin in some other place than where it belongs. For, it is clear that, no matter of the point of view, the temple is the center of the social activity. In this capacity, its use deteriorated in time, because the social life evolved in history. It began to mean, among others, the spiritual alienation of man. Indeed, the center of social activity began to mean, at some point in time, also political center, commercial center and such like, qualities which Jesus explicitly disapproved, trying to resuscitate in us the first attribute of the concept, the natural one. According to this, the social activity should actually mean the preservation of one in the determination of the multiple. The ‘multiple’ is the innate determinative of man. Indeed, even if he is created “in Our image, after Our likeness”, this doesn’t mean that man is God, by the very fact that “image” and “likeness” have, first and foremost, the determination of the multiple. But then, this is also the very meaning of the temple: the place of dwelling of God, the One in His multiple determinations. This should therefore be the human body - the natural temple - and this is apparently what Jesus meant!

  • This episode from the New Testament shows just how much can science separate the man from the true understanding of natural things. But it also shows that a proper awareness when it comes to human body can resuscitate in man that very true understanding of natural things. Indeed, the science as we know it today, was born, and has grown, specifically along the ideas of a “limited multiple”, involving elimination of outsiders by destructive force. This is why the science considers the force - its fundamental concept - only in its particular instance of the destructive capability. This is, on one hand, exactly the message we can draw from the essential steps of science, and on the other hand is surely what Jesus rejected through His acts and teachings.
  • Symptomatic should be therefore for us, first and foremost, the fact that He doesn’t use the force as man understands it, but even rejects it - and everyone of us knows with what painful consequences - as if specifically to make us aware of the fact that the force, in the destructive determination of its concept, is not essential for our life. Following this principle, we can see that the force maintaining the harmony, therefore the one accounting for freedom, is actually endorsed by mathematics as the basis of science. Obviously, this is the kind of force accepted and promoted by Jesus, and therefore this should be the true determination of the concept. What is the reason?
  • The force as conceived by science today is not essential because it cannot destroy the whole. Therefore it cannot be taken as a concept. The truth of this statement is certified by the resurrection of Jesus Christ Himself. This fact is nowhere shown but only in the New Testament, so that only Christians would be able to understand it properly. In our opinion, this is even the reason why the modern science is actually created by Christians. Indeed, as we just said before, the temple is the perpetuation of one in the determination of the ensemble. From the point of view of the perpetuation of the being, the temple has the role of directing the action of ensemble: that action has to be the action of ensemble as a whole. And the image of the whole is, again, the human body - the temple Jesus was talking about. As a whole, it can regenerate in resurrection, for no unnatural force can destroy it.
  • The historical and social case of the New Testament is a natural situation used scientifically, in the manner in which the experiments are used. The man had to go through a certain period of his social evolution, in order to be able to assimilate a message like that of the New Testament. Yet, unlike the case of experiments as they are scientifically understood today - i.e. provoked according to a preconceived image, like creating the landscape from a given map - the “experimental” situation of the New Testament was not one technologically built, but was entirely “natural”, although according to a nature that man created. On one hand, we can say that God gave us another chance of coming back to the status we had by Creation. On the other hand, we can see from this, the way in which an experiment must be conceived - exclusively for observation - and we should take special heed of this teaching. Securing preconceived conditions in order to verify a certain assumption, does not make a natural law from that assumption!
  • By His nature - and this fact is most important for what we have to say here - Jesus Christ cumulates in Himself the world from its beginning, even together with its evolution, because He is also the Son of Man. He holds the godly nature in the form of power of the Creator to act upon His creation, and this is revealed mostly by His relation with the physical world, with that part of Nature, external to human body, and which we sometimes designate as “dead” or “inert”. This is especially the place to pay due attention, because He doesn’t act but according to the natural laws, which He actually summarizes in Himself. If the science is indeed after these natural laws, then it should give up the arrogance of declaring the circumstantial findings as natural laws, and accept the laws as they are presented in the acts and teachings of Jesus Christ.
  • We thus have a firm criterion of choice of the acts and teachings of Jesus, which indicates to us what really are the laws that science must hold as natural laws. This criterion is simply that of following those acts and teaching related to the human body, either His own body or that of the individuals He encountered. The force, according to its natural concept, does correspond actually to the necessities of describing the universe as a whole. And we understand the word “whole” in the meaning first given by Copernicus, when he proposed the heliocentrism: it is the meaning given by harmony, whose unique image is that of the summit of Creation - the human body. It is therefore the time to show that the New Testament, as a new covenant, is mostly a covenant for that science which we call objective. And as Jesus has not expressed Himself but only by acts and parables, His acts are not miracles as usually believed, but plain natural acts, according to natural Laws. This shows, implicitly, what are those Laws indeed.
  • The Laws of Nature According to the New Testament

    The difference between the human natural philosophy and the New Testament is that the first one claims explicit statement of the laws of Nature, while the New Testament gives them implicitly, in order to be understood regardless the particular of human experience. This is the main reason why the natural philosophy falls usually under the spell of axiomatics, having actually no laws. According to the New Testament, ther are but three main laws of Nature. 

                First and foremost we find in the New Testament the law that we like to call of the whole and the part: every part of the world around us is perceived by man as a whole. No other manifestation of Nature can be found but only by respecting this law. This is how the world around is presented to our senses, and certainly this is what Christ respects in the first place. In its moments of truth the modern science came out as we have it today, only respecting such a law. In the very first moments of natural philosophy, the whole was the human body, and it was taken as the expression of harmony, being by itself a whole part of another whole.
                Secondly, the universe is made obvious to our senses by what we call matter. The New Testament then shows that the matter is submitted to the law of accumulation, i.e. of growing, until it can reach our senses or is reached by them. This law, stated in one parable of the New Testament, is obvious by itself in the history of science. Indeed, if this law wouldn’t act, we couldn’t have at our disposal nowadays concepts like substances, elements, and such like; the chemistry as well as the physics, would not be possible.           
                Finally, there is a law that shows how the wholes behave with respect to each other, that we’d like to call the law of density. It shows that the relation between the wholes is actually decided internally. However, the man, being of “little faith” hasn’t even succeeded to bring this law among the basic facts of the modern science, to say the least: it is only a fact that needs to be explained.           
                It is interesting, in the first place, to bring the acts of New Testament which show these laws.

    THE THEORY OF TIRE-ROAD FRICTION

    The rolling tire interacts dynamically with the road surface, under the constant constraint of the load of vehicle. There are a bunch of distinct characteristics of this interaction, reflected in the friction force, for the case of traction and braking properties of the tire, in the noise produced by the tire when rolling, in the hydroplaning properties of the tires, etc. All these depend upon some primary properties of the tire surface, of the rubber of tire, and of the road surface roughness and its physical condition (dry, wet, erroded and so on). In order to decide on the theoretical description of the way all these factors intermingle during the rolling of a tire, it is necessary to start from the basic process involved at any level in such an interaction: the local deformation of a surface.

    A Classical Description of the Surface Deformation

    Let’s therefore describe the so-called infinitesimal deformation of a surface, the way it is classically described (Guggenheimer, 1977). By definition this is the process of deformation in which the position vector on surface varies infinitesimally, something like

    image0014.png

    (1)

    so that the change in the first fundamental form of the surface is always smaller than ε even if this quantity is arbitrarily small:

    image0024.png

    (2)

    As from equation (1) the deformed first fundamental form can be written as

    image0034.png

    (3)

    and therefore

    image0044.png

    (4)

    the condition (2) is satisfied if, and only if, the two infinitesimal vectors are perpendicular:

    image0054.png

    (5)

    The degree of arbitrariness of the deformation vector image0066.png is thus somewhat reduced, in the sense that we can assume the existence of a vector, image0074.png say, such that

    image0084.png

    (6)

    About the vector image0074.png we don’t know anything. However, about its differential we have a wealth of information. Indeed, as in the left hand side of equation (6) we have a vector with components exact differentials, applying the exterior differentiation to it should yield the null vector. This condition comes to

    image0094.png

    (7)

    where the symbol ʻ×˄’ shows that in the cross product of vectors, the regular multiplication is replaced by exterior multiplication of the differentials. Thus the equation (7) can be expanded over, just as usual, in the form of a 3×3 determinant

    image0103.png

    (8)

    provided we replace the regular multiplication by exterior multiplication of the differential forms. The result is a set of three differential equations:

    image0113.png

    (9)

    The first two of these tell us that dp3 is null, therefore the vector image0074.png must be chosen such that its component along the normal to surface is constant. This circumstance is ideal in introducing, for instance the deformation of the tire due to the global loading force given by the dead weight of the vehicle, which is approximately constant. Then, on top of this deformation, the local details on the footprint come into play. Indeed, when using the Cartan’s lemma, the last of equations (9) tells us that there is a symmetric 2×2 matrix with entries A, B, C say, such that:

    image0124.png

    (10)

    As, on the other hand, the vector image0133.png is an exact differential vector, we can write

    image0143.png

    (11)

    Expanding the last of these equations after the manner of the classical theory of surfaces, we get the vectorial equation

    image0153.png

    (12)

    amounting to the system of three differential equations:

    image0163.png

    (13)

    The first two of these equations show that image0133.png is transported by parallelism along the (already deformed) surface, while the third provides a condition we need to impose on the symmetric matrix from equation (10). Indeed, as the equations of definition of the curvature vector of the surface are (Guggenheimer, 1977):

    image0173.png

    (14)

    if we use the equation (10), the third of the conditions (13) comes to

    image0183.png

    (15)

    For an algebraic – and physical – interpretation of this result, let’s notice that, because image0133.png is an intrinsic vector with respect to surface (its component along the normal to surface is null), the cross product of this vector with the elementary displacement on surface is oriented along the normal to surface. This vector is

    image0193.png

    (16)

    Consequently its magnitude is a quadratic form, algebraically conjugated to the second fundamental form of the surface (apolar to that form). One can therefore say that it adds to the second fundamental form, thus changing the local curvature of the surface. In order to assure this condition by default, one can define the coefficients A, B, C, up to a multiplicative factor, by equations

    image0202.png

    (17)

    where λ, μ, ν are three external physical parameters, or even three differentials determined by the second fundamental form itself, when the situation is referred to a previous state of the surface.

          These last parameters may describe the material of the deforming surface from a constitutive point of view. On the other hand, they may be even variations of the first fundamental form of the surface, therefore deformations properly speaking. This leads to the idea that the deformation described by displacements is not equivalent to that described by the variation of the first fundamental form of the surface. Usually there are also necessary some conditions relating the variation of the metric tensor with that of displacement vector. As long as one assumes that the metric tensor must depend exclusively only on the coordinates on surface, the conditions for infinitesimal deformation define, for instance, the Killing vectors of the first fundamental form. The equations (17) show nevertheless that, no matter of the physical nature of the deformation of surface, if this one is infinitesimal, the second fundamental form of the surface is involved directly and explicitly in its description, together with some external, perhaps physical, properties. This purely speculative conclusion mirrors the practical observation that there is no deformation of a surface in a point, that can proceed without being noticed by a variation of its normal direction in that point. Therefore this manner of describing the deformation is particularly useful in characterizing the friction process between two rough bodies, like the tire tread lugs and the road surface. Here the constant normal component of vector image0074.png is simply the normal contact force – the loading force. The other two components don’t even matter but in the final end, and at a finer scale, after we have characterized their variations, possibly as stochastic processes due to roughness, and even after we have integrated these processes. As a matter of fact, the process of friction between two surfaces, offers an example where the force in surface should be specially defined by a differential 2-form, as the following analogy shows.

    A Physical Interpretation by Electrodynamical Analogy

    The equation (6) reminds us of the definition of force in electrodynamics. This science was built upon the ideas of deformations and stresses, but the definition we are talking about does not rely on such concepts, but mainly on the kinematics extracted from experience. That experience shows us that a piece of wire through which an electric current passes, assumes a rotation when submitted to a magnetic field. This rotation is determined by a force proportional to the intensity of the current and the density of the magnetic field lines, and is perpendicular on both the lines of the magnetic field and the element of current:

    image021.png

    (18)

    Here image022.png is the characteristic density of the magnetic flux lines – the so-called magnetic induction – and I is the intensity of current passing through the the piece of wire characterized by the vector image023.png. As well known, the piece of wire is immaterial here, so that the formula (18) can also describe for instance the bending of the trajectories of a current of electrons or ions in vacuum.

          Now, let’s associate to the deformation vector image024.png a force, say by a Hooke-like formula, or by some other mathematical means, involving for instance a ‘matrix of stiffnesses’, regularly used by engineers in the estimation of deformation forces in finite-element calculations:

    image025.png

    (19)

    Then the equation (6) shows that, just like in the case of magnetic field from electrodynamics, the vector image0074.png is related to some density of ‘flux of lines’ entering the physical surface under deformation. This analogy has, in the case of tire, a nice connotation that can be properly speculated in the theoretical physics of tire. Let’s describe it.

          The road roughness is ‘felt’ by the tire through the intermediary of its footprint, which is determined by the dead weight of the vehicle, by the material properties of the tread rubber, as well as by the tread design. In rolling conditions the road roughness influence on tire can be imagined as a flux of tiny ‘pokes’ in the portion of the surface of tire delimited by the footprint. This flux of tiny pokes, acting like the “rain on the roof”, as they sometimes say, is actually the analogous of the induction of the magnetic field as described above. More to the point, an appropriate expression would be ‘rain on the windshield’, because that ‘rain’ is strongly influenced by the motion of the vehicle. As the equation (16) shows, the details of local deformation on the very footprint or, even better, on the tread lugs, are then to be read in the variation of the second fundamental form of the external surface of the tire rubber.

    Description of the Process of Wrinkling

    Obviously, the rubber surface is prone to wrinkling: that much we notice even in the daily life. The wrinkling of a surface, is a process dependent on the state of previous wrinkling, and is physically assisted by the friction force. Therefore, one can assume that the friction force is null when the wrinkling is absent. Let’s show how this observation can be theoretically put into equation. First of all, the state of wrinkling is aptly described by the local curvature of the surface: the more wrinkled the surface the many more variations of curvature we notice in a certain portion of it. Secondly, the variation of curvature is described by the two differential forms of curvature (the components of the variation of normal unit vector of surface). On the other hand the wrinkling is physically determined by the friction (rubbing) force, which is a force acting tangent to surface, therefore a force ‘in-surface’, as they say. Then, according to one of the Cartan lemmas the friction force can be written in the form

    image026.png

    (20)

    where f1 and f2 are two conveniently chosen 1-forms. Indeed, only in this case the friction force is zero if the curvature is zero and reciprocally. If the ‘conveniently chosen’ 1-forms are the components of the first fundamental form s1 and s2, then equation (20) gives us the definition of the curvature matrix as a limiting case. Indeed, it can happen that there are no friction forces, even if the surface is curved. Therefore the friction forces are null, without the curvature being null, so we have zero in the left hand side of equation (20). In this case, another one of the Cartan lemmas, applied to equation (20), shows that we must have

    image027.png

    (21)

    which is the usual definition of the curvature matrix, used in the development above.

          Let’s, however, assume that the friction force is nonvanishing, as surely is the case while the tire is rolling. In that case we can express the friction force as a differential 2-form:

    image028.png

    (22)

    where the indices take the values 1 and 2. The equation (20) can thus be rewritten in the form

    image029.png

    (23)

    Now, according to the same one of the Cartan’s lemmas, we can write, using a compact notation:

    image030.png

    (24)

    where Φ is the skew-symmetric 2×2 matrix having the entry Φ12 ≡ Φ, and b is the usual curvature matrix from equation (21). We do recognize in the second one of these equations a curvature matrix which is no more symmetric. This means that it accounts also for the twist of surface, which is very important in the theory of deformations of a physical surface (Lowe, 1980). From among the usual measures of curvature, friction thus described does not touch the mean curvature, but has an important saying in the Gaussian curvature:

    image031.png

    (25)

    It is therefore proper to say that the associated wrinkling of the surface does not affect the mean curvature of the surface. In this respect, another important connection is to be noticed.

          The road surface is rough, and one needs to model somehow this roughness. A good approximation of such a surface is a minimal surface, i.e. a surface of zero mean curvature, but nonzero absolute curvature. In this case it is proper to say that Φ is a stochastic process, characterizing the road surface in a time decided by the speed of the rolling tire; therefore the Gaussian curvature itself of the road surface is a stochastic process. This further means that the road surfaces can have different roughnesses to be modeled as stochastic processes, and this fact bestows upon them different friction properties. It is also proper to say that the road surface bestows upon a tire specific noise properties, inasmuch as it determines a specific deformation on the footprint rubber. The fact is well known: the tire prediction industry has a whole arsenal of techniques dedicated to it, mainly along the lines of characterizing the tire behavior in connection with standard road surfaces. We shall come again on this subject later on.

          To conclude, in the cases when friction exists and acts continuously, like in the case of rolling tire, instead of equation (14) we must have

    image032.png

    (26)

    The second fundamental form of the surface does not change:

    image033.png

    (27)

    and thus the ‘rubbing force’ does not influence the geometrical nature of the local surface of the tire rubber. On the other hand, if in the equation (20) we choose the ‘convenient’ differential 1-forms f1,2 as the components of the differential of position vector on the deformed surface:

    image034.png

    (30)

    then there is a friction force acting in-surface:

    image035.png

    (29)

    In other words, a friction force is always present whenever the rubber of the tire is deformed. In such cases the deformation vector from equation (6) is known, therefore the vector image0074.png can be known, as we have, by definition

    image036.png

    (30)

    where, as shown before, p3 is a constant. Using equation (30) in equation (29), we have an explicit expression for the 2-form of the friction force acting in the surface of the tire:

    image037.png

    (31)

    One can draw from this formula some well known conclusions about friction. For instance,  the force is directly proportional with the normal load but, more importantly, it is also proportional with the mean curvature of the surface, at a certain scale represented by ε. As the friction force is usually hard to quantify properly, this last property makes formula (31) particularly valuable, of course, when properly used.

    Conclusions

    The last phrase here alludes to a “proper use” of the theory, and we discuss this notion now, by the way of concluding the above developments. If not used correctly, and read ad literam, formula (31) would show that on a surface of zero mean curvature the friction force is zero, which is far from what we observe in practice. As a matter of fact the friction phenomenon involves two surfaces, one of which, at least, has to be in deformation. Indeed, by the way we obtained that formula, we actually appealed to deformation in every step. In order to clarify this point let’s think about the process of friction between the tire and the road.

          When the road surface is perfectly smooth like, say, the face of ice, there is indeed no friction between road and tire. The friction starts to count significantly when the road surface becomes rough. This is because the road roughness determines a local deformation of the tire tread lugs, and it is this local deformation that contributes to friction. It can be diminished again by a lubricant, like the water or snow on the road, having the essential property to fill the little valleys of the road profile, and thus level it at a certain microscopic scale. This indicates the fact, well known to engineers for a long time, that the friction force is a scale property. For instance if the mean curvature is calculated on the whole area of footprint of the tire, it might end up being zero, thus indicating an improper zero friction force. However, we need to consider that formula on regions of such magnitude that the deformation becomes important: such a region is the tread lug surface area. On such regions the deformation imprinted by the road is far from being zero, so that the formula (31) gives actually nontrivial results depending on the tread rubber properties.

          Now let’s take another example, in order to correct the particular choice of an ice surface, whose behavior is in significant measure due to the melting of ice. Assume that the road face is from glass, and the road and tire are perfectly dry. Now there is no question of melting and lubrication, and the glass surface is ideally smooth. On the other hand, there is also no question that the friction properties of the rubber on glass are way different form those on ice. Why? It is here the place where the ‘material’ properties come into play: forces at the molecular level first initiate a small deformation of rubber, which then triggers a subsequent macroscopic deformation enhancing the friction force. In the case of ice, the melting water precludes, or at least delays, the initial small deformation, so that the friction force is appropriately diminished. This is why, when driving on snow we have to act carefully: and the known way to take such care is not to touch the brakes!

    References

    Guggenheimer, H. W. (1977): Differential Geometry, Dover Publications, New York

    Lowe, P. G. (1980): A Note on Surface Geometry with Special Reference to Twist, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 87, pp. 481 – 487

    THE STRESSES IN THE TIRE RUBBER

    From thermodynamical point of view the pressure is a stress, or a density of energy. The differential form thus defined by pressure should be a 3-form. As a matter of fact, the stresses in a material, in general, can be defined likewise, by a formula which generalizes only slightly the 3-form of pressure:

    image0016.png

    (1)

    The origin of the triadic tensor t, which here is no more totally skewsymmetric, can be explained in the engineering terms that follow. The forces transmitted through matter – in our case the tire rubber – can be written in the form

    image0025.png

    (2)

    where p is a triadic tensor, skewsymmetric in the lower indices, and σ is the regular tensor of stresses, as defined classically. In view of the definition of the oriented elementary area as a skewsymmetric tensor of second order, we have the following definition for the tensor p:

    image0036.png

    (3)

    where the summation convention over dummy indices is implicitly understood. In detail, the table defining p is

    image0045.png

    (4)

    One can see that the covariant vector, defined by the contraction of p as

    image0055.png

    (5)

     is zero whenever the stress tensor is a symmetric matrix, and vice versa. If the classically defined stress tensor is not symmetric – for instance if the material has inhomogeneities leading to local internal moments – then the vector pk is not zero. Now, in view of the definition (2) we can define the tensor t from equation (1) by lowering the upper index with the help of the metric tensor reflecting the state of deformation of the rubber:

    image0067.png

    (6)

    Consequently, the thermodynamical transport theorem for the rubber can be written in the form

    image0075.png

    (7)

    in view of the fact that the exterior differential of a 3-form in space is zero. Here tkl is the second order tensor obtained from the triad t by contraction with the velocity field of the surface of rubber:

    image0085.png

    (8)

    Just as in the case of the thermodynamics of the tire cavity air, there is a second derivative of the quantity from the left hand side of equation (7), coming from the transport theorem as applied to the quantity from the right hand side, which varies too, from different obvious reasons: the ‘little cube’ of the definition of stresses is deformed in a rolling tire, mostly in the region of the footprint, the whole tire vibrates, the heat deforms the tire, etc. Thus, we must have a time variation for the right hand side of equation (8), which normally induces a second time derivative for the left hand side:

    image0095.png

    (9)

    This was the general idea, but we can simplify it by noticing that the fifferential 3-form from equation (1) can be written as:

    image0104.png

    (10)

    Using now the definition (6) and the table (4) we get

    image0114.png

    (11)

    This 3-form reduces to the usual mean stress when the metric is cartesian, i.e. when there is no deformation. Otherwise, if the metric is a deformed one but still of constant curvature (Coll, Llosa, Soler, 2002), the rubber stretches enter our considerations, and we have, for instance

    image0125.png

    (12)

    where sn is the classical mean stress (the trace of the stress tensor). The quantity from the curly brackets of this equation is the density of energy of the material due to stress in a state of deformation described by the stretches lk. It explicitly depends on the stresses as well as on the very state of deformation.    Let’s denote by U the density of energy from the curly brackets of equation (12). The equation (7) thus takes the form

    image0134.png

    (13)

    with image0144.png  the velocity field at the surface of the tire material. As known, this surface has the internal part – the surface of the tire cavity – and an external part – the outer surface of sidewalls together with the tread. So, the velocity field image0144.png accounts here for the internal motions of the tire carcass as well as for the external vibrations. The equation (9) can be written as:

    image0154.png

    (14)

    The right hand side here can again be treated by the transport theorem, and yields

    image0165.png

    (15)

    where image0174.png is the velocity vector on the line boundaries of tire rubber, i.e. at the bead reinforcements, but not only there, because there is a subtle catch in this formula.
         Indeed, if we consider, just for the sake of understanding, the tire as a homogeneous structure, with no tread pattern and other accidents, the line boundary of the tire rubber is given not only by the bead reinforcements, where the two parts, internal and external, of the tire surface meet, but also by the limit closed line making the boundary of the footprint. So the line integral from the right hand side of the equation (15) is actually a sum of three cyclic integrals, two for the bead reinforcements and one for the tire footprint contour. This fact copes not only with the known fact that the rolling resistance of the tire is mainly concentrated in the footprint stress cycling, but also with the more subtle fact that, for instance, the flatspotting of the tire should be strongly influenced by the contact between the bead and wheel hub. Using equation (15) and the explicit form of the stress energy density from (12) into equation (14) we get the final form of the ‘acceleration’ of the energy in the tire structure:

    image0186.png

    (16)

    This formula shows not only the intuitive fact that the second variation of the stress energy of the tire structure is given by the behavior of energy density at the rubber surface, but also that it should be decided, to a great extent, by the exchange of this energy through the limit of the footprint and the bead contact between the tire and the wheel.

    Conclusions

     In order to get the gist of the issue, we reasoned, in the development above, on the tire as a kind of homogeneous structure. However, the tire engineers are well aware of the significant difference between different tire constructions. Where is that difference coming from? Well, as the equation (16) shows, it can only come from the particular properties of the bead reinforcements, of the footprint shape, and last, but by no means the least, from the material properties entering the very form of the density of energy.
         Just to illustrate the idea, let’s limit the present discussion to the footprint contour line, for it is by far the most significant in this respect. We’ll deal later on explicitly anyway, with all of the issues involved here, but for now let’s discuss just the footprint, because it is more obvious as an example. In a ideally homogeneous tire, and on an ideally smooth road, the footprint contour line is a continuous one, so that the line integral in equation (16) makes plainly sense, and can be performed as usual. In a real tire, however, the specific tread pattern interrupts the continuity of the contour of the footprint to only its segments on the tread lugs, while the road roughness interrupts the continuity of those very segments. This is an issue that can compel us to consider higher time variations of the right hand side of equation (16), inasmuch as the very interrupted contour varies. Therefore, the variation of energy from the left hand side of that formula may be decided by the third or even higher time derivatives. The bottom line is that the behavior of the tire depends strongly on its construction (tread pattern) as well as on the road roughness, and the fundamental formula (16) takes into consideration all of the possible cases. 
         But the things get even more complicate in the real tire. Indeed, by the very same token, one can figure out that, in a real tire, the ‘surface delimiting the material’ is to a large extent a matter of construction of the tire. In this construction we ought to decide what is the main material, i.e. the one producing the variable density of energy. One can say right away, by a century or so of practice, that this is the rubber. However, in the tire construction there are quite a few kinds of rubber having wildly varying material properties. Let’s therefore push the idea of homogeneity to still another level, again, just for the sake of argument. Assume, thus, that the tire construction involves only a kind of rubber and only the metal of wires of the belt and bead reinforcements. This way the argument that the rubber is the main material in this construction, having a variable energy density, remains indeed still in force. However, in this case it is obvious that the integral from the right hand side of the equation (13) involves not only the two parts of the outer surfaces of the tire rubber, but also the internal closed tiny surfaces between the rubber and metal, accounting for the loss of energy from rubber through the metal, which thus warms up. In case we treat the wires as structural inhomogeneities, the behavior at the limit interface metal-rubber is decided by the covariant vector from equation (5), for there the stress tensor is nonsymmetric. This is also the main microscopic fact that links the flatspotting of the tire at the footprint to the properties of contact between the bead and the hub. As we said before, we’ll have to deal with these issues separately later on. 

    Reference

     Coll, B., Llosa, J., Soler, D. (2002): Three-Dimensional Metrics as Deformations of a Constant Curvature Metric, General Relativity and Gravitation, Vol. 34, pp. 269 – 282

    THE INTERNAL PRESSURE OF TIRE

    The tire is a structure complicated not only by the details of design, but by the very physics involved in its working conditions. In the matters of tire design details, a century of designing and building tires has taught the manufacturers that the tires should have just about the same standard design features. The different tire brand names basically designate details in dimensions of the tire parts, materials, belt construction, tread patterns, bead reinforcements, etc. There is no tire performing ideally in all of the working conditions. However, in order to assess the basic tracts of the performance of the tire, one needs to understand the physics governing its working conditions. And the fundamental laws of this physics reside in the concept of force. This much is well known and understood by everybody. What is not well known and understood, and therefore not properly used when occasions require that use, is the fact that the concept of force in a rolling tire involves, like in no other moving earthly structure created by man, all of the aspects of the concepts of force. Let’s therefore elaborate a little, on this occasion, on the concept of force itself, in order to show what we mean by ‘all aspects’.

         First, while working, the tire is in a rolling motion, and this requires the representation of the force by differential forms, in order to be able to apply a transport theorem helping in dynamical calculations. Physically the force is represented either by a 1-form (the mechanical work) or by a 2-form (the flux), as in the case of the pressure of the tire cavity. The difference between the two representations is given by the fact that, in the first case the force is a vector upon which we construct the 1-form of elementary mechanical work, while in the second case the 2-form itself is the force, while the flux defining the 2-form is the pressure. Correctly speaking, the pressure is therefore a skew-symmetric second order tensor, which defines a 2-form representing the force, exactly the way in which the vector force defines the 1-form of mechanical work. Speaking of  the tire cavity air, there is a third approach of the pressure, that from a thermodynamical point of view, whereby the pressure is a tensor of the third order, generating a differential 3-form, which physically is an energy. It is this representation that allows us to understand the thermodynamics of the tire cavity and to connect it with the behavior of the materials from the construction of tire.

         As the rolling tire is in motion, we can reckon that there should be a theorem of transport for every physical quantity, so much more for the force in all its aspects presented above, analogous to the classical one of Osborne Reynolds (Reynolds, 1903). This theorem is accounting for both the variation of the physical quantity itself, and for the variation of the space support of the force (the footprint, the matter volume of rubber, the surfaces – internal and external – of the tire). In the form we need it here, such a theorem was given by David Betounes (Betounes, 1983) who noticed that the general transport theorem is actually an explicit expression of recovering the Lie derivative of a certain differential form. Even though we presented this theorem elsewhere for the necessities of electrodynamics and solar physics, let’s repeat it here for the sake of completeness and ‘local use’ so to speak.

         The evolution of a certain domain D, like the internal cavity, the structural material volume or the surfaces of the tire, can be mathematically represented by a family of applications indexed by a real parameter ‘t’, playing the part of ‘time’: {ft}tÎR. The time derivative of a differential form, w say, is then replaced by the Lie derivative accounting for both the variation of the physical quantity represented by the differential form, and for the evolution of the domain itself. First, the evolution of the domain generates a velocity field:

    image0013.png (1)

     where image0023.png is the position vector in the initial domain D. Then, the Lie derivative of the differential form can be written as the time variation of the differential form when dragged by this vector field:

    image0033.png (2)

    A star denotes here the so-called pullback, i.e. the result of replacing the new coordinates of the domain in the form ω. The actual computation of the Lie derivative can be performed by the “golden formula”

    image0043.png (3)

    The symbol ‘Ù’ means here the exterior multiplication or exterior differentiation as the case may be, while the symbol image0053.png means the internal product between vector and any differential form (see Arnold 1976). The transport theorem describes the rate of variation of the quantity represented by ω – not by the quantity it defines, i.e. the quantity represented by its coefficients – in the domain D, while this very domain varies. Betounes gives the transport theorem in the form

    image0065.png (4)

    where ∂ft(D) denotes the border of ft(D) and the Stokes theorem was used. Let’s illustrate this theorem for the case of tire.    We consider first the case of internal pressure of the tire cavity. The pressure is physically represented by the force exerted by the air from the cavity upon the internal wall of the tire cavity, which is usually given by a double integral written as

    image0073.png (5)

    Here we neglected the global effect of pressure which, if considered, would ask for a closed surface of integration. But we shouldn’t be interested in this when it comes to the dynamical considerations on the rolling tire. For instance, a part of the surface bordering the tire cavity belongs to the rim, it is metallic. Consequently we expect no variations of this portion of surface that might qualify as deformation, by comparison with the internal surface of the tire carcass. Therefore, the integral from equation (5) is simply done over the internal surface of the tire per se, which is delimited by the circles of the bead reinforcements. This is an open surface.

         Perhaps the most important consequence of this line of thought is the fact that the pressure should be treated as a skew symmetric second order tensor, because the force given by pressure at the wall is a actually a differential 2-form. Indeed, the element of oriented surface of the interior wall of the tire is a skew symmetric tensor which, in view of the three-dimensionality of space, is equivalent to a vector that can be written in the form of a column matrix

    image0083.png (6)

    where εijk is the Levi-Civita totally antisymmetric symbol. Consequently the force exerted by pressure at the wall should rather be represented as a differential 2-form:

    image0093.png (7)

    The matrix p is here a skew symmetric second order tensor. This makes out of ‘p’ from equation (5) the projection of the vector equivalent to p along the direction of the normal to surface, which is the natural way to consider the pressure. This is, for instance, the case in the kinetic theory of ideal gases. The bottom line is that the variation of force at the internal wall of the tire is given, according to the transport theorem, by the equation

    image0102.png (8)

    Here we applied the formula (4) whereby we considered that the exterior differential of a 2-form oves a surface is zero. Consequently we consider this effect of pressure intrinsic to the surface. The vector ‘pressure’ in equation (8) is defined by equation

    image0112.png (9)

    as the vector whose projection along the normal to surface is the internal pressure of tire. In the right hand side of equation (8) the cyclic line integral is performed over the circles of the bead reinforcements, and therefore the velocity field image0123.png is the internal tire surface velocity field at the bead reinforcements. 

        On the other hand, the pressure should be conceived here thermodynamically, through the elementary thermodynamical work (pdV), as the tire is heating and its working and life are strongly dependent on the heat generated in rolling. In this case the pressure should be taken as a differential 3-form of energy, i.e. we should have

    image0132.png (10)

    Here pijk is a totally antisymmetric third order tensor representing the pressure, while εijk is the Levi Civita’s totally antisymmetric symbol. The transport theorem has now the form

    image0142.png (11)

    where D is the space domain of the tire cavity, and we used the property of the differential 3-forms of yielding zero when exterior differentiated in space. Here image0123.png is the velocity at the limiting surface of the tire cavity, i.e. the internal surface of the tire carcass plus the surface of the rim. Under the cyclic integral from the right hand side of equation (11) we have the differential form

    image0152.png (12)

    which, again, can be treated according to the transport theorem. Mention should be made that in the right hand side of equation (11) we have a skew-symmetric second order tensor representing a power:

    image0162.png (13)

    This makes sense physically: the rate of variation of energy in a volume is the power dissipated or absorbed in that volume. However, as the tire rotates, even the integral of the 2-form from the right hand side of equation (11) varies, at the very least due to the vibration of the internal surface of the tire cavity, and this variation is described by the same transport theorem. Thus, it turns out that actually we must have even a rate of variation of the dissipated power, due to the fact that the border of the tire cavity varies during rolling:

    image0172.png (14)

    Let’s concentrate on the integral from the right hand side here. First, we have reasons to believe that the metallic rim contribution to that integral has no variation. For instance the rim does not vibrate due to the road surface as much as the tire itself vibrates. Thus we are left with the time variation of an integral over the internal surface of the tire carcass which, by the Betounes’ transport theorem, can be expressed by a sum involving the variation of energy due to the variation of the internal surface of the tire and a cyclic line integral over the bead reinforcements:

    image0182.png (15)

    Here image0192.png is the velocity of tire at the bead level. Thus the thermodynamics of the internal cavity of the tire is decided by an equation of the form

    image0201.png (16)

    If the velocity vector of the surface of the tire at the level of bead reinforcements coincides with the velocity of the rim itself, which is a first thought about the two vectors, and can be true in certain situations, then the line integral from the right hand side of equation (16) is zero and the thermodynamical contribution of the air inside tire cavity is limited to the dissipation through the internal surface of the tire. However, in general, we can expect a certain jump of the velocity field of the tire internal surface close to the rim reinforcement, so that the two velocity fields can very well be different, and the line integral from equation (16) should be taken into consideration.

    Conclusions

    Let’s read the conclusions of the present part of the work, starting from the last equation and going towards the beginning of the developments. The air pressure inside the tire cavity – simply known to technicians as the tire pressure – is usually assumed to be constant. This may not be the case in view of the heat production of the rolling tire: the air evolves in a closed cavity, and the least we can assume is that it is an ideal gas. It is therefore to be expected that the elementary work involved in the thermodynamics of the air inside the tire cavity is a dynamic physical quantity, in the sense that it varies in time, due to the rolling conditions.

         The main result here is that, accounting for the paths of dissipation of the energy of the air inside the tire cavity, the basic quantity measuring the variation of the elementary thermodynamical work is the second time derivative of that quantity – an ‘acceleration of energy’ so to speak. Usually, in case they ever think of it, the engineers concentrate here on the first derivative, according to the classical Reynolds’ transport theorem. That theorem reflects a ‘genuine statical’ situation, whereby the details of the dissipation of energy do not include the evolution of the very channels of dissipation due to rolling. One can see that the Betounes’ formulation of the transport theorem has the advantage of forcing us to explicitly consider the time evolution of the borders of regions in which the variation of energy takes place. Thus, the ‘acceleration’ of the energetic content of the air inside cavity is dictated, even under ideally constant pressure, by the space variation of the internal surface velocity field, in the form of its divergence, and by the behavior of that surface at the bead reinforcements.

         In order to understand the interrelation between these phenomena in the rolling tire, let’s assume that we are in a situation where we can neglect the rim effects, which are transmitted through the bead reinforcements. Then the second rate of dissipation of the cavity content of energy is calculated only through the internal surface of the tire. In rolling conditions, that surface is in vibration, and this vibration can be modeled by the field of velocities of each and every one of its points. Due to a certain degree of randomness of the local vibrations, one can guarantee that the divergence of the velocity field is always nonnull on the internal surface of the tire. Therefore the energy in and out of the tire cavity is guaranteed to be at least a quadratic function of time, depending on the vibrational properties of the tire itself. In other words, the tire vibrations strongly influence the heat transfer in and out of the tire cavity.

         One way to correlate the variations of pressure with the local deformations of the internal tire surface, is through the force developed by pressure upon a small portion of surface. In this respect, the pressure is to be considered as a skew-symmetric tensor, described strictly in connection with the intimate geometry of the surface. As we will show here, this tensor contributes to the local deformation of the surface. This representation of pressure is different from the ‘thermodynamical’ one, but it is surely connected to it, as we will also show here.

    References

    Arnold, V. (1976): Les Méthodes Mathématiques de la Mécanique Classique, Editions MIR, Moscou

    Betounes D. E. (1983): The Kinematical Aspect of the Fundamental Theorem of Calculus, American Journal of Physics, Vol. 51,  pp. 554–560

    Reynolds, O. (1903): The Sub-Mechanics of the Universe, Cambridge University Press, UK

    QUADRATIC FORMS ON A SURFACE

    There are quite a few phenomena related to the theoretical physics of the tire that can be modeled by quadratic forms on its surface. First, one needs to account for the surfaces delimiting the tire. These are basically two: the internal surface of the carcass delimiting, together with the metallic part of the wheel (the rim), the tire cavity, and sustaining the internal pressure. While rolling, this surface vibrates, producing a sound which is amplified by the tire cavity. The problem is to describe these vibrations and the phenomena related to them. Another surface delimiting the tire is the external surface of the rubber, comprising roughly two regions: the main external (crest) region of the tread, and the lateral regions of the sidewalls. While rolling, these surfaces vibrate too, producing the exterior sound, but this is not the only phenomenon to be taken into consideration here. There are also phenomena related to the footprint determined by the dead weight of the vehicle, the most important of which is the rolling loss of energy. This is the phenomenon mostly responsible for the consumption of fuel. The two main surfaces of the tire are connected by design, and this connection is provided by the tire bead, whose reinforcement is instrumental in the so-called phenomenon of flat spotting. However, in order to properly understand the flat spotting phenomenon we need to take into consideration the fact that the two surfaces of the tire are also naturally connected by the material structure of the tire itself, whose main physical component is the rubber. It is through this material structure that the dead weight and its variations are transmitted to the footprint. It is also through this material structure that the high frequency vibrations of the internal surface of the tire cavity are transmitted to the environment. By the cycling thus induced, this material structure produces heat which in the short term is dissipated, but in the long run contributes to the aging of the tire rubber.
         These are, in broad lines, the phenomena related to a ‘working’ tire, selected starting from the criterion that the tire communicates with the environment through the surface of its physical structure. When talking here of ’surface’ we have in mind mainly its geometrical definition, and a certain way to introduce the physics in this definition. To start with, it is quite obvious that the overall process through which a surface ‘works’ is a deformation process. When a surface is deformed, one can recognize this phenomenon locally, through the variation of the two fundamental forms of the surface, and this is how we came up with the subject matter of the present work: both of them are quadratic forms. The first fundamental form, i.e. the metric of surface, gives the infinitesimal distance around a point of the surface, as measured in the tangent plane in that point. According to the common geometrical wisdom, the second fundamental form represents the curvature of the surface in a point, taking as reference the tangent plane in that point (Struik, 1988). Practically, however, the second fundamental form is the height of the surface above or beneath tangent plane in a point. This is the interpretation to be considered, for instance, when the intimate profile of the road is to be accounted for, or when this profile penetrates the rubber of the tire tread, producing the tiny local cycling spots of the rubber in rolling. The physical properties of a surface are thus to be embedded somehow in the six coefficients of the two fundamental forms, and more importantly, in their variations. In what follows we describe the channels through which this embedding can be done.
         Let’s consider the deformation of a surface in its utmost generality. This can be described by the variation of both fundamental forms of the surface, and by the relation of these variations. Therefore, we have to consider in general the variation of a quadratic form, which can be the first or the second fundamental form, or some other physical or geometrical properties for that matter, and show how this is described in connection with the surface. Let’s denote this quadratic form by

    image0011.png (1)

    and assume that it is defined in any point of the surface. This ‘definition’ means that we know the coefficients X, Y, Z as functions of the coordinates on the surface, and also the differentials s1 and s2 of the position vector of the points on surface, which play the part of coordinates in the tangent plane. These differentials are the main tool in problems of the local theory of surfaces. They allow for instance the definition of the curvature matrix (Guggenheimer, 1977): this is the transition matrix between the vector (s1, s2) from the tangent plane and the components of the variation of the normal vector to surface.
         When there is a variation of this quadratic form, it is described by both the variation of the coefficients: X, Y, Z → X+dX, Y+dY, Z+dZ, and by the second variation of the position vector itself. The total variation of the quadratic form can then be obtained according to the known rules, by differentiating it to obtain

    image0021.png (2)

    One can then decide if the variation of the quadratic form is strictly due to the variation of its coefficients, by a simple test: the second part of the equation (2) should be zero. Thus the variation of the quadratic form is strictly due to the variation of the coefficients along the curves from the tangent plane in a point, given by the differential equation

    image0031.png (3)

    Along these curves the equation (2) becomes:

    image0041.png (4)

    One can say that along these curves the variation of the quadratic form is ‘perceived’ only through the variation of its coefficients. Therefore, along these curves, the quadratic form is constant only if its coefficients are constant. The equation (3) represents a Hamiltonian motion in the tangent plane of a point of surface. If the quadratic form is defined, and its coefficients are constant, the equation (3) can represent two harmonic oscillators on the surface. Indeed, assuming that we discovered a time parameter with respect to which we can characterize the motion continuously, the differential equation (3) can be written in the form of a Hamiltonian system in the ‘phase plane’ of coordinates x ≡ s1 and y ≡ s2:

    image0051.png (5)

    Now, either by direct exponentiation or by finding the second order differential equation for each component, one can see that indeed, we have to deal with two harmonic oscillators having the same frequency. We choose the exponentiation, because it makes more obvious the fact that the quadratic form (1) is both the generator of motion and is conserved along motion. Indeed, denoting ω2 ≡ XZ – Y2, the solution by exponentiation is

    image0063.png (6)

    where E is 2×2 identity matrix, and the index ʻ0’ marks the ‘initial conditions’ at t = 0. One can directly verify the equality

    image0071.png (7)

    which is the clear expresion of the conservation of the generator along the motion it generates.
         If the coefficients are not constants, but vary with the point of surface, in order to integrate equation (3) they must satisfy some integrability conditions, amounting to the fact that they can be expressed as functions of the local coordinates in the tangent plane, or that these coordinates can be expressed with respect to them or their variation. In these integrability conditions it is implicitly understood that the differentials ds1 and ds2, i.e. the second order differentials with respect to position on surface, are taken as fundamental, while the first order differentials s1 and s2, which are fundamental in the regular theory of surfaces, should be expressed with respect the differentials of the second order. Practically the integrability condition amounts to the vanishing of the exterior differential of the left hand side of equation (3):

    image0081.png (8)

    Then, by one of the Cartan lemmas one has

    image0091.png (9)

    where λ, μ, ν are three external parameters, the entries of a matrix ‘assisting’ in integrability. Therefore, assuming that the matrix here is nonsingular, s1 and s2 must satisfy the system of differential equations

    image0101.png (10)

    Performing the matrix multiplication, we get

    image0111.png (11)

    The elementary ‘second order’ area – the so-called symplectic form – is given by equation

    image0122.png (12)

    This quadratic form is algebraically conjugated to the variation due to coefficients of the original quadratic form (1)

    image0131.png (13)

    and to the quadratic form introduced by the matrix ‘assisting’ in integrability:

    image0141.png (14)

    It is this quadratic form that accounts for the physical conditions determining the integrability.
         Now, in order to solve (11) the matrix of evolution is essential. It can be written as

    image0151.png (15)

    where by ω1,2,3 we denoted the differential forms

    image0161.png (16)

    In cases where the quadratic forms (13) and (14) are algebraically apolar, the matrix of evolution (15) reduces to

    image0171.png (17)

    The parameters λ, μ, ν may represent new conditions of a geometrical or physical nature, affecting the deformation process of the surface. These conditions may be external (for instance the road surface) or internal to the surface (air pressure variation), reflecting its geometry (the first and second fundamental forms) or physics (for instance the deformation matrix, or the material condition of the tire).

    Conclusions

    Two main points are worth fixing in mind, as a conclusion of the previous exercise.
         First, is the fact that the frequency of vibration of the tire is a surface phenomenon, and needs to be treated as such. The usual wisdom puts first a simple model (Kelvin-Voigt or Maxwell) describing it together with the material underneath, in a time that has nothing to do with the real phenomenon, and then trying to fit experimental data to results. These are then improved by complicating the things with series of simple models. This philosophy is mainly entertained, in the prediction ‘industry’, by the preexisting commercial calculational instruments. It has, however an essential shortcumming: the frequency model is one-dimensional, and there is no way to take into consideration the complicate structure of the spectrum which is mainly a statistical process in a real time. In other words, the phase space associated to the classical constitutive models is actually four-dimensional, not two-dimensional, as the usual wisdom holds true. 
         Secondly, the variations of a quadratic form, be it the first or the second fundamental form, needs an external quadratic form, ‘assisting’ in integrability, in order to clarify physically the mechanics of deformation of the surface. The integrability is not just a mathematical process, but mainly a physical one. We think that, in order to understand this point, an example will do best. Consider the tread of the rolling tire on the rough surface of the road. First we have a stationary deformation due to the load of vehicle. This is a first-order process of deformation, according to which we need to ‘update’ the first and second fundamental forms of the tire surfaces snd thus describe the tire footprint shape. The road surface can be modeled by such a quadratic form which, naturally, is external to tire tread surface. Then the coefficients of this quadratic form go, in the manner shown above, into the variation of the first and second fundamental forms of the tire tread from the footprint region, thus imprinting a statistical variation of the frequency of vibration of the tire. This statistical variation is characteristic to the road surface.

    References

    Guggenheimer, H. W. (1977): Differential Geometry, Dover Publications, New York
    Struik, D. J. (1988): Lectures on Classical Differential Geometry, Dover Publications, New York

    GEOMETRIC THEORY OF SURFACES

    From a purely geometrical point of view, the variation of a quadratic form characterizing a surface cannot be assisted but by one of the two fundamental forms of that surface. This has a certain natural connotation, inasmuch as the quadratic forms that vary in this case are not any different from those ‘assisting’ that variation. That is to say that the bending of a surface – the variation of the second fundamental form – is assisted by a deformation – the variation of the first fundamental form. Reciprocally, the pure deformation – the variation of the first fundamental form – may be assisted by a bending of surface – the variation of the second fundamental form – which is the usual process of ‘wrinkling’ of surface due to its local deformation. In a complex process of deformation, however, like the ones we may expect in the rolling tire, which is a certain combination of such simple processes of wrinkling and bending, both of the fundamental forms of the surface vary simultaneously. One thus has to face the problem of describing this variation so as to include all geometrical conditions under which the deformation is done. This way we should recover, at least partially, the classical geometrical results.

         Let’s say that the first fundamental form of the surface is given by

    image0012.png (1)

     while the second fundamental form, which is the second variation in position along the normal to surface, is given by

    image0022.png (2)

    where (,) means the usual dot product. Let’s take first the case where the second fundamental form varies: the pure bending of surface. This means that the normal to surface varies, perhaps due to some external perturbations, like local roughness of the road or some wave on the tire determining the local vibration in the case of tire. Applying the general considerations, the second fundamental form variation due exclusively to the variation of the curvature parameters can only be noticed along the curves from the tangent plane given by the differential equation

    image0032.png (3)

    If the coefficients α, β, γ are constants, the equation can be integrated giving the quadratic form from the right hand side of equation (2). This means that along this conic, the surface shows the same distance from the tangent plane. The conic is known as the Dupin indicatrix of the surface in a point - a classical geometrical result. Assume now that the coefficients α, β, γ vary, but that the quadratic form ‘assisting’ in the integrability is given by the first fundamental form of the surface so that

    image0042.png (4)

    This simply means that the bending of the surface is only compatible with certain states of deformation represented by the first fundamental form of the surface. Therefore the condition of integrability of the equation (3) becomes

    image0052.png (5)

    The matrix of evolution in (5) is thus given by

    image0064.png (6)

    with the differential forms given by

    image0072.png (7)

    Now, the trace of matrix (6) gives the known geometrical result regarding the mean curvature of the surface:

    image0082.png (8)

    On the other hand, the determinant of the matrix (6) gives us the absolute or Gaussian curvature:

    image0092.png (9)

    These formulas contain, again, the known classical results in case where the metric of the surface is not euclidean, provided we consider the euclidean case as a reference and identify the metric tensor and the curvature matrix with their variation.

    Conclusion

    It is to be noticed that considering the coefficients of the fundamental quadratic forms as variations rather than finite quantities, is as general as it can be, comprising all the classical results in matter of geometry of surfaces. Indeed, if we consider the reference plane in a point of a surface, other than the tangent plane, for instance the mean plane of a rough road surface in the case of a rolling tire, then the variations of the curvature parameters are indeed these very parameters, and the formulas (8) and (9) are the usual ones from the classical theory of surfaces. Consequently, for physically realistic situations, we are entitled to take any quantity referring to the surface as being a variation rather than an absolute quantity. More than this, the theory provides the opportunity of describing the ‘wrinkling’ of a surface as a local deformation assisted by bending (remember that, in this jargon, the case above is bending assisted by deformation, inasmuch as the assisting matrix is that of the first fundamental form of the surface, which usually contains the deformation). However, the pure ‘geometrical’ description of the deformation of surfaces is obviously not sufficient from a physical point of view. This leads us, as will be shown elswhere on this page, to an interesting way of describing the friction forces between two surfaces in relation to their roughness and the physical properties of the materials they are delimiting. Such is plainly the case of the tire-road interaction, so important in braking and skidding properties of the tire.