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Special Relativity: Einstein's Spherical Waves Versus Poincaré's

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Special Relativity: Einstein's Spherical Waves Versus Poincaré's Annales de la Fondation Louis de Broglie, Volume 30 no 3-4, 2005 353 Special Relativity: Einstein’s Spherical Waves versus Poincar´e’sEllipsoidal Waves Yves Pierseaux Physique des Hautes Energies Universit´eLibre de Bruxelles (ULB) e-mail: [email protected] ABSTRACT. We show that the image by the Lorentz Transformation of a spherical light wavefront, emitted by a moving source, is not a spherical light wavefront but an ellipsoidal light wavefront. Poincar´e’s elongated ellipse is the direct geometrical representation of Poincar´e’s relativity of simultaneity. Einstein’s circles are the direct geometrical representation of Einstein’s convention of synchronisation. Poincar´e’s ellipse supposes another convention for the definition of space-time units involving that the Lorentz Transformation (LT) of an unit of length is directly proportional to the LT of an unit of time: This is Poincar´e’sdefinition of isotropic elongated distance (with dilated travel time). The historical (polemical) problem of priorities is therefore sci- entifically solved because Einstein’s explicit kinematics and Poincar´e’s implicit kinematics are not the same. It is generally admitted that the observed expansion of space (Hub- ble) can only be considered in the framework of General Relativity (Robertson-Walker’s metric) and not in the framework on Special Rel- ativity (Minkowski’s metric). We show that Poincar´e’srelativistic kine- matics, with the invariance of the quadratic form by LT, predicts both a dilation of time and an isotropic expansion of space as well. 1 Introduction: Einstein’s Spherical Wavefront & Poincar´e’s Ellipsoidal Wavefront Einstein writes in 1905, in the third paragraph of his famous paper: At the time t = τ = 0, when the origin of the two co- ordinates (K and k) is common to the two systems, let a spherical wave be emitted therefrom, and be propagated 354 Y. Pierseaux -with the velocity c in system K. If x, y, z be a point just attained by this wave, then x2 + y2 + z2 = c2t2 (1) Transforming this equation with our equations of trans- formation (see Einstein’s LT, 29), we obtain after a simple calculation ξ2 + η2 + ζ2 = c2τ 2 (2) The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system k. [Einstein A.1905] Poincar´ewrites in 1908 in his second paper on ”La dynamique de l’´electron” with the subtitle ”Le principe de relativit´e”: Imagine an observer and a source involved together in the transposition. The wave surfaces emanating for the source will be spheres, having as centre the successive positions of the source. The distance of this centre from the present po- sition of the source will be proportional to the time elapsed since the emission - that is to say, to the radius of the sphere. But for our observer, on account of the contraction, all these spheres will appear as elongated ellipsoids. The compen- sation is now exact, and this is explained by Michelson’s experiments. [Poincar´eH. (1908)] We can further find in Poincar´e’stext the equation (two dimensions) of an elongated ellipse. The observer at rest (let us call him: O) is situated at the centre C and the source S in moving (together with ”our observer”, ”Michelson”, let us call him O’) is situated at the focus F of the ellipse. The contrast between both great relativists, Einstein et Poincar´e, is very clear: according to Einstein, the image of a spherical wave is a spherical wave whilst according to Poincar´ethe image of a spherical wave is an ellipsoidal wave. Does the latter not know the invariance of the quadratic form? Not at all because he does demonstrate, with the structure of group, in his first paper on ,”La dynamique de l’´electron” Special Relativity: Einstein’s Spherical Waves . 355 [Poincar´eH. 1905], that the Lorentz Transformation (LT) ”doesn’t mod- ify the quadratic form x2 + y2 + z2 − c2t2”. We must point out that Poincar´e’s lengthened ellipse has been completely ignored for a whole century by the scientific community1. We note also that Poincar´edoesn’t use LT in the previous quotation and directly deduces the ellipsoidal shape of the light wavefront (in the system of the source !) from the principle of contraction of (the unit) of length (see conclusion). So who is right: Einstein or Poincar´e? The best thing that we can do, to solve this dilemma, is to apply a LT to a spherical wavefront. 2 Image by LT of the Object ”Circular Wave” What is the image, by LT, in the system K of a spherical wave emitted in t0 = t = 0 by a source S at rest in the origin O’ of the system K’ ? The LT defined by Poincar´e(K’ is in uniform translation with respect to K) is: x0 = k(x − εt) y0 = y t0 = k(t − εx) (3) We keep Poincar´e’s notations where ε, k correspond to Einstein- Planck’s notations β, γ because, according to Poincar´ein his 1905 work about the theory of relativity, ”I shall choose the units of length and of time in such a way that the velocity of light is equal to unity” [Poincar´eH. 1905]. The deep meaning of Poincar´e’s choice of space- time units with c = 1 will be specified in §6. In order to have one only 0 wavefront, we have to define a time t as unit of time 1t0 . The equation of the circular wave front in the system of the source K’ (the geometrical locus of the object-points in K’) in t0 = 1 is : 02 02 02 x + y = t = 1t0 (4) The unprimed coordinates of the image-points are given by the in- verse LT: x = k(x0 + εt0) y = y0 t = k(t0 + εx0) (5) 1Poincar´e’s ellipsoidal wavefront was mentioned in his course of 1906 ”Les limites de la loi de Newton” [Poincar´eH. (1906)]. We also find them in ”La M´ecanique Nou- velle” [Poincar´eH. (1909)] and in ”La dynamique de l’´electron” [Poincar´eH. (1912)]. In fact it was in 1904, at a talk in Saint Louis, that Poincar´e first introduced the elongated ellipsoidal wave as an alternative (non relativistic alternative, see note 12) and not as a consequence of the contraction of the unit of length [Poincar´eH. (1904)]. Poincar´e’snon-relativistic 1904 ellipsoid is, very curiously, de- velopped by Guillaume[Poincar´eH. (1909)], Leroux [Le Roux J.] and Dive[Dive P.]. 356 Y. Pierseaux The coordinates (0, 0, 1) in K of the source in t0 = 1 are (kε, 0, k) and (kεt0, 0, kt0) in t0 6= 1. As LT is a punctual transformation (a transformation of events), let us determine the images (x, y, t) in K of different events or object-points in K’ with t0 = 1 : (1, 0, 1), √ √ 2 2 (−1, 0, 1), (0, 1, 1), ( 2 , 2 , 1) etc. (see figure 1). The image-point E, k(1 + ε), 0, k(1 + ε), is on the large dotted circle x2 + y2 = t2 with radius r = t = k(1 + ε). The image-point A, k(ε − 1), 0, k(1 − ε), is on the small dotted circle x2 + y2 = t2 with radius r = t = k(1 − ε). The image-point C, (kε, 1, k), is on the dotted circle x2 + y2 = t2 √ √ √ with the radius k. The image-point D, k( 2 + ε), 2 , k(1 + 2 ε), is on 2 2 √ 2 2 2 2 2 the dotted circle x + y = t with radius r = t = k(1 + 2 ε) etc... Special Relativity: Einstein’s Spherical Waves . 357 The images in K, by LT, (5) of the simultaneous events in K’, con- trary to what one might expect, are not situated on one circular wave- front but, given the invariance of the quadratic form (the dotted circles x2 +y2 = t2), in a circular ring between k(1−ε) ≤ r ≤ k(1+ε), figure1). By introducing , in the system K’, the angle θ0 determined by both the radius vector r’ and the Ox’ axes, we have x0 = r0 cos θ0 et y0 = r0 sin θ0. So with r0 = t0 6= 1 we have: t = kt0(1 + εcosθ0) (6) which is the temporal LT (5), t = k(t0+ εx0), with x0 = r0cosθ0 = t0cosθ0. We can also write (r = t) the locus of the images-points: r = kr0(1 + εcosθ0) (7) If r0 = t0 = 1 (figure1, normalization in K’), we then have: t = r = k(1 + εcosθ0) (8) We will show now that this ”temporal equation” (6) or ”radial equa- tion”(7) is the equation of an elongated ellipse in polar coordinates if we define the polar angle θ (see figure 2) as the relativistic transformation of the angle θ0(§3.1). 3 Poincar´e’s Elongated Ellipse deduced from LT 3.1 Poincar´e’sellipse (I) and the relativity of simultaneity Let us first determine Poincar´e’s elongated ellipse in Cartesian coordi- nates. We are seeking for the shape of the wavefront t0 = 1 in K, given that the quadratic form (4) is invariant: x2 + y2 = t2 (9) If the time t were fixed (see paragraph 4 on Einstein’s synchronisation), we would obviously have a circular wavefront; but t depends by LT on x0. If t is written in function of x’, we would not have the image of the wave in K. We must write t in function of x. By using the first and the third (x and t) LT (5), we have respectively if r0 = t0 6= 1 or r0 = t0 = 1: t = k−1t0 + εx or t = k−1 + εx (10) 358 Y.
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