On the dynamics of the electron: Introduction, 1, 9

Transcription

1 On the ynamics of the electron: Introuction,, 9 Henri oincaré Introuction. It seems at first that the aberration of light an relate optical an electrical phenomena will provie us with a means of etermining the absolute motion of the Earth, or rather its motion with respect to the ether, as oppose to its motion with respect to other celestial boies. Fresnel pursue this iea, but soon recognize that the Earth s motion oes not alter the laws of refraction an reflection. Analogous experiments, like that of the water-fille telescope, an all those consiering terms no higher than first orer relative to the aberration, yiele only negative results the explanation was soon iscovere. But Michelson, who conceive an experiment sensitive to terms epening on the square of the aberration, faile in turn. It appears that this impossibility to etect the absolute motion of the Earth by experiment may be a general law of nature we are naturally incline to amit this law, which we will call the ostulate of Relativity an amit without restriction. Whether or not this postulate, which up to now agrees with experiment, may later be corroborate or isprove by experiments of greater precision, it is interesting in any case to ascertain its consequences. An explanation was propose by Lorentz an FitzGeral, who introuce the hypothesis of a contraction of all boies in the irection of the Earth s motion an proportional to the square of the aberration. This contraction, which we will call the Lorentzian contraction, woul explain Michelson s experiment an all others performe up to now. The hypothesis woul become insufficient, however, if we were to amit the postulate of relativity in full generality. Lorentz then sought to exten his hypothesis an to moify it in orer to obtain perfect agreement with this postulate. This is what he succeee in oing in his article entitle Electromagnetic phenomena in a system moving with any velocity smaller that that of light (roceeings of the Amsteram Acaemy, May 7, 904). The importance of the question persuae me to take it up in turn the results I obtaine agree with those of Mr. Lorentz on all the significant points. I was le merely to moify an exten them only in a few etails further on we will see the points of ivergence, which are of seconary importance. Translate by Scott Walter from Reniconti el Circolo Matematico i alermo, 906, To appear in J. Renn (e.), The Genesis of General Relativity Vol. 3: Theories of Gravitation in the Twilight of Classical hysics art I (Boston Stuies in the hilosophy of Science 0). Springer, in press. The original notation is faithfully reprouce, incluing the use of for both orinary an partial ifferentiation. The translator s footnote calls are brackete. For alternative translations of oincaré s memoir see C. W. Kilmister (Special Theory of Relativity, Oxfor: ergamon, 970, 45 85), an H. M. Schwartz (American Journal of hysics 39: :86 87, 8 87).

2 Lorentz s iea may be summe up like this: if we are able to impress a translation upon an entire system without moifying any observable phenomena, it is because the equations of an electromagnetic meium are unaltere by certain transformations, which we will call Lorentz transformations. Two systems, one of which is at rest, the other in translation, become thereby exact images of each other. Langevin ) sought to moify Lorentz s iea for both authors, the moving electron takes the form of a flattene ellipsoi. For Lorentz, two axes of the ellipsoi remain constant, while for Langevin, ellipsoi volume remains constant. The two scientists also showe that these two hypotheses are corroborate by Kaufmann s experiments to the same extent as the original hypothesis of Abraham (rigi-sphere electron). The avantage of Langevin s theory is that it requires only electromagnetic forces, an bons it is, however, incompatible with the postulate of relativity. This is what Lorentz showe, an this is what I foun in turn using a ifferent metho, which calls on principles of group theory. We must return therefore to Lorentz s theory, but if we want to o this an avoi intolerable contraictions, we must posit the existence of a special force that explains both the contraction, an the constancy of two of the axes. I sought to etermine this force, an foun that it may be assimilate to a constant external pressure on the eformable an compressible electron, whose work is proportional to the electron s change in volume. If the inertia of matter is exclusively of electromagnetic origin, as generally amitte in the wake of Kaufmann s experiment, an all forces are of electromagnetic origin (apart from this constant pressure that I just mentione), the postulate of relativity may be establishe with perfect rigor. This is what I show by a very simple calculation base on the principle of least action. But that is not all. In the article cite above, Lorentz juge it necessary to exten his hypothesis in such a way that the postulate remains vali in case there are forces of non-electromagnetic origin. Accoring to Lorentz, all forces are affecte by the Lorentz transformation (an consequently by a translation) in the same way as electromagnetic forces. It was important to examine this hypothesis closely, an in particular to ascertain the moifications we woul have to apply to the laws of gravitation. We fin first of all that it requires us to assume that gravitational propagation is not instantaneous, but occurs with the spee of light. One might think that this is reason enough to reject the hypothesis, since Laplace emonstrate that this cannot be the case. In reality, however, the effect of this propagation is compensate in large part by a ifferent cause, in such a way that no contraiction arises between the propose law an astronomical observations. Is it possible to fin a law satisfying Lorentz s conition, an reucing to Newton s law whenever the spees of celestial boies are small enough to allow us to neglect their squares (as well as the prouct of acceleration an istance) with respect to the square of the spee of light? To this question we must respon in the affirmative, as we will see later. Moifie in this way, is the law compatible with astronomical observations? It seems so on first sight, but the question will be settle only after an extene iscussion. Suppose, then, that this iscussion is settle in favor of the new hypothesis, what Langevin was anticipate by Mr. Bucherer of Bonn, who earlier avance the same iea. Bucherer, Mathematische Einführung in ie Elektronentheorie, August, 904. Teubner, Leipzig). (See:

3 shoul we conclue? If propagation of attraction occurs with the spee of light, it coul not be a fortuitous accient. Rather, it must be because it is a function of the ether, an then we woul have to try to penetrate the nature of this function, an to relate it to other flui functions. We cannot be content with a simple juxtaposition of formulas that agree with each other by goo fortune alone these formulas must, in a manner of speaking, interpenetrate. The min will be satisfie only when it believes it has perceive the reason for this agreement, an the belief is strong enough to entertain the illusion that it coul have been preicte. But the question may be viewe from a ifferent perspective, better shown via an analogy. Let us imagine a pre-copernican astronomer who reflects on tolemy s system he will notice that for all the planets, one of two circles epicycle or eferent is traverse in the same time. This fact cannot be ue to chance, an consequently between all the planets there is a mysterious link we can only guess at. Copernicus, however, estroys this apparent link by a simple change in the coorinate axes that were consiere fixe. Each planet now escribes a single circle, an orbital perios become inepenent (until Kepler reestablishes the link that was believe to have been estroye). It is possible that something analogous is taking place here. If we were to amit the postulate of relativity, we woul fin the same number in the law of gravitation an the laws of electromagnetism the spee of light an we woul fin it again in all other forces of any origin whatsoever. This state of affairs may be explaine in one of two ways: either everything in the universe woul be of electromagnetic origin, or this aspect share, as it were, by all physical phenomena woul be a mere epiphenomenon, something ue to our methos of measurement. How o we go about measuring? The first response will be: we transport soli objects consiere to be rigi, one on top of the other. But that is no longer true in the current theory if we amit the Lorentzian contraction. In this theory, two lengths are equal, by efinition, if they are traverse by light in equal times. erhaps if we were to abanon this efinition Lorentz s theory woul be as fully overthrown as was tolemy s system by Copernicus s intervention. Shoul that happen some ay, it woul not prove that Lorentz s efforts were in vain, because regarless of what one may think, tolemy was useful to Copernicus. I, too, have not hesitate to publish these few partial results, even if at this very moment the iscovery of magneto-cathoe rays seems to threaten the entire theory.. Lorentz Transformation. Lorentz aopte a certain system of units in orer to o away with 4 factors in formulas. I will o the same, an in aition, select units of length an time in such a way that the spee of light equals. Uner these conitions, an enoting electric isplacement f, g, h, magnetic intensity, ˇ,, vector potential F, G, H, scalar potential, charge ensity, electron velocity,,, an current u, v, w, the funamental formulas 3

4 become: u D f C D t y D g h z y t D ˇ z D H y t C x D 0 G z f D F t f x D t x t D D F D : x t C F x D 0 An elementary particle of matter of volume xyz is acte upon by a mechanical force, the components of which are erive from the formula: 9 >= > () D f C. ˇ/: () These equations amit a remarkable transformation iscovere by Lorentz, which owes its interest to the fact that it explains why no experiment can inform us of the absolute motion of the universe. Let us put: x 0 D kl.x C "t/ t 0 D kl.t C "x/ y 0 D `y z 0 D `z (3) where ` an " are two arbitrary constants, such that Now if we put: we will have: k D p " : 0 D x 0 t 0 0 D ` : Let a sphere be carrie along with the electron in uniform translation, an let the equation of this mobile sphere be:.x t/ C.y t/ C.z t/ D r an the volume of the sphere be 4 3 r3. Œ The transformation will change the sphere into an ellipsoi, the equation of which is easy to fin. We thus euce easily from (3): x D k`.x0 "t 0 / t D k`.t 0 "x 0 / y D y0 ` The equation of the ellipsoi then becomes: z D z0 ` : (30 ) k.x 0 "t 0 t 0 C "x 0 / C.y 0 kt 0 C k"x 0 / C.z 0 kt 0 C k"x 0 / D `r : This ellipsoi is in uniform motion for t 0 D 0, it reuces to k x 0. C "/ C.y 0 C k"x 0 / C.z 0 C k"x 0 / D `r Œ The original reas: 4 3 r. 4

5 an has a volume: 4 `3 3 r3 k. C "/ : If we want electron charge to be unaltere by the transformation, an if we esignate the new charge ensity 0, we will fin: 0 D k`3. C "/: (4) What will be the new velocity components 0, 0 an 0? We shoul have: whence: 0 D y0 t 0 D 0 D x0 D t 0 y k.t C "x/ D.x C "t/.t C "x/ D C " C " k. C "/ 0 D k. C "/ 0 0 D k`3. C "/ 0 0 D `3 0 0 D `3 : (40 ) Here is where I must point out for the first time a ifference with Lorentz. In my notation, Lorentz put (l.c., page 83, formulas 7 an 8): 0 D In this way we recover the formulas: k`3 0 D k. C "/ 0 D k 0 D k: 0 0 D k`3. C "/ 0 0 D `3 0 0 D `3 I although the value of 0 iffers. It is important to notice that the formulas (4) an (4 0 ) satisfy the conition of continuity 0 t 0 C 0 0 x 0 D 0: To see this, let be an unetermine coefficient an D the Jacobian of t C x C y C z C (5) with respect to t, x, y an z. It follows that: with D 0 D, D D t C x D 0. Let 0 D `4 0 Πthen the 4 functions D D D 0 C D C D C D 3 3 C D 4 4 t 0 C 0 0 x 0 C y 0 C z 0 C (5 0 ) are relate to the functions (5) by the same linear relationships as the ol variables to the new ones. Therefore, if we enote D 0 the Jacobian of the functions (5 0 ) with respect to the new variables, it follows that: ΠThe original reas: 0 D ` 0. D 0 D D D 0 D D 0 0 C D0 0 C C D

6 an thereby: Œ3 D 0 0 D D 0 D D 0 D ` 4D D 0 D 0 t 0 C 0 0 x 0 : Q.E.D. Uner Lorentz s hypothesis, this conition woul not be met since 0 has a ifferent value. We will efine the new vector an scalar potentials in such a way as to satisfy the conitions 0 0 D 0 0 F 0 D 0 0 : (6) From this we euce: 0 D k`. C "F/ F 0 D k`.f C " / G0 D `G H 0 D `H: (7) These formulas iffer noticeably from those of Lorentz, although the ivergence stems ultimately from the efinitions employe. New electric an magnetic fiels are now chosen in orer to satisfy the equations f 0 D F 0 t 0 0 x 0 0 D H 0 y 0 G 0 z 0 : (8) It is easy to see that: D k` " t 0 t x D k` x 0 x " t D ` y 0 y D ` z 0 z an we euce thereby: f 0 D ` f g0 D k`.g C " / h0 D 0 D ` ˇ0 D k`.ˇ "h/ 0 D k`.h "ˇ/ k`. C "g/: 9 >= > (9) These formulas are ientical to those of Lorentz. Our transformation oes not alter (). In fact, the conition of continuity, as well as (6) an (8) were alreay feature in () (neglecting the primes). Combining (6) with the conition of continuity, we obtain: It remains for us to establish: f 0 C 0 0 D 0 t 0 y 0 ˇ0 z 0 0 t 0 C F 0 x 0 D 0: (0) 0 t 0 D g0 z 0 h 0 y 0 f 0 x 0 D 0 an it is easy to see that these are necessary consequences of (6), (8) an (0). We must now compare forces before an after the transformation. Œ3 The original reas: D 0 D ` D. 6

7 Let, Y, Z be the force prior to the transformation, an 0, Y 0, Z 0 the force after the transformation, both forces being per unit volume. In orer for 0 to satisfy the same equations as before the transformation, we must have: 0 D 0 f 0 C Y 0 D 0 g 0 C Z 0 D 0 h 0 C 0. 0ˇ0 0ˇ0/ 0 0 / 0 0/ or, replacing all quantities by their values (4), (4 0 ) an (9), an in light of (): 0 D k`5. C " 9 / >= Y 0 D Y `5 () Z 0 D `5 Z: > Instea of representing the components of force per unit volume by, Y, Z, we now let these terms represent the force per unit electron charge, an we let 0, Y 0, Z0 represent the latter force after transformation. It follows that: D f C ˇ 0 D f 0 C 0 0 0ˇ0 D 0 D 0 an we obtain the equations: 0 D k`5. 0 C " / Y 0 D `5 Y 0 Z 0 D `5 Z : 0 Lorentz foun (page 83, equation (0) with ifferent notation): 9 D ` 0 `". 0 g 0 C 0 h 0 / ` `" >= Y D k Y 0 C k 0 g 0 ` `" Z D k Z0 C k 0 h 0 : > 9 >= > ( 0 ) ( 00 ) Before going any further, it is important to locate the source of this significant ivergence. It obviously springs from the fact that the formulas for 0, 0 an 0 are not the same, while the formulas for the electric an magnetic fiels are the same. If electron inertia is exclusively of electromagnetic origin, an if electrons are subject only to forces of electromagnetic origin, then the conitions of equilibrium require that: D Y D Z D 0 insie the electrons. Accoring to (), these relationships are equivalent to 0 D Y 0 D Z 0 D 0: 7

8 The electron s equilibrium conitions are therefore unaltere by the transformation. Unfortunately, such a simple hypothesis is inamissible. In fact, if we assume D D D 0, the conition D Y D Z D 0 leas necessarily to f D g D h D 0, an consequently, to f D 0, i.e., D 0. Similar results obtain for the most x general case. We must then amit that in aition to electromagnetic forces there are either non-electromagnetic forces or bons. Therefore, we nee to ientify the conitions that these forces or these bons must satisfy for electron equilibrium to be unisturbe by the transformation. This will be the object of an upcoming section. 9. Hypotheses Concerning Gravitation. In this way Lorentz s theory woul fully explain the impossibility of etecting absolute motion, if all forces were of electromagnetic origin. But there exist other forces to which an electromagnetic origin cannot be attribute, such as gravitation, for example. It may in fact happen, that two systems of boies prouce equivalent electromagnetic fiels, i.e., exert the same action on electrifie boies an on currents, an at the same time, these two systems o not exert the same gravitational action on Newtonian masses. The gravitational fiel is therefore istinct from the electromagnetic fiel. Lorentz was oblige thereby to exten his hypothesis with the assumption that forces of any origin whatsoever, an gravitation in particular, are affecte by a translation (or, if one prefers, by the Lorentz transformation) in the same manner as electromagnetic forces. It is now appropriate to enter into the etails of this hypothesis, an to examine it more closely. If we want the Newtonian force to be affecte by the Lorentz transformation in this fashion, we can no longer suppose that it epens only on the relative position of the attracting an attracte boies at the instant consiere. The force shoul also epen on the velocities of the two boies. An that is not all: it will be natural to suppose that the force acting on the attracte boy at the instant t epens on the position an velocity of this boy at this same instant t, but it will also epen on the position an velocity of the attracting boy, not at the instant t, but at an earlier instant, as if gravitation ha taken a certain time to propagate. Let us now consier the position of the attracte boy at the instant t 0, an let x 0, y 0, z 0 be its coorinates, an,, its velocity components at this instant let us consier also the attracting boy at the corresponing instant t 0 C t, an let its coorinates be x 0 C x, y 0 C y, z 0 C z, an its velocity components be,, at this instant. First we shoul have a relationship '.t x y z / D 0 () in orer to efine the time t. This relationship will efine the law of propagation of gravitational action (I o not constrain myself by any means to a propagation velocity equal in all irections). Now let, Y, Z be the three components of the action exerte on the attracte boy at the instant t 0 Œ4 we want to express, Y, Z as functions of What conitions must be satisfie? Œ4 The original reas: à l instant t. t x y z : () 8

9 ı The conition () shoul not be altere by transformations of the Lorentz group. ı The components, Y, Z shoul be affecte by transformations of the Lorentz group in the same manner as the electromagnetic forces esignate by the same letters, i.e., in accorance with ( 0 ) of section. 3 ı When the two boies are at rest, the orinary law of attraction will be recovere. It is important to note that in the latter case, the relationship () vanishes, because if the two boies are at rest the time t plays no role. ose in this fashion the problem is obviously ineterminate. We will therefore seek to satisfy to the utmost other, complementary conitions. 4 ı Since astronomical observations o not seem to show a sensible eviation from Newton s law, we will choose the solution that iffers the least with this law for small velocities of the two boies. 5 ı We will make an effort to arrange matters in such a way that t is always negative. Although we can imagine that the effect of gravitation requires a certain time in orer to propagate, it woul be ifficult to unerstan how this effect coul epen on the position not yet attaine by the attracting boy. There is one case where the ineterminacy of the problem vanishes it is the one where the two boies are in mutual relative rest, i.e., where D D D I this is then the case we will examine first, by supposing that these velocities are constant, such that the two boies are engage in a common uniform rectilinear translation. We may suppose that the x axis is parallel to this translation, such that D D 0, an we will let " D. If we apply the Lorentz transformation uner these conitions, after the transformation the two boies will be at rest, an it follows that: 0 D 0 D 0 D 0: The components, Y, Z shoul then agree with Newton s law an we will have, apart from a constant factor: 0 D x r 03 Y 0 D y r 03 Z0 D z r 03 r 0 D x 0 C y 0 C z 0 : (3) But accoring to section we have: We have in aition: x 0 D k.x C "t/ y 0 D y z 0 D z t 0 D k.t C "x/ 0 D k. C "/ D k. " / D k D " 0 D k 0. C " / D k. " / D Y 0 D k 0 Y D ky Z 0 D kz : x C "t D x t r 0 D k.x t/ C y C z 9

10 an which may be written: D k.x t/ Y r 03 D y k r Z 03 D z I (4) k r 03 D V x Y D V y Z D V z I V D k r 0 : (40 ) It seems at first that the ineterminacy remains, since we mae no hypotheses concerning the value of t, i.e., the transmission spee an that besies, x is a function of t. It is easy to see, however, that the terms appearing in our formulas, x t, y, z, o not epen on t. We see that if the two boies translate together, the force acting on the attracte boy is perpenicular to an ellipsoi, at the center of which lies the attracting boy. To avance further, we nee to look for the invariants of the Lorentz group. We know that the substitutions of this group (assuming ` D ) are linear substitutions that leave unaltere the quaratic form Let us also put: D ıx ıt D ı x ı t x C y C z t : D ıy ıt D ı y ı t D ız ıt D ı z ı t I we see that the Lorentz transformation will make ıx, ıy, ız, ıt, an ı x, ı y, ı z, ı t unergo the same linear substitutions as x, y, z, t. Let us regar x y z t p ıx ıy ız ıt p ı x ı y ı z ı t p as the coorinates of 3 points, 0, 00 in space of 4 imensions. We see that the Lorentz transformation is merely a rotation in this space about the origin, assume fixe. Consequently, we will have no istinct invariants apart from the 6 istances between the 3 points, 0, 00, consiere separately an with the origin, or, if one prefers, apart from the two expressions x C y C z t xıx C yıy C zız tıt or the 4 expressions of like form euce from an arbitrary permutation of the 3 points, 0, 00. But what we seek are invariants that are functions of the 0 variables (). Therefore, among the combinations of our 6 invariants we must fin those epening only on these 0 variables, i.e., those that are 0th egree homogeneous with respect both to ıx, ıy, ız, ıt, an to ı x, ı y, ı z, ı t. We will then be left with 4 istinct invariants: x t t x p t q x q : (5) 0

11 Next let us see how the force components are transforme we recall the equations () of section, that refer not to the force, Y, Z consiere at present, but to the force per unit volume:, Y, Z. We esignate moreover T D I we will see that () can be written (` D ): 0 D k. C "T / T 0 D k.t C " / Y 0 D Y Z 0 D ZI ) (6) in such a way that, Y, Z, T unergo the same transformation as x, y, z, t. Consquently, the group invariants will be T x T t ıx T ıt ı x T ı t: However, it is not, Y, Z that we nee, but, Y, Z, with We see that T D : D Y Y D Z Z D T T D : Therefore, the Lorentz transformation will act in the same manner on, Y, Z, T, as on, Y, Z, T, except that these expressions will be multiplie moreover by 0 D k. C "/ D ıt ıt : 0 Likewise, the Lorentz transformation will act in the same way on,,, as on ıx, ıy, ız, ıt, except that these expressions will be multiplie moreover by the same factor: ıt ıt 0 D k. C "/ : Next we consier, Y, Z, T p as the coorinates of a fourth point Q the invariants will then be functions of the mutual istances of the five points Q an among these functions we must retain only those that are 0th egree homogeneous with respect, on one han, to Y Z T ıx ıy ız ıt (variables that can be replace further by, Y, Z, T,,,, ), an on the other han, with respect to Œ5 ı x ı y ı z ı t (variables that can be replace further by,,, ). Œ5 The original reas ı x, ı y, ı z,.

12 In this way we fin, beyon the four invariants (5), four istinct new invariants: T x T t p T p q T : (7) The latter invariant is always null accoring to the efinition of T. These terms being settle, what conitions must be satisfie? ı The first term of (), efining the velocity of propagation, has to be a function of the 4 invariants (5). A wealth of hypotheses can obviously be entertaine, of which we will examine only two: A) We can have x t D r t D 0 from whence t D r, an, since t has to be negative, t D r. This means that the velocity of propagation is equal to that of light. It seems at first that this hypothesis ought to be rejecte outright. Laplace showe in effect that the propagation is either instantaneous or much faster than that of light. However, Laplace examine the hypothesis of finite propagation velocity ceteris non mutatis here, on the contrary, this hypothesis is conjoine with many others, an it may be that between them a more or less perfect compensation takes place. The application of the Lorentz transformation has alreay provie us with numerous examples of this. B) We can have t x q D 0 t D x : The propagation velocity is therefore much faster than that of light, but in certain cases t coul be positive, which, as we mentione, seems harly amissible. Œ6 We will therefore stick with hypothesis (A). ı The four invariants (7) ought to be functions of the invariants (5). 3 ı When the two boies are at absolute rest,, Y, Z ought to have the values given by Newton s law, an when they are at relative rest, the values given by (4). For the case of absolute rest, the first two invariants (7) ought to reuce to x or, by Newton s law, to r 4 r I in aition, accoring to hypothesis (A), the an 3 r invariants in (5) become: r x p r q x that is, for absolute rest, r r: Œ6 The original reas t pourrait être négatif.

13 We may therefore amit, for example, that the first two invariants in (7) reuce to Œ7 q.r C x / 4 r C x although other combinations are possible. A choice must be mae among these combinations, an furthermore, we nee a 3 r equation in orer to efine, Y, Z. In making such a choice, we shoul try to come as close as possible to Newton s law. Let us see what happens when we neglect the squares of the velocities,, etc. (still letting t D r). The 4 invariants (5) then become: 0 r x r x an the 4 invariants (7) become:.x C r/. / 0: Before we can make a comparison with Newton s law, another transformation is require. In the case we consier, x 0 C x, y 0 C y, z 0 C z, represent the coorinates of the attracting boy at the instant t 0 C t, an r D p x. With Newton s law we have to consier the coorinates of the attracting boy x 0 C x, y 0 C y, z 0 C z at the instant t 0, an the istance r D p x. We may neglect the square of the time t require for propagation, an procee, consequently, as if the motion were uniform we then have: x D x C t y D y C t z D z C t r.r r / D x ti or, since t D r, x D x r y D y r z D z r r D r x I such that our 4 invariants (5) become: 0 r C x. / r an our 4 invariants (7) become: Œx C. /r. / 0: In the secon of these expressions I wrote r instea of r, because r is multiplie by, an because I neglect the square of. For these 4 invariants (7), Newton s law woul yiel x. / x. / 0: r 4 r r Therefore, if we esignate the n an 3 r of the invariants (5) as A an B, an the first 3 invariants of (7) as M, N,, we will satisfy Newton s law to first-orer terms in the square of velocity by setting: M D B 4 Œ7 The original has (4) instea of (7). r 3 N D CA B D A B B 3 : (8) 3

14 This solution is not unique. Let C be the 4 th invariant in (5) C is of the orer of the square of, an it is the same with.a B/. The solution (8) appears at first to be the simplest, nevertheless, it may not be aopte. In fact, since M, N, are functions of, Y, Z, an T D, the values of, Y, Z can be rawn from these three equations (8), but in certain cases these values woul become imaginary. To avoi this ifficulty we will procee in a ifferent manner. Let us put: k 0 D p k D which is justifie by analogy with the notation k D p q feature in the Lorentz substitution. In this case, an in light of the conition r D t, the invariants (5) become: 0 A D k 0.r C x/ B D k.r C x / C D k 0 k. /: Moreover, we notice that the following systems of quantities: x y z r D t k 0 k 0 Y k 0 Z k 0 T k 0 k 0 k 0 k 0 k k k k unergo the same linear substitutions when the transformations of the Lorentz group are applie to them. We are le thereby to put: D x 9 k C ˇ C k 0 k 0 Y D y k 0 C ˇ C k k 0 Z D z k 0 C ˇ C k k 0 T D r k 0 C ˇ C k k 0 : It is clear that if, ˇ, are invariants,, Y, Z, T will satisfy the funamental conition, i.e., the Lorentz transformations will make them unergo an appropriate linear substitution. However, for equations (9) to be compatible we must have T D 0 which becomes, replacing, T, Z, T with their values in (9) an multiplying by k 0 : >= > (9) A ˇ C D 0: (0) 4

15 What we woul like is that the values of, Y, Z remain in line with Newton s law when we neglect (as above) the squares of velocities, etc. with respect to the square of the velocity of light, an the proucts of acceleration an istance. We coul select ˇ D 0 D A C : To the aopte orer of approximation, we obtain k 0 D k D C D A D r C x. / B D r x D x C t D x r: The st equation in (9) then becomes D.x A /: But if the square of is neglecte, A can be replace by r, or by r, which yiels: D.x C r/ D x : Newton s law woul yiel D x : r 3 Consequently, we must select a value for the invariant which reuces to aopte orer of approximation, that is,. Equations (9) will become: B3 D x 9 k A k 0 B 3 B 3 C k 0 Y D y k A k 0 B 3 k 0 B 3 C Z D z k A k 0 B 3 k 0 B 3 C r k A T D k 0 B 3 k 0 B 3 C : >= > r 3 in the We notice first that the correcte attraction is compose of two components: one parallel to the vector joining the positions of the two boies, the other parallel to the velocity of the attracting boy. Remember that when we speak of the position or velocity of the attracting boy, this refers to its position or velocity at the instant the gravitational wave takes off for the attracte boy, on the contrary, this refers to the position or velocity at the instant the gravitational wave arrives, assuming that this wave propagates with the velocity of light. I believe it woul be premature to seek to push the iscussion of these formulas further I will therefore confine myself to a few remarks. ı The solutions () are not unique we may, in fact, replace the the global factor by B 3 C.C /f.a B C / C.A B/ f.a B C / () B 3 5

16 where f an f are arbitrary functions of A, B, C. Alternatively, we may forgo setting ˇ to zero, but a any complementary terms to, ˇ, that satisfy conition (0) an are of secon orer with respect to the for, an of first orer for ˇ an. ı The first equation in () may be written: D k B 3 C h x an the quantity in brackets itself may be written: C r C i x ( 0 ).x C r / C. y x / C. z x / () such that the total force may be separate into three components corresponing to the three parentheses of expression () the first component is vaguely analogous to the mechanical force ue to the electric fiel, the two others to the mechanical force ue to the magnetic fiel to exten the analogy I may, in light of the first remark, replace B 3 in () by C B 3, in such a way that, Y, Z are linear functions of the attracte boy s velocity,,, since C has vanishe from the enominator of ( 0 ). Next we put: ) k.x C r / D k.y C r / D k.z C r / D k. z y/ D 0 k. x z/ D 0 k. y x / D 0 I (3) an since C has vanishe from the enominator of ( 0 ), it will follow that: D B 3 B 3 Y D >= 0 0 B 3 B 3 an we will have moreover: Z D 0 0 I B 3 B 3 > (4) Now,,, or 0 B 3, 0 B 3, B 3, B D 0 : (5) B 3, is an electric fiel of sorts, while 0, 0, 0, or rather B, 0 is a magnetic fiel of sorts. 3 B3 3 ı The postulate of relativity woul compel us to aopt solution (), or solution (4), or any solution at all among those erive on the basis of the first remark. However, the first question to ask is whether or not these solutions are compatible with astronomical observations. The eviation from Newton s law is of the orer of, i.e., 0000 times smaller than if it were of the orer of, i.e., if the propagation were to take place with the velocity of light, ceteris non mutatis consequently, it is legitimate to hope that it will not be too large. To settle this question, however, woul require an extene iscussion. aris, July, 905. H. oincaré 6