ELECTROMAGNETIC RADIATION

Transcription

1 LUCIANO BUGGIO ELECTROMAGNETIC RADIATION On the basis of the (unpreedented) dynami hypothesis that gives rise to yloidal motion a loal and deterministi model of eletromagneti radiation is onstruted, possessing both partile and wave harateristis. The terms of Speial Relativity are intrinsi to the model. Venie, November 2004 buggiol@libero.it

2 Abstrat On the basis of the (unpreedented) dynami hypothesis that gives rise to yloidal motion a loal and deterministi model of eletromagneti radiation is onstruted, possessing both partile and wave harateristis. The terms of Speial Relativity are intrinsi to the model. CHAPTER 1 - ORDINARY CYCLOIDAL MOTION 1 - Definitions Ordinary yloidal motion is defined as the motion of any point of a irumferene that rolls with onstant veloity on a straight line of the plane. The geometrial trajetory desribed by the point in the ourse of a single rolling is the ordinary yloid. It goes without saying that the motion resulting from several rollings is equipped with periodiity, and implies a translation of the point along an overall diretion, whih is that of the straight line on whih the irumferene rolls. A yle, both in the spatial and the temporal sense, is the single yloidal "jump," from one extreme to the other. More preisely, a period is the duration of the yle, and the length of the "jump" (d) is the distane overed in eah yle between two suessive extremes, whih we shall all uspidal points, or usps. Frequeny ( f ) is the number of yles performed in the unit of time. The greater the veloity of rotation of the irumferene, i.e., of rolling, the greater is the frequeny. The overall diretion of propagation is that of the line r on whih the uspidal points lie (or on a line parallel to it). We also define the instantaneous veloity ( v i ) of a generi point of the trajetory, in both the modular and the vetorial sense, represented by an arrow, tangent in that point to the trajetory, whose length indiates the modulus, and whose orientation indiates the diretion and the sense. The useful instantaneous veloity ( v iu ) is the projetion of v i onto the overall diretion of propagation; i.e., the omponent of v i that is useful for produing advanement in that diretion. The mean useful veloity in the diretion of propagation ( v ) is the mean of the useful instantaneous veloities, alulated on the basis of a whole or semi-whole number of yles; it is equal to the "length of the jump" divided by the period ( v=d / p ). 1

3 The tangential aeleration (a) is the variation of the modulus of the instantaneous veloity, i.e., its derivative, represented by an arrow tangent to the trajetory, whose length gives the modulus and whose diretion, oiniding point by point with that of the instantaneous veloity, qualifies it also as vetor. Initial aeleration ( a i ) is the tangential aeleration at the instant in whih it begins a yle, equal, in modulus, to the tangential deeleration with whih the previous yle terminates, registered at the same uspidal point and at the same instant. 2 - Kinematis of ordinary yloidal motion The instantaneous veloity varies from zero, at the beginning of the yle, to double the mean useful veloity of propagation. Maximum veloity is attained half way through the yle, at the peak of the yloidal jump. The equation for the instantaneous veloity as a funtion of time (with all the parameters unitary) is, in the ourse of eah yle: y= 2 2 os x Also the useful veloity varies between these extremes, but its values are always lower than those of the instantaneous veloity, apart from the extremes and the middle point, where they oinide. The equation for the useful veloity, in the ourse of eah yle, is: y=2sin 2 x 2 In Figure 2 the two graphs, that of instantaneous veloity and that of useful veloity, are referred to the same frame of axes for omparison. The mean useful veloity is one half of the maximum veloity attained. For the first half of the yle the point aelerates, for the seond half it deelerates. The maximum aeleration is the initial aeleration and the maximum deeleration is the final deeleration, relative to eah of the two uspidal points, respetively. The aeleration is zero where the veloity is greatest. The equation for the tangential aeleration in the ourse of eah yle is: y= sin x 2 2os x Figure 3 shows its progress. 2

4 Initial aeleration, mean useful veloity and frequeny are linked by the following relation: (1) a i=2 v f 3 - Dynamis of ordinary yloidal motion If, in the vauum of elementary dynamis, the point is a material point possessing inertial mass m, and there is a vetor fore F applied to it apable of making it move, in what onditions is ordinary yloidal motion obtained? We show that: Given a frame of referene, a material point of mass m initially at rest, to whih a vetor fore F in rotation on a plane with onstant angular veloity is applied, moves with ordinary yloidal motion. Demonstration: m = mass = modulus of the vetor fore F = fore = angular veloity of the rotation a = aeleration The dynami hypothesis is: m a=f x, y,t = os t, sin t To obtain the trajetory it is neessary to integrate this dynami onfiguration twie. mx ' ' = os t my ' '= sin t r os t 2 2 x= m r sin t t 2 y= m If we render the parameters m, and unitary, we obtain: x= os t 1 y= sin t t 3

5 whih is the parametri equation of the ordinary yloid of the line. Observation If, given a frame of referene, the initial state of rest is simpler (= has a smaller quantity of information) than the initial state of motion, and if the simplest (least "informed") way of making the diretion of a vetor vary over time while maintaining its modulus onstant is rotation with onstant angular veloity on a plane, then ordinary yloidal motion stems from the simplest (least informed) dynami hypothesis after the one that gives rise to uniformly aelerated retilinear motion, whih is distinguished from the former by the lak of the rotation of the fore. In other words, given a frame of referene and, in it, a material point initially at rest, the ordinary yloidal trajetory is, dynamially speaking, the trajetory that is simplest to obtain after the retilinear trajetory. 4 - Relations among parameters In the ourse of the yle, the arrow that represents the thrust (fore F) is oriented as we see in Figure 4 in the five most signifiant positions, at the beginning of the yle, at one fourth, at mid-yle, at three fourths, and at the end. We show that, given a fixed value for the mass m: a) - Veloity of rotation - i.e., frequeny f - being equal, inreasing the thrust length of the jump d and, onsequently, inreases the mean useful veloity v. b) F being equal, inreasing, d dereases and onsequently so does v. ) Proportionally inreasing F and, d dereases and v remains onstant. The equation that links the parameters in play is, by virtue of (1) and given (2) F=m 2 v f 4 F=m a i : F inreases the

6 CHAPTER 2 - ELECTROMAGNETIC RADIATION 1 - The elementary onstituent of eletromagneti radiation (photon) The defining terms that will be adopted here (for the most part borrowed from the physial nomenlature in fore) are purely onventional: in general the physial desription of the objets and of the properties to whih they refer is not given. The elementary onstituent (or elementary partile) of eletromagneti radiation (photon) is defined as a puntiform objet (P in the figure) having inertia (m) to whih a vetor is applied (represented by the arrow F), apable of translating it in spae; the vetor is in rotation with onstant angular veloity on a plane (ab in the figure). In the figure the rotation, with its sense, is symbolized by the urved arrow; the length of the arrow is proportional to the veloity of rotation, just as the length of the retilinear arrows is proportional to the intensity of the vetors that they symbolize. The diretion and the sense of vetor F are the same as those of a here not better defined vetor E, eletri vetor of the photon, elementary onstituent of the eletri field of radiation. The axis of rotation, i.e., the perpendiular to the plane of rotation passing through the point, is onventionally equipped with a sense. This axis is defined as the magneti axis, and is represented in the dynami diagram by another arrow, B, it too issuing from the point, orthogonal to the first arrow and to the plane of rotation, with its tip, i.e., the sense, oriented in the diretion in whih the rotation of the eletri vetor is seen to be lokwise, as in our figure. Hypothesis on the nature and the struture of E and of F For a moment, let us step outside our shema to hazard - parenthetially and in a first approximation - an intuitive hypothesis on the physial struture of the entities we are onsidering. Think of a "perturbation" of spae, loalized - as far as we are onerned here - in a "small surrounding of the point", integral with the point itself and stable in the frame of the rotation. In this sense "material point" is redefined simply as geometri enter of the perturbation. The perturbation may be desribed in terms of variation (as a funtion of its position) of the potential - always positive - of the spae (where the lak of perturbation - the vauum - would be desribed by a onstant potential, with zero derivative at every point). At its entral point the potential is zero (puntiform "blak hole"): it has no derivative, whih from all diretions tends toward infinity. Suh a point is thus a point of disontinuity of spae, a singularity. The fat that the perturbation represents an elementary "eletri field" possessing a diretion and a sense implies that it has no radial symmetry around its enter: in terms of the behavior of the derivative of the potential to tend toward 5

7 zero from all diretions, this means that there is a diretion and a sense (whih rotate with onstant angular veloity) along whih the potential dereases more rapidly toward zero (and its derivative inreases, in absolute value, more rapidly toward infinity). The fore that auses the enter to move (dragging the surrounding struture of the potential, whih is integral with it) may be seen as the effet of the unbalane due to the asymmetry of the struture itself, analogously with what ours in fluid dynamis: a useless attempt to reinstate the perturbation's radial symmetry. The attempt is useless beause, just as the arrot at the tip of the stik always maintains the same distane from the donkey in motion to reah it, so the unbalaned struture of the potential, whih is rigid and integral with the enter, does not modify with its motion, and thus ontinues to be translated. Analogous to the terminology used for the eletri vetor, we attribute the oriented diretion of the magneti axis to another "loalized perturbation of spae" (whose nature we shall not investigate here, but whih in all likelihood is due to the fat that the first perturbation is in rotation) that we all magneti vetor of the photon, elementary onstituent of the magneti field of radiation. 2 - The motion of the photon By virtue of what was said in Chapter 1, in partiular in Setion 3, the photon (the enter of the perturbation), starting off at rest in the given frame of referene, desribes ordinary yloidal motion. We assume that: a) - All the photons have the same inertial mass m. b) - For eah photon F and f are proportional to one another; i.e., that the relation between fore and frequeny (whih is to say, the angular veloity of rotation of the fore) is linear. In terms of the just-formulated intuitive hypothesis, this would mean that the greater the deviation of the distribution of the potential from spherial symmetry - i.e., the greater the unbalaning of the stable perturbation - the greater is the veloity of its rotation. It follows that - as we saw at Point () of Setion 4 - the mean useful veloity (v) is onstant for the motion of all photons, independently of their frequeny. The photons with greater frequeny - the ones that make more jumps in the unit of time - make their jumps proportionally shorter, so that there be no gain in the veloity (v) with whih they advane as a whole. In the onditions posed there is thus an inverse relation between d (length of the jump) and f (frequeny). 3 - Pereptibility of the photon's eletri and magneti fields Let us suppose that the photon's eletri and magneti fields are "pereptible" (on the part of an "observer", an "exploratory objet sensitive to the field"), and the more so the more slowly the point proeeds in the ourse of its trajetory. In fat, the greater the instantaneous veloity, the shorter is the time the reeptor (imagined to be puntiform) of the field stays in the field itself, the less does it interat with the field and, thus, the less does it pereive it. We thus have maximum sensitivity to the fields on the part of their respetive exploratory objets at the beginning (and at the end) of eah "jump", when for an instant the veloity is zero. Let us assume a mean useful veloity of displaement so great () that for most of the distane overed by eah jump the instantaneous veloities are so high that the two fields prove impereptible, at least where ertain effets are onerned. Let us go so far as to assume, as a working hypothesis, that the vetors are pereptible only at the uspidal points, where veloity is annulled. However great its mean veloity, the photon will nonetheless have to present itself at its appointment with the usps with zero veloity, whih it will attain by passing through all the gradually dereasing values of veloity: the two fields will be perfetly pereptible, for an instant, only in orrespondene with the uspidal points of the yloidal trajetory. It follows that when both the magneti and the eletri fields are pereived they are always headed in the same respetive sense, whih for both will be orthogonal to the overall diretion of propagation: this onsideration is pertinent espeially in the ase of the eletri vetor (see Fig. 4), whose orientation varies ontinually in the ourse of a yle. Thus the instantaneous appearing and disappearing of a single photon will be pereived at points aligned in the diretion of propagation, along a straight line, as far apart in the spae between them as the length of the jump, in suessive instants separated by the duration of the period of the yloidal motion. At those points and at the instants aforesaid the double elementary perturbation E and B of the spae will be loalized, represented by the two vetors orthogonal to one another, and orthogonal to the diretion of propagation 6

8 along whih the disrete events follow one another. 4 - Constrution of eletromagneti radiation We have seen how the single elementary onstituent of radiation (the photon) moves (ordinary yloidal motion) and the phenomenology (the suession in spae and time of oriented loalized perturbations E and B) to whih it gives rise. To onstrut a full and proper radiation, whih also has wave harateristis, all we have to do is aggregate an extremely great number of the onstituents ("photon gas"), making them hop in an orderly way. To arrive at the onstrution of the (linearly polarized) plane wave we shall first define the progress of the perturbation along a straight line: the extension to three-dimensional spae will be immediate Many points on a irumferene that rolls On a irumferene that rolls on a straight line r advaning with veloity we fix a great number of points, not neessarily spaed the same distane apart, but in suh a way that, sine their distribution is also random, in the suffiiently small surroundings (ar) of any given point of the irumferene they will in any ase be plentiful. Let suh points be the enters of as many photons and let their eletri vetor be oriented toward the enter of the irumferene. Consequently their magneti vetor will point up from the page in the diretion of the figure's viewer who sees the irumferene roll from left to right. For the sake of simpliity only a few points are depited in the figure; for eah point the yloidal trajetory has been traed (with a solid line for the trajetory already traversed and a dotted line for the trajetory remaining to be traversed). We adopt this kinemati diagram for larity's sake; it is evident that the idential onfiguration may be obtained presinding from the irumferene that rolls and exlusively employing the dynami hypothesis of the rotating fore applied to an ensemble of photons that have been emitted in an orderly way. In fat eah of the positions in the figure is that whih would be oupied at a ertain instant by a photon emitted from a preise point at a preise instant, in flight in the ourse of its yloidal motion indued by the rotating fore applied to it. For example the position (C in the figure) antipodal to the irumferene's point of ontat with the straight line is that of the photon that has reahed its maximum veloity (2) and is half way through its jump: we know in fat that in that position the fore (vetor E) is perpendiular to the trajetory and direted downward, as our diagram shows (see Figure 4). Given what was said in Setion 3 of this hapter, our observer will see a "perturbation" translated with veloity along the straight line on whih the ideal irumferene rolls. This will not be a single perturbation that moves retilinearly, but rather a "ontinual" suession along the line, in time and in spae, of the "landings" of always different perturbations (in the ambit of a rolling). Eah perturbation, moreover, follows not a retilinear but rather the ordinary yloidal trajetory whih, however, is not pereived by the observer, sine he an interat with the perturbation only when it is at rest relative to him. 7

9 The perturbation that advanes onsists, pereptively, in an "eletri vetor" always oriented in diretion and sense orthogonal to the diretion of the advane and ontained in the plane of the rolling; assoiated with this vetor is a "magneti vetor" (not shown in Figure 6, nor in the two figures that follow) with diretion and sense orthogonal to the plane itself A train of irumferenes that roll We now add a series of irumferenes that roll on the same straight line, one following the other at a very lose distane, suh that a great number of them are in ontat with the line within a suffiiently small interval fixed on it; eah one, like the previous one that advaned by itself, will have a great number of "elementary perturbations" entered on it and oriented toward its enter. Let the train, measured from the point of ontat of the first irumferene to the point of ontat of the last one (i.e., between the respetive enters) be as long as the jump made by eah individual photon, namely d. The point of ontat of eah of the great many irumferenes with the line orresponds to the perturbation that is pereived by the observer from one time to the next during the rolling, with the advane of the train: a perturbation in the form of a retilinear segment of length d will be pereived to be translated with veloity, onstituted instant by instant by the ensemble of eletri and magneti vetors stably oriented as we stated above. Now, let us suppose that the distribution of the irumferenes that roll (the density of the points onsidered on eah of them remaining equal) is not uniform along d, but is denser at the enter and less dense toward the extremes of the segment. Let the law of the distribution be "bell-shaped", typial of normal distributions, sine here in any event we have a finite domain, of the type sin 2 x. Consequently sin 2 x will be the intensity of the field pereived along the segment of length d that is translated on the diretion of propagation with veloity. 8

10 4.3 Alternate trains of irumferenes that roll Following on the train in the figure ( d 1 ) we set another train ( d 2 ) rolling, omposed of the same number of irumferenes, distributed in the same way, but this time below the line, on a segment equal in length to that of the first train. If in the streth d 1 the eletri vetor was oriented upward, in the streth d 2 it is oriented downward; and if in the first the magneti vetor pointed up perpendiularly from the page, in the seond it enters into it. We then plae, one after the other, in any quantity, other trains of length d of irumferenes that roll alternately above and below the line in eah train Transition to three-dimensional spae: the plane wave We divide the three-dimensional spae in parallel setors individuated by planes perpendiular to the diretion of propagation and passing through the "nodes" (the extremities of the segments of length d), and we fill eah setor - with a density distribution equal to the one fixed on the linear dimension of d - with irumferenes that roll (keeping to planes parallel to that of the irumferenes already given) in one setor in the same sense (lokwise) and in the next in the opposite sense (ounterlokwise); whih is to say that eah irumferene rolls, aording to its setor, below or above its line, parallel to the irumferene with whih we began. We have thus onstruted the plane wave. The wavelength is 2d= The period is 2p=P The frequeny is f /2= 5 - Emission of eletromagneti radiation Soure of eletromagneti radiation, or emission antenna, is defined as an objet (not more preisely speified here) that periodially assumes two states, with a polarity reversal produed in the transition between them. Spatial polarity implies an axis, whose orientation is onstant over time. Complete yle, onsisting in two half-phases, is defined as the ourrene of two events, after whih the initial onditions for the beginning of a new yle are reinstated. Period is the duration of the omplete yle, and frequeny the number of omplete yles in the unit of time. As a result of dynamis we shall not investigate here, the osillation defined provokes the emission from this "antenna" of photons in ordinary yloidal motion. For the sake of simpliity we provisionally assume that the emission be spherially isotropi in the surrounding spae, and that eah photon emitted follow a trajetory of ordinary yloids lying on a meridian plane - a plane, i.e., that ontains the axis of the dipole. Let the dimensions of the soure be negligible relative to d. Now, we assume that in the first half-phase of every yle of the soure's ativity the yloidal trajetory of the photons emitted shall have their onavity direted downward, and in the seond direted upward. We also assume that the photons an leave the antenna with any veloity (understood also vetorially) inluded 9

11 in the interval of the instantaneous veloities assumed in the ourse of the ordinary yloidal jump, so that they an then reah a value of zero veloity, at the end of the jump, at any distane from the antenna inluded in the interval of length d of a omplete jump. In Figure 9 we see the trajetories of different photons, emitted by the antenna with different veloities from a point of their yloidal trajetory more or less distant from the onlusion of the jump. No photon, with the first jump, an go farther than distane d; it follows that the period of the wave that will be emitted will be equal to that of the ativity of the antenna. In the ourse of eah half-period, of duration p, the emission takes plae "with ontinuity", from the beginning to the end of that interval of time; and, here again by dynamis we have not demonstrated, we assume that the distribution of the emission's intensity over time, probably linked to the progress of the veloity of the osillatory motion, is the one established previously, viz., sin 2 x. For a visual rendition of this "asade", let us analyze the emission's first half-phase in detail, breaking it down into its suessive instants. Let us onsider separately the ideal asade of photons that at the instant in whih the first half-period begins detah themselves simultaneously from the antenna, eah one departing from it from a different point of its yloidal trajetory, in order to over the entire span of positions. This instant is represented in Figure 9. In the figure the different levels of the vertial frame from whih the individual photon detahes itself is unimportant: what ounts is the horizontal frame. Stritly speaking, we should have drawn a puntiform soure; but, here, we have adopted this representation in order to obtain the alignment of the landings. Moreover, all the yloidal paths in the figure are the "future" trajetories of the individual photons, whih have been drawn to translate the information of their veloity and aeleration at the moment of detahment, when the first photons all "show themselves" outside the antenna "simultaneously". Sine the figure is designed to represent a moment of time, viz., the first instant of the ativity of the dipole, what is important, as we said, are the points of departure in the horizontal frame, with whih a veloity and an aeleration are assoiated that will produe the trajetory indiated. We know that the eletri field is pereived when the photon reahes the usp of its trajetory, with zero veloity, and we have assumed that the onavities of the yloidal path shall point downward in this first half-phase. If follows that the eletri field vetor that will be pereived around the antenna in the first half-phase points upward. Let us now onsider the future outome, i.e., the atual trajetories. The photons that show themselves outside the antenna with very low veloity in their deeleration phase (suh as photon A) will immediately reah their "dead-point", making their fields felt in the viinity of the antenna. The photons that show themselves with greater veloities, albeit they too in their deeleration phase (suh as B), make their fields felt a little later and farther along, when and where they in turn reah their uspidal points; and the greater their distane from the antenna, and thus after a longer time, the greater will be their points of departure from the onlusion of the yle. The photons that detah themselves at maximum veloity - the veloity registered half-way through the jump (suh as C) - will glide down and land at a distane equal to one half of the jump, after a time p/2; and the photons that fall even farther along, from the mid-point of this spatial interval to its extreme (suh as D), will be those that detahed themselves at a veloity less than maximum, but in aeleration. 10

12 The photon that loses the asade, photon E, making the longest jump and, after a time p, oming to a halt for an instant at the distane of the length of its jump, will be the one that left the antenna with zero veloity in aeleration: this, moreover, is photon A, seen taking off instead of landing. As the photons "touh down" they take off again immediately, no longer making their eletri field be felt; and sine the photons that suessively touh down are gliding to a greater and greater distane, the eletri field, determined moment by moment by always different photons, will "sweep" with onstant veloity, always pointing upward, the spae of a half-phase in the time needed by any given photon to make a omplete jump. Sine we have onsidered only the instantaneous asade at the beginning of the yle, the perturbation in motion is limited to a very restrited region of spae - ideally to a point, as represented in Setion 4.1 of this hapter, on the ontat with a straight line of a irumferene that rolls - and ontinues to move with the same veloity and, of ourse, even beyond the distane of half a wavelength: one they have left the antenna, the photons ontinue to hop in the same way endlessly, if they meet no obstales. When the last one has touhed down, the others are in the air and will glide down farther along, one at a time and one after the other, both in the spatial and the temporal sense. But we are just at the beginning of the half-phase emission. After the asade of photons emitted at the first instant (whih are few indeed or, stritly speaking, none at all, given the temporal law of the emission's intensity) other (ideal) asades will follow, for the entire duration of the half-period. Of ourse they are not distint, disrete "asades": as we said earlier, for the sake of onveniene we have broken up the emission, whih is to be understood as ontinual for the entire duration of the half-yle. The emission from the first instant on, in aordane with the temporal law established ( sin 2 x ), will be on the inrease for the first half of the half-phase, and on the derease for the seond half. At the end of the half-period, at the moment in whih the polarity of the antenna is reversed, the instantaneous photo of the perturbation of the spae, whih until then gave us a restrited region of spae (a point) affeted by the eletri field at distane d from the antenna, now gives us an extensive perturbation, whih affets the entire interval of length d beginning at the antenna, aentuated at the enter and toned down toward the extremes (see the density distribution of the arrows, or the progress of the irumferenes, in Figure 8, where the origin, the soure, is fixed at the left extreme of the segment of length d). To avoid misunderstandings, it is important to understand that the order in whih the photons "land" one after the other, eah one ontributing to the onstrution of the half-wave that advanes with veloity while onserving its length d, effetively stems from their random emission: in the span of a half-period of the soure's ativity they detah themselves at all veloities (ompatible with ordinary yloidal motion: but we shall see later on how even this obligation will have to be broken), with a distribution that is in fat perfetly random over time. In the subsequent half-period the asade will have the opposite sense, and so forth. 6 - Generalization: the soure also emits photons in non-ordinary yloidal motion Thus far we have taken the emission to be onstituted only by photons that make ordinary jumps. The model makes provision for a soure emitting photons that make yloidal jumps of all kinds, hene also non-ordinary (whih shall presently be defined). The generalization is right and proper if we onsider the fat that ordinary yloidal motion - the motion with usps - represents a most partiular ase, dynamially speaking. To have this trajetory, it is in fat neessary that at a given instant (whih will be the initial instant of the yle) the photon have zero veloity, and there is no reason why in a physial entity this most partiular ondition has to enjoy speial privilege. For photons to detah themselves from a soure and then glide with zero veloity to a ertain distane from it, the diretion of the fore, at the moment of take-off, must neessarily diverge from that of the veloity in progress, given the modulus of the veloity itself, by a preise angle, and this angle must neessarily lie on the fore's plane of rotation. All the possible ombinations between this angle and the value of the modulus of the veloity are legible in the dynami sheme of the ordinary yloidal trajetory with the instantaneous veloity vetor assoiated at every point, with its modulus and its orientation: other ombinations, inluding those in whih the veloity vetor does not lie on the fore's plane of rotation, belong to non-ordinary yloidal trajetories, whih differ from the ordinary trajetory beause, among other things, in a non-ordinary yloidal trajetory at no point does the photon ever take on zero veloity. Thus the soure has maximum freedom of emission (respeting the given restraints, viz., the sense of the 11

13 onavity for the two respetive half-yles and the bell-shaped distribution): the photons detah themselves from the soure with any veloity and with any orientation relative to the fore and to the plane of its rotation. We an say in advane that, with this generalization, from the standpoint of the observer at rest nothing will hange relative to what we have said in the preeding setions, sine he will only be able to observe the photons that make the ordinary jumps previously onsidered, for the instant in whih they stop at the uspidal points of their trajetory: he unonsiously selets these photons in a gas that is enormously more dense but that will not appear to him to be so, sine all the other photons will never have zero veloity relative to him The dynamis of non-ordinary yloids of the straight line in the plane We show that if the fore rotating with onstant angular veloity on a plane is applied to a material point already possessing a veloity lying on the same plane, the trajetory desribed by the point is not, in general, an ordinary yloid: it is so only at the onditions speified in Setion 6. If suh onditions are not satisfied (and, I repeat, we are on the plane) we shall have yloids termed either "prolate" or "urtate", aording to the diretion and the sense of the initial veloity relative to those of the fore vetor. Let us examine their respetive kinemati onstrutions. 1 - Prolate (or "strethed", "lengthened") yloids The prolate yloid is the trajetory of a point of a irumferene that rotates on a straight line in the plane while "reeping" uniformly forward: whih is to say, of a point that moves with onstant veloity on a irumferene that is translated with uniform retilinear motion, with the (peripheri) veloity of the point less than that of the translation of the irumferene. Another equivalent definition, and the best known: it is the trajetory of a point within a irle that rolls (without reeping). 2 - Curtate (or "urved", "looped", "shortened") yloids The urtate yloid is the trajetory of a point of a irumferene that rolls on a straight line in the plane while reeping uniformly bakward: whih is to say, of a point that moves with onstant veloity on a irumferene that is translated with uniform retilinear motion, with the (peripheral) veloity of the point greater than that of the translation of the irumferene. Another equivalent definition, and the best known: it is the trajetory of a point loated on the outer extension of a ray integral with a irumferene that rolls on the straight line (without reeping). In Figure 10 their respetive paths (b and ) are ompared with that of the ordinary yloid (a). In all three ases the irumferene is translated with uniform retilinear motion toward the right while being traversed at onstant veloity (v) by a point. The urved arrows indiate the (peripheral) veloity of revolution of the inner point at the enter, and the retilinear arrows applied to the enters indiate the veloity of translation of the irumferenes. The lengths of the arrows are proportional to the modulus of the veloity. The urved arrows are all of the same length, to indiate that the veloity of rotation (v) is the same in all three ases. This length is equal to that of the retilinear arrow in the ase of the ordinary yloid (a), it is greater ( v 1 v ) in the ase of the prolate yloid (b), and less ( v 2 v ) in the ase of the urtate yloid (). 12

14 While the prolate yloid is lengthened relative to the ordinary yloid, the urtate yloid is shortened, due to the presene of "eyelets" or "loops", in the ourse of whih the motion is retrograde for a streth (see the arrow drawn under one of the loops in Figure 10-). If d is the "length of the jump" in the ase of the ordinary yloid, d 1 d is the length of the jump in the ase of the prolate, and d 2 d in the ase of the urtate yloid (lengths measured between two suessive points of minimum veloity), as an be seen from the urves in the figure. If, as in the figure, then v 1 v = v v 2 d 1 d = d d 2 We repeat that these non-ordinary yloidal trajetories are distinguished from the ordinary trajetory by an important fat: at no point of suh trajetories is the veloity annulled, sine the "usps" have now vanished. What is more, at the extremities of every "jump" a minimum veloity is registered, whih depends, dynamially speaking, on the initial veloity inherently possessed by the point driven by the rotating fore. Getting bak to dynamis, the non-ordinary yloid is the more "shortened" or "lengthened" the greater (in one sense or in the other) is the value of the omponent of the initial veloity (whih in the ase of the ordinary yloid is zero, and shall therefore be referred to as "exess" veloity) in the overall diretion of the motion. A notable non-ordinary yloid is represented by the irular trajetory, the limit of the looped yloids, an element of separation between the two lasses with opposite onavity: it is a urtate yloid that is "only loop": kinematially it is obtained, in aordane with the riterion of Figure 10, by assuming v 2 =0. The simplest way to obtain it dynamially is with an initial veloity direted orthogonally to the fore vetor (pointed upward) at the moment in whih it is ativated, oriented to the left if the rotation is lokwise, and with the modulus equal to that whih one would have for the mean useful veloity of the yloidal motion if the point were initially at rest. As we see (in onfirmation of the observation made in Setion 3 of Chapter 1) it is, dynamially speaking, less simple to obtain the irular than the ordinary yloidal trajetory. In terms of "length of the jump" the value here is zero (d = 0), there is no overall "translation" of the motion: we an, however, ontinue to speak of temporal frequeny and of period, whih - as is the ase with all the non-ordinary yloidal trajetories - are equal to those of the ordinary yloid, sine (temporal) frequeny and period depend only on the angular veloity of the rotation of the fore, whih up to now we have assumed, together with the fore, to be equal for all the photons (monohromati radiation). 13

15 6.2 - Three-dimensional yloidal trajetories If the "exess" veloity in ourse does not lie on the fore's plane of rotation, we find ourselves in the presene of trajetories (whih we shall term yloidal in any ase) that are far more omplex: they develop in three dimensions, and do not lie on a plane as the ordinary, prolate and urtate yloids do. In suh trajetories, too, a minimum of the veloity is periodially attained, in a diretion that generally is not the overall diretion of the motion, nor is it loated in a plane that ontains it. For now we limit ourselves to this brief remark; we shall, however, have something more to say on the subjet. 7 - Overlapping waves of all lengths and veloities Thus the soure of eletromagneti radiation emits, at random, photons that desribe all the possible yloidal trajetories, inluding the non-ordinary, in two and in three dimensions. Suh trajetories, in their "landing" phase (we employ the term also to define the attaining of the minimum of the veloity in the ourse of their motion, and not neessarily zero value), order themselves in "asades" like the above-mentioned photons that desribe the ordinary trajetory. It is understood that the photons emitted, in our hypothesis (one again: we are oupied with perfetly monohrome radiation), are "struturally" all equal, in the sense that the mass (i.e., the inertia) and the rotating fore is equal for all of them, as is the veloity of rotation, whih, as we have seen, is orrelated with the unbalaning of the struture (whih gives the fore). What hanges is only the initial veloity, whih, being able to assume all the values, zero inluded, gives the asades of the dipole the freedom to be omposed of photons that make yloidal jumps of all kinds. To make this lear, let us fix a speifi non-ordinary trajetory, for example a well-defined urtate trajetory, orresponding to a ertain value of the initial veloity of the material point. The length of the jump, measured between two suessive points with minimum veloity, is less than that of the ordinary yloidal jump (but greater than that of the prolate yloid). In the ourse of a half-osillation the dipole emits - among others - photons all of whih desribe the urtate trajetory, detahing themselves from the soure at "all" the veloities (in the vetorial sense) that are registered in the ar of that speifi "looped" trajetory, and with the same distribution sin 2 x in the time of their quantity, in suh a way that the points with minimum veloity, at the extremity of the loops, are themselves also distributed in aordane with the same law in the spae of a half-wavelength. In the suessive half-osillation, whih ompletes the yle, the asade has its onavities turned the opposite way. Let us onsider a photon that detahes itself with the minimum of its veloity from the dipole, at the instant in whih the first half-osillation begins. In the time of a half-period of the antenna's ativity this photon will "land" (will reah the suessive minimum of veloity, ompleting its ar) at a distane from the dipole less than that at whih the photon that makes the ordinary jump does, but it will land in the same time, equivalent to the ommon period of rotation of the fore. In the time of the period - the time employed by the photon to "land" - the first half of the omplete osillation will be exhausted, and thus when the photon will have reahed its minimum veloity one again the emission of photons with the trajetory with downward onavity will have eased, and with the suessive half of the yle the trajetory of photons with upward onavity will begin. Hene the wavelength will be less than. And it will be the less, the shorter are the individual jumps - i.e., the greater is the loop. Here we have limited ourselves to onsidering, for the sake of example, a wave produed by photons that make urtate jumps of a ertain length: the fat is, as we have seen, that the antenna emits photons that desribe all possible trajetories (evidently not in the ontinuum, but with an extremely high density of values, suffiient to lend ontinuity to the phenomenal effets that interest us here), and thus every osillation of the antenna produes an emission of overlapping "waves" of all possible lengths, within the limits of the field of variability of the initial veloity the photons may possess at the at of emission. Thus "waves" of "all" lengths -, < and > - will be present, and overlapping, in the spae. The wavelength, equal to 2d, travels with veloity, the waves of less length with veloity less than, and those of greater length with veloity greater than. The equation Veloity = /P must in fat hold, and sine P is onstant, the greater is the greater is the veloity. We have rendered the piture far more omplex but our observer has not notied, sine he sees the photon 14

16 only when it is at rest (relative to him), and this is never the ase with the photons that make non-ordinary yloidal jumps. He pereives only the wave attributable to the photons that make the ordinary jumps. He does not interat with the others in any way. Let us say that he "detahes" from a bakground furrowed by an "infinity" of indistinguishable trajetories only the points at whih the veloity of the ordinary yloidal trajetories is annulled - trajetories whih he an thus reonstrut theoretially; and those points (elementary onstituents of the eletri and magneti fields) are distributed in the spae as a wave that advanes with veloity. Contemporaneously other "waves" advane, some shorter, others longer, with veloities lesser and greater than : we put them in quotes beause if a preise value of their length is not fixed, they remain indistinguishable in the spatial "ontinuum" in whih they are all present with their different lengths; whih is to say, with the different minimums of veloity of the photons that onstitute them - minimums, moreover, with different orientations. 8 - The observer in motion sees the photons with non-ordinary yloidal motion The observer, however, an pereive these other "waves" as well. It is suffiient that he move in the diretion of the forward or of the bakward movement of the points of their minimum veloity: beause if he does so, at the extremes of their yle, for an instant, the waves will have zero veloity relative to him. Now he will no longer see the photons that make the ordinary jumps in his previous referene frame, but rather those that in that frame make the so-alled non-ordinary jumps giving rise to a wave that is propagated with a veloity that differs from the of the veloity with whih he moves. Let us reommene, for the sake of simpliity, with the individual photon. Given a frame of referene, let the point move desribing ordinary yloidal jumps, and let the observer be at rest in the same frame: he will "see" ordinary yloidal jumps and will measure mean useful veloity (whih he will obtain by dividing the distane d between two points that he sees suessively illuminated in the spae for the time p that intervenes between the two events). If the observer moves with veloity v, we suppose first in the same sense as the overall motion of the point, in the new frame the ordinary yloidal trajetory will be transformed into a urtate ("looped") yloidal trajetory, with a distane between the points of minimum veloity less than d and a mean useful veloity of advane less than : - v, to be preise. But our observer, who interats only with the point at rest, will now see nothing at all. Analogous reasoning holds for his movement (again at a veloity with modulus v) in the opposite sense: he will "see" (but he will not see them) prolate trajetories, with distane greater than d between the points with lower veloity and mean useful veloity in the diretion of the propagation equal to + v. In the same frame of referene the point now moves desribing the "looped" trajetory, advaning with a mean useful veloity v (with, dynamially speaking, -v the suitably oriented "exess" additional veloity that the point inherently possesses), and with the distane between the extremities of two loops thus equal to p v. This is what the observer at rest relative to the frame of referene would measure (if he saw the photon also when it was in motion). If, however, he moves in the overall diretion of the motion of the point, with opposite sense, at veloity v, in his new referene frame the urtate yloid beomes ordinary: he sees the ordinary yloidal motion, with mean useful veloity ( = v v ), distane between the extremities of the loops d ( = p ), and the usps at zero veloity. The observer will find himself periodially, for an instant, proeeding with the same minimum veloity as the photon does when its turns bakward: at that instant he sees it at rest. By virtue of this last onlusion, if he is the observer who interats only with things at rest - the observer who, in the new hypothesis, at rest did not see anything - he will now be able to see the dots suessively illuminated, and thus measure and d. Exatly like when he was at rest in a frame of referene relative to whih the ordinary yloidal trajetory developed. To give the piture fuller detail, and to broaden it, we now fill the plane with all the possible yloidal trajetories referred to that frame, all of them direted as a whole like the two trajetories already onsidered, in suh a way that they possess all the additional initial values possible (within the limits of the "pseudoontinuity" we find onvenient). At any veloity v (positive or negative) with whih the observer moves in the given diretion he will be onfronted with a non-ordinary trajetory arising from that very "exess" veloity v, and, transforming it into an ordinary trajetory, he will ontinue to observe the same thing: a series of "dots turning on and off" 15

17 separated by spatial distane d and by temporal interval p, and thus, again, in "propagation" with veloity. Whih is equivalent to saying - if that whih he sees represents everything from whih he an obtain information experimentally - that he annot say whether he is in motion or whether he is at rest, and, if he is in motion, with what veloity, diretion and sense he is in motion relative to the provenane of the photon. 9 - Definitive generalization Only the motion of the observer in the diretion (x) of the overall advane of the photon has been onsidered here. Generalizing, we need to onsider his motion in any diretion whatever, not only in the domain of the non-ordinary yloids on the plane (xy) but also in that of ordinary yloids in three dimensions (xyz). In that way, with the onavity of the ar always oriented in the right sense half-phase by half-phase, with the observer moving in the same diretion and sense as the exess veloity, now direted in any way whatever, he will "reonstrut" the yloidal trajetory in his frame of referene and, for an instant, he will pereive the photon at rest. Let us now return to the wave proper. 16

18 CHAPTER 3 - SOURCE OR/AND OBSERVER IN MOTION 1 - Motion of the observer relative to the soure at rest Given a frame of referene, the observer an move in any diretion and sense relative to the soure at rest, just as the soure an move in any diretion and sense relative to the observer at rest. We shall onsider here the two most instrutive ases, those of approahing and of moving away along the fixed line joining observer and soure, with just a brief remark on movement along any diretion Observer approahing the soure along the fixed line that joins them The observer approahing the soure pereives the wave (shorter, as we have seen) formed by photons that make the looped jumps: in his frame of referene in motion the individual jumps are pereived as ordinary, of length d, and the mean useful veloity of the advane of the photon, and thus of the wave, is, again, : however, he measures the length 1 of the perturbation as that whih it is in the given frame, less than. Preisely, if - v is the veloity of the wave in the given frame, it will be 1=P v or, as a funtion of v 1= Let us onsider the frequeny. In the frame of referene of the soure at rest the frequeny ( u 1 ) of the shortest wave, of length P v that advanes with the lowest veloity - v, is the same ( u ) as the ordinary wave (length that travels at veloity ). In fat, the frequeny being equal to the veloity divided by the wavelength, we will have: u 1= v = = u v But in the frame of referene of the observer in motion toward the soure the frequeny ( v 2 ) inreases. In fat: v v 2 u 2= = = = u v v v As a funtion of u : u 2=u v Observer moving away from the soure along the fixed line Analogously in the ase of moving away from the soure: the observer will measure the length of the wave as that whih objetively it is, >, and a frequeny <. 1=P v or, as a funtion of 17