aa r X i v : . [ g r- q c ] M a y Three lectures on Newton’s laws
Sergey S. Kokarev ∗ RSEC ”Logos” , Yaroslavl, Russia
Abstract
Three small lectures are devoted to three Newton’s laws, lying in thefoundation of classical mechanics. These laws are analyzed from the view-point of our contemporary knowledge about space, time and physical in-teractions. The lectures were delivered for students of YarGU in RSEC”Logos”.
Coordinate method of describing physical phenomena is based upon two impor-tant and seemingly alternative notions. The first is the idea that all coordinatesystems are equal in rights , the second that a coordinate system is being chosenaccording to the practical convenience arising when solving a specific physicalproblem. When worded mathematically, that is in general covariant equationsin terms of tensor bundles, this seeming contradiction is resolved: invariant op-erations on tensors (tensor products, contractions, covariant differentiation) donot depend on a coordinate system while their expression in components can besimplified if one chooses a coordinate system which reflects existing symmetries.Relativity theory allows us to place reference frames among four-dimensionalcoordinate systems where reference frames world lines coincide with time coor-dinate [1]. Here again, as in the case of coordinate systems, ideas of equality inrights of all reference frames and practical convenience of a given system canbe applied. The correctness of the second can be easily established in numerousexamples from classic mechanics while the first idea is not so obvious in respectto Newton’s mechanics. Moreover,
Newton’s first law also known as the lawof inertia , states that there exist special reference frames called inertial frames where mechanical bodies free from the influences of external forces persevere intheir state of being at rest or of moving uniformly straight forward. As a rule,during a school or even college course of physics this law is seldom paid due at-tention to. Sometimes, regrettably, one can find outright wrong interpretations ∗ [email protected] Changes in the path’s scale of elaboration — both decreasing (to the left)and increasing (to the right) can show the path’s deviations from a straight line. of it. Meanwhile, this law (that is better to call a principle, as we will showlater), when duly restated suddenly becomes a general principle valid not onlyfor mechanics but for all modern physical-geometrical theories of nature as well.For this reason we will examine its role in mechanics more closely.Galilean law of inertia postulates existence of inertial frames of reference.Our daily experience can tell us nothing about the motion of bodies not subjectto external forces or when these forces are balanced simply because we cannotisolate a body from the outside world. Nor can we fully balance all the forcesconcerned (for logically there always remains a possibility of forces in existenceperfectly unknown to us as yet) or exactly define a geometrical straight line inphysical space. The latter aspect was pointed out by H. Poincar´e in 1909 [2].It’s evident that under such circumstances Galilean law of inertia simply cannotbe proven experimentally.
That’s the reason we stressed the law’s postulativecharacter in the beginning.But let us for the time being forget about the complication concerning therealization of geometric straight lines. Let us assume that we have at our dis-posal a certain device that can clearly and with any chosen degree of precisionshow us whether the body in question moves straight forward or not, withoutinterfering with the motion of the bodies studied. Upon eliminating the bodies”foreign” to our experiment and balancing the impacts of those remained, ifwe find that the body in question moves straight forward by any possible degreeof precision, then we can safely conclude that our reference frame is an inertialone, thus having proved
Galilean law of inertia experimentally. However, suchoutcome is hardly probable. Experience shows that as the degree of precisionincreases there would always appear new and more complex details of the exper-iment which complicates the previous picture and ultimately leads to the newrevisiting of the whole conception behind it (Fig. 1).Now we can almost ascertain that our device will show some deviations froma straight line. But according to the law of inertia that means the body is eithersubject to some forces distorting the path or our reference frame is not inertial.
The choice between these two alternatives in not an easy one!
Let2 ~F R ω Figure 2:
Hypothetical ”fundamental force” of nature on a planet with opaque at-mosphere is in fact centrifugal force of inertia related to the planet’s rotation. us imagine a planet, Poincar´e writes, that rotates between relatively immobilestars and has an atmosphere opaque to the starlight (Fig. 2).Physicists of this planet are not able to observe their rotation around thestars and the resulting uninertiality of their reference frame. Hence in their thor-ough experimentation they would consider the first alternative. While studyingthe motion of different bodies they could at last arrive at a conclusion that de-viations from a straight line shown by test bodies indicate existence of a certainforce that at two points on the surface — ”repulsion poles” — turns to zero,increases in proportion R sin( s/R ) where s — distancing from these points onthe planet’s surface, R — planet’s radius; this force is always directed at anangle π/ − s/R to the vertical and is proportional to body mass. The generalformula looks like: | ~F | = αmR sin( s/R ) (1)For the planet’s inhabitants it would hold the same basic meaning as the law ofgravity, and α constant estimable by means of experiment would be consideredone of nature’s basic constants. This state of affairs would last until it woulddawn upon an alien Copernicus, already familiar with the law of inertia as statedby an alien Galilei: ”May be there is no force F, but the reference frame relatedto their planet is not inertial? If we are to make the simplest assumption that theplanet is in rotation around an axe which pass through the ”repulsion poles” ,with constant angular velocity ω , the expression for F acquires a perfectly clearphysical sense. That would simply indicate the usual centrifugal force of inertiaas daily observed by the planet’s inhabitants, only on a far more modest scale.In that case the formula (1) should be rewritten as: | ~F | = mω r, (2)where r means distance from a point on surface to the rotation axe . And In the formula (1) this distance is expressed by means of distance to the ”repulsion poles” α means squared angular velocity with whichthe surface rotates around some invisible outward bodies with larger mass.”Such or like would be the train of thought of our alien Copernicus. For obviousreasons he would surely have to face serious trouble in persuading his fellowaliens to agree with this concept.This example of Poincar´e’s indicates the general way of reasoning charac-teristic for researchers who seek to test Galilean law of inertia experimentally.Explaining of the deviations from a straight line would be done in terms ofinteraction by means of forces; then, if it were possible, the problem would bereinterpreted in terms of uninertial reference frames. Newton’s first law allowsboth approaches. But which is better? Or, more precisely, which is more cor-rect? Before answering this questions let us first sum up. What do we gainby switching our focus from forces interactions to uninertiality? Firstly, weeliminate the ”excessive nature’s forces” : in the example with the planet, afterthe discovery of our alien Copernicus force ~F is ”transferred” from dynamicsto kinematics. Secondly, description of outside motions in an open uninertialreference frame looks sufficiently simpler ! Simplicity and beauty in this caseare not mere abstract principles so dear to a philosophically inclined mind. Itis hardly coincidental that only after Copernicus’ discovery Newton toocould formulate a very simple and basic law of nature — namely, thelaw of gravity. He was able to do that upon observing simple elliptic trajec-tories proposed by Kepler while it would surely prove to be a task impossiblefor human mind to arrive at the same conclusion through analyzing countlessPtolemaios’ epicycles. But it is beyond powers of today’s computers, even mostadvanced among them, to perform inductive generalization, let alone grasp thenotions of beauty and simplicity!Thus it should be apparent that the law of inertia is not a law in the propersense of the word. It says nothing of the world that Newton’s classical mechanicsdescribed or attempted to describe. It only suggests a general rule of reasoning which can be applied to this world. In fact, it wants us to assume that thereare certain so called inertial reference frames, where the simplest situation offree motion is represented by simplest possible kind of trajectory — straight line.
It’s a basic assumption of classical mechanics. Situation as described above can be changed only by forces or uninertial motions. The decision which is thecase can be made only after we’ve analyzed the whole situation, found forces atwork, estimated the possibility of some unknown forces interfering and definedwhether the motion of outside forces looks simpler when described in uninertialterms.We can thus conclude that, its special status aside, the law of inertia is themost basic link of the whole logical structure of classical mechanics. In on spherical surface. The point is, on the planet with constantly hidden stars, inhabitantswould have little use of the system with parallels and meridians. Distances on the surface, onthe contrary, would serve as a convenient coordinates. Unfortunately, it is not a very sound argument for the inhabitants of our twilight planet.For they have no picture of celestial bodies motion whatsoever! The argument would get moresound with the appearance of alien cosmonautics Figure 3:
Trajectories of free particles in a world with modified Galilean law of inertia.There is no gravitational force near the Earth’s surface in this world. As one movesfurther from the surface, gravity appears and increases and come close to a constantvalue that equates − m~g on great distances. view of this we would like to stress that those authors who argue that Newton’sfirst law is the effect of the second are entirely wrong. Their argumentationlooks like follows. Let consider a particular case of the second law:¨ ~r = ~F /m, (3)where ~F = 0 . Body moves with zero acceleration along a straight line, in fullaccordance with the law of inertia. The mistake of this argument lies in the factthat Newton’s second law in the form as cited above holds true only in inertialreference frames, which existence is postulated in the first law.
Were it notfor Newton’s first law (and even were it otherwise formulated), no one would beable to write down Newton’s second law (or it would look absolutely different)!
Were it not for the first law, how could we know that the equation describingthe motion of a free body looks like ¨ ~r = 0 , that it takes the form of a straightline equation!For example’s sake let us imagine a slightly ”modified” world where the lawof inertia would read as follows: There exist special reference frames called in-ertial frames where mechanical bodies free from the influences of external forcesmove along parabolas with permanent vector coefficient in front of squared time.
In latter case Newton’s second law would look like that: ¨ ~r − ~g = ~F /m, where ~g stands for constant vector which sets the direction of the axes of all parabo-las. Then, insofar as forces amount to zero we get ¨ ~r − ~g = 0 , which means butparabolic motion (Fig. 3). If then we ”recognize” gravitational acceleration invector ~g , then we’d see that this ”modified” world is fairly identical with theworld of Newton’s classical mechanics near the earth’s surface, but all aspects offree fall and gravitation are being described without reference to forces. Whatexactly did we do? By modifying Newton’s mechanics, we’ve ”transferred” oneof the forces from dynamics to free body kinematics, or, more precisely, to theforms of their trajectories. Near the Earth’s surface this kind of mechanics wouldbe even more convenient than ours! We could postulate some other curves in-stead of parabolas, and then some other forces would have left dynamics forfree body kinematics. When building his system of classical mechanics we’vebecome so accustomed to, Newton chose the most radical path: made his freebody kinematics simplest possible while his force dynamics is as rich as it canbe! In this sense, Newton’s form of mechanics is the simplest one can find.This simplicity has the other end as well. For example, we could try to do5t the other way round: to convert the whole force dynamics into kinematics.For gravity forces this problem has been already solved within general relativitytheory [3], for other interactions there were more or less successful attempts atsolving it within gauge principle of interactions [4]. In both cases the curvingof the paths is not caused by forces, but comes from non-euclidian geometryof the physical space-time which can have extra dimensions as well [5]. Such”anti-newtonian” approach dubbed the problem of geometrization of the physicalinteractions [6] grows still more popular within modern theories of physics.As a conclusion we would like to show that there exist a quantum-mechanicsanalog of the Galilean law of inertia. In nonrelativistic quantum mechanics wedeal with states | t i , that are vectors of a certain Hilbert space and observables— Hermitian operators acting on it [7]. General equation of nonrelativisticquantum mechanics: i ~ ˙ | t i = ˆ H| t i (4)called the Schroedinger’s equatuion is the quantum-mechanical analog of New-tonian dynamics equation (3). In (4) operator ˆ H is the differential evolutionaryoperator, or Hamiltonian for short. The transition to a common wave function ψ ( ~r, t ) is analogous to the choice of reference frames and transition from vectorsto their projections in Newtonian classical mechanics. Within quantum theorythis procedure is called the transition to certain representation of a Hilbert spaceand operators in it. Common wave functions are obtained via transition to the ~r -representation: ψ ( ~r, t ) ≡ h ~r | t i . Now the quantum-mechanical ”law of inertia’can be formulated as follows: ”There are certain reference frames called inertial where the state of a freequantum particle of mass m and momentum ~p in coordinate representation isdescribed by a wave function proportional to exp[( εt − ~p · ~r ) / ~ ] , where ~p and ε are connected by a standard relation: ε = ~p / m ” . It is this statement that allows us to concretize the Hamiltonian: ˆ H = ˆ T + ˆ U . With this in mind we are able to prove that for the differential operators of thesecond order the quantum mechanics law of inertia as formulated by us earlier,would lead to the only standard expression:ˆ T = − ~ m ∇ , while the states described by plane waves thus appear (in coordinate represen-tation of the quantum mechanics) to be quantum-mechanical analogs of theuniformly straight forward motions of bodies in Newton’s classical mechanics. This analogy takes more apparent form in a known Ehrenfest theorem for average valuesof classical dynamic qualities. Two approaches to Newton’s second law
Newton’s second law states that in inertial reference frames all changes in thebody velocity are caused by the influence of external forces . The aim of dy-namics is to study the properties of nature’s forces and to solve the problemsconcerning body motion under influence of given forces. The vectorial equationof a point-like body motion has the form (3). Let us consider a possibility of itsbeing proven experimentally. We’ve already discussed the conditional characterthat the notion of force has in the previous section. Depending on the chosenformulation of the Newton’s first law, some forces can ”dissolve” into kinemat-ics and vise versa — ”crystallize” inside the dynamics. Let us first accept thetraditional Newtonian formulation of the Galilean law of inertia. The experi-mental testing of the Newton’s second law entails that during the experiment wewill probably need to measure simultaneously and independently all three values ~a, ~F , m of the expression (3) as well as substituting them into the expression(3) and to verify its identical fulfillment at the set level of precision. We mustalso bear in mind that, as experience shows, in experiments of all kinds all ourqualitative measurements are always linked to the length measurement or theinteger counting of time . F − ~F ~F ~F Figure 4:
The work of the dynamometer is based upon the calibrating dependency(linear, as a rule) and Newton’s third law
Thus even if we ignore the difficulties arising from the measuring of thederivatives as boundary limit relations, only acceleration remains to be measuredby pure experiment. How then do we measure force and mass? Traditionallythe force is measured with the help of a dynamometer. But the work of a dy-namometer (aside from the calibrating dependency specific for its type) is basedon the condition of the equilibrium of the bodies and Newton’s third law (Fig.4). If such a body remains in mechanical equilibrium while being connected tothe dynamometer, the force applied from the side of the dynamometer balancesthe measured force as applied by the outside bodies. According to Newton’s Here and below we assume that body mass remains the same in the process of the motion.The case of bodies with changing mass can be always reduced to the motion of bodies withconstant mass, if we conceive that the mechanical system consists of subsystems the overallmass of which remains constant. Indeed, apart from measuring lengths with the help of a ruler we use special rulers calledclock-faces or scales of the multimeter, manometer, thermometer to measure time intervals,voltage, pressure, temperature and so on. Measuring of time by the means of counting oscilla-tions of a pendulum is an example of the second type of quantitative measurements with helpof whole counting. Pendulum’s modern modifications — electric watch or counter of electricimpulses device perform the same operation, only with the help of special electronic schemes.Simple analysis shows that the work of all the other measuring devices is based on space scalesor integer time counting. ma would be the same by modulus for any two bodies interacting.For our reasoning this kind of ”proof” would not hold because it is based on thesecond law — the same we’re trying to prove. The other way of ”measuring”the forces would be via mass and acceleration. But then equation (3) becomes amere definition of force. Besides, a simple analysis shows that any other meth-ods of defining the mass will inevitably rely on the second and third Newton’slaw (for example, rotation on a thread and weighing).Then, if viewed logically, the ”experimental” verification of the second New-ton’s law turns into a logical circle. It doesn’t mean much for practical purposes:Newtonian axioms are consistent and make it possible to pose and solve a lot ofproblems. But forces and masses usually stay behind the scene, so to speak: allobservations concern motions and trajectories. Trajectories showing deviationsfrom predicted on the basis of second law, which exceed the limits of possibleerror can be explained either by the interference of some additional forces orexistence of additional masses.Scientists of different generations have pointed out this logical incomplete-ness in the experimental basis of the classical mechanics, as well as the contro-versial status that force and mass have. Here are some examples:H. Herz: ”To find a logical fault in the system that has been worked out by thebest of brains appears almost impossible. But before we give up further investigationit is worth asking whether everybody, the best of brains included, were satisfied bythis system. . . As for me, I would especially point out the difficulty one meets whenexplaining the introduction to mechanics to the thoughtful listeners, because here andthere an excuse is needed and one with some embarrassment hastens to turn to theexamples which can speak for themselves”. [8] (English translation from Russian textin [9]). H. Poincar´e: ”We meet the worst difficulties trying to define the basic notions.What is mass? — That, Newton says us, is the product of volume and density. —”Better said, that density is the quantity of mass in a unit of volume” — answerThompson and Tait. — What is force? — ”That, Lagrange would say, is the causethat makes the body to produce motion or tend to produce motion”. — ”That,answers Kirchhoff, is the product of mass to acceleration”. But why not say thatmass is the quantity of force calculated for a unit of acceleration? This problemis unsolvable. . . Thus we return to the definition given by Kirchhoff: force equatesmass multiplied by acceleration. This ”Newton’s law” ceases to be considered anexperimental law and becomes a mere definition. But the definition too is not sufficientfor we don’t know what is mass. . . We’ve acquired nothing, all our efforts were in vain— and thus we are forced to resort to the following definition which only shows our efeat: masses are coefficients used in calculations for convenience. . . . We have toconclude that it is impossible to give a fair idea what force and mass are withinclassical system” [10] (English translation from Russian text in [9]). A. Einstein: ”To connect force and acceleration becomes possible only after thenew notion of mass is introduced — which, by the way, is explained through a seemingdefinition” [11].
It is clear that the source of our problems lies in the theory, not the experi-ments. Is it possible to formulate the laws of classical mechanics so as to escapethe logical circle and to make the status of the quantities introduced clear? Ineffort to find the answer to this question we’ve arrived at two views on forceand mass which will be discussed later.1.
Force and mass are superfluous notions and it is possible toformulate the laws of mechanics without them. Force and mass are not two essences, but one which manifestitself differently.
First view should be attributed to the operational approach to the principlesof physics [12]. One of the operationalism’s basic notions reads: definitionsof the physical quantities should be constructive, i.e. should in fact set therules for measuring the quantity defined. Besides, the law of nature should beformulated in terms of the constructively defined physical quantities only.
Aswe’ve discovered in previous section, the standard formulation of the classicalmechanics is far from being operational. Since acceleration is the only oneconstructively defined quality, the operational formulation of mechanics shouldbe based on the acceleration only. Below we give the operational formulation ofthe classical mechanics as stated by Y. I. Kulakov, who took it up within thescope of his (meta)physical theory concerning physical structures [9]. Withinthis formulation the basic notions are set B of bodies and set F of forces. Letus examine the mapping B × F → R, which image we will write down as a iα for any b i ∈ B and f α ∈ F . We’ll call a iα acceleration of the body b i caused bythe force f α . Let us now consider an arbitrary body pair { b i , b j } and force pair { f α , f β } . It appears possible to restate the Newton’s second law in the terms ofaccelerations only: a iα a jβ − a iβ a jα = (cid:12)(cid:12)(cid:12)(cid:12) a iα a iβ a jα a jβ (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (5) The term ”metaphysical” here is applied in a sense synonymic to the term ”metamathe-matics” , i.e. theory of cathegories, functors and toposes which allow to operate with wholemathematical theories as mathematical objects of a higher order. Therefore metaphysics is(or maybe will in future) a theory of physical theories. Here we cite a fragment of the theoryof physical structures by Y. I. Kulakov, slightly modified and adapted for our purposes. The vectorial aspect of this quantity here is irrelevant. The reasoning below is true for one-dimensional motion. The three-dimensional case is obtained via repetition of the reasoningfor the remaining two projections. B and F respectively. We’ve formulated the second law in terms of accelerations, withthe help of images of bodies and forces that have to satisfy the only condition —to exist. Qualitative characteristic of bodies (mass) and of forces (their values)are not included into the expression (5).Let us show now that the equation (5) is in some sense equivalent to theequation (3). In order to prove that we would need a mathematical theoremwhich was rigorously proven by G. G. Mikhailichenko in his series of worksdevoted to the theory of physical structures [13]. The theorem reads: if theequation (5) is invariant in respect to the change of bodies and forces (the authorscalled this kind of invariance phenomenological symmetry), then accelerationshave the form: a iα = λ i F α , (6)where λ i = λ ( b i ) , F α = F ( f α ) are some mappings B → R F → R respectively.It is easy to verify one side of the theorem by simple substitution of this form tothe equation (5). The basic result of the theorem obtained via solving the com-plicated functional-differentional equations lies in the conclusion (6) from (5).There remains only to recognize force F α in the right side of the equation (6),as well as the reversed mass λ i = 1 /m i sometimes called movability. Thereforeif the accelerations defined through sets of bodies and sets of forces satisfy thephenomenological invariant equation of the form (5), then bodies have masses,forces — their values so that accelerations can be expressed through them bythe relations of the form (6). This formulation of the Newton’s law fol-lows as the rigorous mathematical concordance from the metalaw (5).
The operational formulation of the classical mechanics was obtained, but at the”cost” of a certain abstractization of the language and loss of visuality. It is in-teresting to note that for this formulation there’s no need for the first Newton’slaw: set of forces includes forces of inertia, therefore the equation (6) remains al-ways correct. Third Newton’s law in its traditional formulation is not consistentwith the operational formulation as described above: forces and bodies belongto different sets, but the third law stated traditionally inevitably mixes thesenotions together. However, with the help of an additional structure (algebra ofbodies) it becomes possible to include the third law too: it will be expressed bythe additivity of the mapping F in respect to the specific composition of bodies(see lecture 3). In order to explain the other approach to the laws of classical mechanics wewould need to use some data from the special relativity (SR) [14]. The SR’s basicidea was first distinctly stated by Herman Minkowsky in 1909. It postulatesthe affinity between space and time which, according to this theory, form theunite scene for the events called space-time.
Sometimes the SR’s space-time is10 tX SS ′ ∆ t ∆ ℓ t = t = onst t = t = onst YY ′ ct ct ′ X c ∆ t ∆ ℓ t ′ = t ′ = onst t = t = onst t ′ = t ′ = onst t = t = onst Figure 5:
Spaces of events in the classical Newtonian mechanics and SR. Diagram tothe left shows the space line of a point-like body, world line of some inertial referenceframe S ′ , and a part of a world tube that represents a formal unity of space positionsof a certain three-dimensional body (shown by transversions t = const). Diagram tothe right shows the world line of a point-like body, axis ct ′ and Y ′ that moves in thedirection OY of the inertial reference frame and a four-dimensional body — relativisticworld tube that has the same measure — four-dimensional length — in all directions.The sections of this four-dimensional body into sequence of three-dimensional bodiesdepend on the choice of the reference frame. That’s why it would be natural to callworld tube in SR absolute history. also called four-dimensional Minkowsky space. In order to clear the differencebetween the absolute space and time of Newton’s mechanics and the space-timeof SR let us look at the figures 5.Both consist of elementary events — points with coordinates ( t, ~r ) in the firstcase and ( ct, ~r ) in the second. The motion of the point on both diagrams will berepresented by a certain curve (called world line): in the first case the curve’stilt to the time axe can be at any angle (from null to π/ π/
4. The lattercomes out of the finiteness of the maximal velocity that bodies and signals canacquire in SR, i.e the finiteness of the velocity of light. Sections of the worldlines by planes t = const will result in a momentous position of a point in somereference frame fixed in space. The motion of the extended bodies in both caseswill be represented by world tubes, while the section of the world tubes by thelines t = const would define the momentous position of these bodies in space(in a fixed space reference frame). But the similarity between these two spacesof events remain at this purely visual level. The time in classical mechanics isabsolute and therefore, universal for all reference frames. Figure to the left showstwo reference frames: conditionally ”stationary” S and conditionally ”moving”11 ′ . Intervals of time being counted by clocks in S and S ′ are identical regardlessof their movement character and can be defined on this diagram by projectingthem on the axe of absolute time unique for all reference frames and bodies. Thediagram to the right shows the relativity of time in SR — there exist infiniteset of times, every one of which moves in its own direction in four-dimensionalspace of time. The intervals between a pair of events depend on the choice of atime line, i.e. reference frame and are related to each other through specific SRformulae.The difference between the diagrams becomes even more apparent if welook at the interval between events. In classical mechanics it can be calculatedonly for simultaneous pairs of events while in space-time of SR distance (4-dimensional interval) is defined for any pair of events. If in some referenceframe the events got divided by a time interval ∆ t and distance interval ∆ l, thefour-measure distance between these events can be defined by the formula:∆ s = c ∆ t − ∆ l . (7)The value of the interval does not depend on the choice of the reference frame.That allows us to deduce the laws of transition from one reference frame to an-other (Lorenz transformation), while the division of this interval into time andspace parts appears to some extent conditional. When we change the referenceframe, a part of the space projection of the interval pass into time projectionand visa versa, approximately as in the rotating system of coordinates OXY the rigid bar’s XY -projections on a plane can pass into one another accord-ing to certain laws. Considering the world tubes it can be said that the worldtube on the diagram to the left is an artificial object because its vertical size ismeausured in seconds, while horizontal (thickness)— in meters. These quanti-ties are perfectly incomparable in Newton’s classical mechanics, that’s why inclassical mechanics it is reasonable to consider the world tube, as well as thewhole space of events to be a formal unity of momentous positions of the bodyor momentous spaces (a pack of sheets, separated but closely stuck together,the width and thickness of which is measured in different units that have ab-solutely no relation to each other). On the other side, in SR the world tubeis one four-dimensional extended object that can be measured by the interval(7) in united units (for example, meters) in all directions, time as well as space.This four-dimensional body is a history of a certain three-dimensional body”frozen” in four-dimensional world. This four-dimensional body stays alwaysextended along the time direction, thus reminding a bent thin bar. Every refer-ence frame defines a combination of sections by planes of simultaneous events.The sections are three-dimensional and define momentous positions of a certainthree-dimensional in a chosen reference frame. In another reference frame thesame bar would define another sequence of sections. Since the appearance ofthe bar in four-dimensional space does not depend on the system (only the waythe bar gets dissected in three-dimensional sections), it would be natural to callthe bar absolute history. The sequence of the three-dimensional bodies defining relative history of a certain three-dimensional body can be defined only after12e’ve chosen reference frame and set the planes of simultaneous events asso-ciated with it. It is absolute histories of bodies that will be the object of ourstudy. Note that in existing literature on SR scientist usually confine themselvesto the consideration of one-dimensional world lines, leaving out four-dimensionalextended bodies.If we assume the four-dimensional approach and consider four-dimensionalworld bars a new physical reality of relativistic nature, it would be naturalto start by examining its four-dimensional physical qualities. Since in four-dimensional world the bars are at rest, here we face the four-dimensional kindof statics. This statics can be reconstructed analogically to the statics of simplethree-dimensional bars. As we know, three-dimensional bars can be subject tostrains of stretching-compressing, twisting and bending [15]. In case of moderatestrain these can be considered independently. Each kind of strain has its ownexpression for respective elastic energy. So for the common three-dimensionalbar the energy of twisting can be expressed as follows: E tw = Z Cτ dl, (8)where τ is the angle of the bar sections’ relative turn to a unit of length (thisquantity is called twist ), C is twist rigidity that depends on elastic constantsand the form of section, and the integration is being taken along the bar axe.In order to calculate bending energy E b1 for longitudinally unstrained bar thereis a more complex expression which we won’t need. Finally, a strongly tensedbar bent by cross load has the elastic energy: E b2 = T Z dl, (9)where T is the bar’s tension and the integration is taken along its axe in bentstate. It should be noted that strongly tensed bars are called strings. Unlikeuntensed bars, their bend resistance is defined by tension and not bend rigiditywhich depends on elastic constants of the bar’s material. Therefore, for strings E b2 ≫ E b1 . In order to formulate the laws of four-dimensional statics we would need togeneralize the relations mentioned above to some extent. Initially it is unclearwhich of the energies would be dominant for description of the four-dimensionalbars statics consistent with Newtonian mechanics as being observed in three-dimensional world. We let out technical details and present here the resultonly [16]. The following conditions should be fulfilled, for equations of four-dimensional statics in nonrelativistic limit to represent Newton’s classic me-chanics:1. The bars should be considered tensed strings, and the four-dimensionaltime-like force of tension is related to the three-dimensional mass by re-lation : T = mc . Thus the expression for the bend elastic energy of the It is worth reminding that in 4-dimensional world force must have the dimension of 3-dimensional energy because simple 3-dimensional force is 4-dimensional force related to the4-dimensional bar length unit. m : E b2 = cS progr = − mc Z ds (10)2. The three-dimensional mass density ρ is connected to the shear modulus ζ of the bar’s four-dimensional material by a relation: ρc = ζ. The four-dimensional twist energy becomes proportional to the rotational part ofaction for a rigid body in classical mechanics: E tw = cS rot = c Z J ( ω, ω )2 dt, (11)where J is the inertia tensor, ω is angular velocity of the body rotationlinked to the twisting τ of a four-dimensional bar by this relation: τ = ω/c.
3. The proper bend rigidity of the bars is not important because the barsbehave as strings. Identification of ζ = ρc = T /V, where V is the three-dimensional section volume in the bar’s reference frame shows that shearrigidity (i.e. rotational inertia) is fully determined by the tension of thestring.So, in a picture reconstructed by us the mass appears as none other than(accurate within dimension factor) time-like force that tenses bar withsuch intensity that its elastic characteristics become determined bythis tension. In a four-dimensional world both simple force and mass appearto be different projections of four-dimensional forces. R R σσ σp + ∆ pσ p Figure 6:In order to make clear the force nature of the mass let us turn to a knownformula for Laplas pressure (see fig. 6):∆ p = 2 σR − , (12)that links pressure difference in gas or liquid on both sides of tensed membraneto the quantity of local surface tension and average curvature k = R − =14 n RTT ~F~F~F ~F Figure 7:(1 /R + 1 /R ) /
2, membrane being in the state of equilibrium. This formulaehas its one-dimensional analog (see fig. 7) dFdl = TR (13)for normal (related to the tensed thread) bending force F, force of tension ofthe thread T and the bent thread’s curvature radius R, in a given point. Here dF/dl stands for ”one-dimensional pressure”.Now it’s time to show that the equation (13) is in fact somewhat simplifiedform of the Newton’s second law. Indeed, the four-dimensional world line ve-locity vector of a particle U is unit, so the acceleration vector dU/ds is noneother than curvature vector of the world line, and its modulus equates k = 1 /R modulus of the curvature [17]. Common forces acting upon a particle are alwaysspace-like, that is, they act in a direction orthogonal U. They play the role ofthe bending forces linear density in (13). If we rewrite the three-dimensionalside of Newton’s second law in the form: ~f = mc ~k = mc ~nR , where ~n is unit vector of the curvature direction (bar protrusion) and compareit with (13), we are now able to see that quantity mc indeed plays the role ofthe tension applied to the world line or world bar. Note that in this pictureNewton’s first law becomes equivalent to a well-known statement that a stronglytensed string, at regions where there is no bending or twisting forces, remainsrectilinear. Newton’s third law stays the same. The law of mass conservation isits rough consequence for the time-like forces.Here we are not going to discuss other interesting consequences of the four-dimensional statics which put a lot of things in a whole new light. One can findthem in the original work [16]. 15 The nature of Newton’s third law
In previous lecture we’ve already discussed the logical incompleteness of me-chanical experiments and dynamic equations without Newton’s third law. Atthe conclusion we’ve proven that the laws of three-dimensional dynamics canbe rearranged into laws of four-dimensional statics of absolute histories in four-dimensional Minkowsky space. As three-dimensional statics, this one is alsobased on the laws of equilibrium (the resultant of the forces and momentaequates zero) and Newton’s third law.In order to make clear the nature of this law let us turn to axiomatics thathas been developed in the works of W.Noll and his school in the 1950-60-ies.We’ll adapt the explanation of necessary axioms of bodies and forces for ourpurposes [18]. Axiomatics of bodies and forces contains in abstract form generalcharacteristics of all bodies and forces to be found in classical mechanics. Letus consider bodies A , B , C , . . . as elements of some universal set Ω , called themechanical universe. There exist between bodies usual relations of inclusion (forexample,
A ⊆ B ”body A is a part of body B ” ), of superposition A∩B (commonpart) and joint
A ∪ B (”composite body” ), with all usual characteristics .Empty body we’ll denote by the symbol ∅ , universal one by ℵ . These bodieshave specific characteristics: ∅ ⊆ A A ∈ Ω; A ⊆ ℵ A ∈ Ω . If two bodies have no common part except ∅ , they are called segregate. For anybody
A ∈
Ω there is an unique body A ext , called exterior of the body A , so that A ∪ A ext = ℵ ; A ∩ A ext = ∅ . The following relations become apparent: ∅ ext = ℵ ; ℵ ext = ∅ , as well as relations( A ext ) ext = A ; from A ⊆ B it follows
A ∩ B ext = ∅ . (14)The reverse for the latter is also true in the universe Ω: the only bodies segregatefrom A ext are the parts of the body A . It is easy to prove the validity of deMorgan’s relations: ( A ∪ B ) ext = A ext ∩ B ext ; ( A ∩ B ) ext = A ext ∪ B ext ; (15)We have an important formula of expansion: A = B ∪ ( A ∩ B ext ) , (16) Instead of standard Boolean operations ⊆ , ∩ , and ∪ in [18] was used (cid:22) , ∧ , and ∨ respec-tively, that are, in fact, more abstract. Author give some examples, when difference betweenthese abstract operations and Boolean ones becomes apparent. Since such examples are veryexotic from the viewpoint of application of classical mechanics, we, for the simplicity sake,will use more accustomed Boolean operations. B ⊆ A . And the components of the expansion are segregate:
B ∩ ( A ∩ B ext ) = A ∩ B ∩ B ext = ∅ . Let us now consider vector-valued functions on the pairs of segregate bodiesof the kind −→ F ( A , B ) . Let us call the vector a force, with which body B acts onbody A . In classical mechanics forces satisfy the principles of superposition and additivity.
Both principles are reflected in the additivity qualities of the forcefunction, by the second and first arguments respectively: −→ F ( A , B ∪ C ) = −→ F ( A , B ) + −→ F ( A , C ); −→ F ( B ∪ C , A ) = −→ F ( B , A ) + −→ F ( C , A ) (17)for any bodies A , B , C segregated in pairs. Assuming that in additivity relations B = ∅ or C = ∅ , we obtain the following relation for empty body: −→ F ( ∅ , A ) = −→ F ( A , ∅ ) = −→ A ∈ Ω . Let us now consider force −→ F ( A , A ext ) , with which the exterior of the body A acts on it. In mechanics this force is called resultant. Let us consider twosegregate bodies A and B . The second quality (14) in combination with deMorgan’s identities (15) results in: A ext = B ∪ ( A ∪ B ) ext ; (18) B ext = A ∪ ( A ∪ B ) ext . According to the principle of forces superposition we get: −→ F ( A , A ext ) = −→ F ( A , B ) + −→ F ( A , ( A ∪ B ) ext ); −→ F ( B , B ext ) = −→ F ( B , A ) + −→ F ( B , ( A ∪ B ) ext ) . Adding the equations together with the use of the forces additivity principle, andafter we’ve gathered the expressions of exterior on the right side, the followingequation can be obtained: −→ F ( A , B ) + −→ F ( B , A ) = −→ F ( A , A ext ) + −→ F ( B , B ext ) − −→ F ( A ∪ B , ( A ∪ B ) ext ) (19)It is the main identity necessary to analyze the nature of Newton’s third law.From relation (19) it follows that in the mechanical universe, where the forceprinciples of superposition and additivity are fulfilled, Newton’s third lawtakes place only when the resultant also is an additive function of thefirst argument by segregate bodies : −→ F ( A , B ) + −→ F ( B , A ) = −→ ⇔ −→ F ( A ∪ B , ( A ∪ B ) ext ) = −→ F ( A , A ext ) + −→ F ( B , B ext )(20)17or all A ∈ Ω , B ∈
Ω and
A ∩ B = ∅ . This statement is the essence of
Noll’stheorem.
Let us define the expression −→ F ( A , B ) + −→ F ( B , −→ A ) through −→ ∆( A , B )and call it discrepancy of forces for bodies A and B . Noll’s theorem states thatdiscrepancy is the measure of nonadditivity concerning interactions of the com-posite body with its environment. Then let us turn back to statics, where forany A we have −→ F ( A , A ext ) = 0 . It follows from the Noll’s theorem that in theworld of statics the discrepancy of forces for any pair of bodies is identical zero.In other words, in statics Newton’s third law is in effect according to the gen-eral principles of forces superposition and additivity.
It is easy to notice thatNewton’s third law is stronger than any principle of superposition or additiv-ity taken independently. Indeed, if we apply the third law to any of the itemsunder condition (17), we can see that the condition of additivity becomes prin-ciple of superposition and visa versa, by any three bodies segregated in pairs.That means the validity of the third law and one of the principles result in thevalidity for the second principle while the third law follows from the principlesof superposition and additivity only on an extra condition that is additivity ofthe resultant. This condition in its turn does not follow from the principles ofsuperposition and additivity.Let us now try to generalize the formulations in order to consider the forces ofinteractions and principles of superposition and additivity not only on segregatebodies. First we’ll redefine relations (17) on bodies with the superposition thatdiffers from zero body: −→ F ( A , B ∪ C ) = −→ F ( A , B ) + −→ F ( A , C ) − −→ F ( A , B ∩ C ); (21) −→ F ( B ∪ C , A ) = −→ F ( B , A ) + −→ F ( C , A ) − −→ F ( B ∩ C , A ) . (22)Under B ∩ C = ∅ these relations transform into (17) and in fact take intoconsideration that the force interaction B ∩ C if it differs from zero, has beentwice accounted for in the formulae.Let us now consider the force of interaction on bodies A and B with A ∩ B = C 6 = ∅ . Using the representations: A = A ′ ∪ ( A ∩ B ); B = B ′ ∪ ( A ∩ B ); (23)where A ′ = A \ B and B ′ = B \ A are bumps of A above B and B above A respectively, (all the characteristics of theoretical set \ -operation taken intoaccount), we define the interaction force of non-segregate bodies this way: −→ F ( A , B ) = −→ F ( A ′ ∪ ( A ∩ B ) , B ′ ∪ ( A ∩ B )) = −→ F ( A ′ , B ′ ) + −→ F ( A ′ , A ∩ B ) + −→ F ( A ∩ B , B ′ ) + −→ F ( A ∩ B , A ∩ B ) . (24)In the case of segregate bodies our definition transforms into identity of the kind −→ F ( A , B ) = −→ F ( A , B ) . In the case of non-segregate it reads thus: the force with hich body B acts on the not segregated from it body A is formed by the force withwhich the bump B ′ acts on the bump A ′ plus the force of the superposition onbump A ′ , force of B ′ on superposition and force of superposition self-action. Thethree first components are simple forces acting on segregated bodies, meaningthat all that is new in the forces interactions with non-segregate bodies lies inthe quantities of the self-acting forces of the kind: −→ F ( A , A ) ≡ −→ F ( A ) . Let us make clear the nature of the force. First we’ll examine the interactionforce of a certain body A with the universal body ℵ . In accordance to ourdefinition (24) we get: −→ F ( A , ℵ ) = −→ F ( A , A ext ) + −→ F ( A ) . Therefore, the self-acting force can be described as difference: −→ F ( A ) = −→ F ( A , ℵ ) − −→ F ( A , A ext )of the force of interaction with universal body and the resultant force. Since allthe common bodies in classical mechanics are in a sense ”small bodies” whencompared to the environment as well as the universal body, we have A ext ≈ ℵ and therefore −→ F ( A ) ≈ . However, the self-acting force can appear resulting from incomplete compensa-tion of two approximately equal values.Now let us show that the mechanical universe with additive resultant andnon-trivial interaction of non-segregate bodies is impossible. In order to do thatwe’ll need to generalize identities (18), considering the case of non-segregatebodies and put to use the principle of superposition for the bumps. It is easy toprove that for non-segregate bodies A and B identities (18) acquire the form: A ext = B ∪ ( A ∪ B ) ext \ ( A ∩ B ); B ext = A ∪ ( A ∪ B ) ext \ ( A ∩ B ) . (25)The principle of superposition for the bumps, in accordance to their definition,has the form: −→ F ( A , B \ C ) = −→ F ( A , B ∪ C ) − −→ F ( A , C ) . (26)Then, after we’ve gone through the calculations analogous to those made underthe conclusion (24) and with regard to (25) and (26), we obtain: −→ F ( A , B ) + −→ F ( B , A ) − −→ F ( A ∪ B , A ∩ B ) − −→ F ( A ∩ B ) = (27) −→ F ( A , A ext ) + −→ F ( B , B ext ) − −→ F ( A ∪ B , ( A ∪ B ) ext ) − −→ F ( A ∩ B , ( A ∪ B ) ext ) In order to compare bodies on Ω we need to introduce a measure. For simple geometricalmeasure — volume and physical measure — mass the conditions of smallness are reduced tostrong inequalities V ( A ) ≪ V ( A ext ) or M ( A ) ≪ M ( A ext ) , which are almost always valid. A ∩ B 6 = ∅ . From it directlyfollows Noll’s generalized theorem: additivity of the resultant on all bodies isequivalent to the expression for discrepancy: −→ ∆( A , B ) = −→ F ( A ∪ B , A ∩ B ) + −→ F ( A ∩ B ) . (28)Assuming that the resultant is additive on all bodies, let us now examine theexpression obtained in detail. With the help of the decomposition (23) andupon denoting A ∩ B = C , after some elementary redenotations we obtain from(28): −→ ∆( A ′ , B ′ ) + −→ F ( C , A ′ ∪ B ′ ) = −→ . (29)Bodies A ′ , B ′ and C can be considered arbitrary segregate bodies, so for A ′ = ∅ (29) gives us: −→ F ( C , B ′ ) = −→ . for all segregate bodies. Therefore, in a universe where there is interactionbetween non-segregate bodies and the resultant is additive, segregate bodies donot interact at all (hence the resultant equates zero), while non-segregate interactby means of self-acting force of its common part. For such system of forces −→ F ( A , B ) = −→ F ( B , A ) = −→ F ( A ∩ B ) . Besides, self-acting force has additive quality: −→ F ( A ∪ A ) = −→ F ( A ) + −→ F ( A )for any subdivision A = A ∪ A A ∩ A = ∅ . In fact, in a world of self-acting the role of resultant would be played by fullforce −→ F ( A , ℵ ) , not by the resultant −→ F ( A , A ext ) . The condition of this full forceadditivity will have the form: −→ F ( A , ℵ ) + −→ F ( B , ℵ ) − −→ F ( A ∪ B , ℵ ) − −→ F ( A ∩ B , ℵ ) = −→ . Adding to and subtracting from the right side (27) necessary components of self-acting, of the kind −→ F ( A , A ) , . . . , and after all necessary simplifyings we wouldcome to the same conclusion: in a universe where the full force is additive thereis no interaction between segregate bodies while non-segregate interact only asa result of their supposition self-acting. It is apparent that in such a worldconsistent statics would be impossible, because if −→ F ( A , ℵ ) = −→ A , then F ( A ) = −→ A , meaning forces vanish at all.It is interesting that the unusual universe we got can be interpreted differ-ently. Let us consider a broader universe Ω ′ = Ω ∪ { i } , where bodies from Ω donot interact between themselves but do interact with body i (Fig.8). Assuming −→ F ( A ) ≡ −→ F ( A , i ) for all A ∈ Ω , where −→ F satisfies only the principle of additiv-ity. We can see that in such extended universe the observer, being the part of Ω , perceive the force of interaction with body i , that lies outside the universe Ω as20Ω ′ i Figure 8:
Universe Ω with self-acting looks similar to universe Ω ′ = Ω ∪ { i } withoutself-acting. Instead of the acceleration force there is the force of interaction betweenbodies Ω and body i , which is inaccessible for observation. For that reason thisinteraction will be viewed as additive self-acting by observers from Ω self-acting. While experimenting with body i in universe Ω are fundamentallyimpossible, the investigator can always choose between the terms of ”self-acting”or ”transcendental” interacting, according to personal philosophical beliefs!I am grateful to Evgeniy Stern for assistance with figures and to Anna Tu-runtaeva for translation of the text in English. References [1] Yu.S.Vladimirov,
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