From Special Relativity to Feynman Diagrams.pdf

The effect on the velocity of the motion of s

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, the effect on the velocity of the motion of S relative to the medium is an effect of order V 2 v 2 ( s ) since: v 2 ( s ) + V 2 v ( s ) 1 + V 2 v 2 ( s ) v ( s ) 1 + 1 2 V 2 v 2 ( s ) . (1.31) The example of a sound wave illustrates the general fact that the velocity of prop- agation of a mechanical wave is isotropic , that is the same in every direction, only in a reference frame at rest with respect to the transmission medium . In any other inertial frame, the wave velocity is not isotropic, but depends on the direction.
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12 1 Special Relativity Fig. 1.3 Sound wave propagating, with respect to S , along the y-axis Fig. 1.4 Same sound wave as seen from S If we now consider the theory of electromagnetism, and in particular the propaga- tion of electromagnetic waves, we immediately note some peculiarities with respect to ordinary material waves. Electromagnetism, ignoring quantum processes, is de- scribed, with extremely good precision, by the Maxwell equations . Maxwell’s theory predicts that electric and magnetic fields can propagate, in the form of electromag- netic waves, in the vacuum , that is apparently without a transmission medium, with a velocity, denoted by c , which is related to the parameters of the theory: c = 1 0 μ 0 = 2 . 997925 × 10 8 m / s . We refer to this velocity as the speed of light since, as is well known, light is just an electromagnetic wave with wavelength in the approximate range between 380 and 780 nanometers. According to the principle of relativity, this velocity, being determined only by the parameters of the theory, should be the same for all the inertial observers. On the other hand, we have learned that the velocity of a wave should change by a change in the (inertial) reference frame. How can we resolve this apparent contradiction? We note, first of all, that not only the velocity of electromagnetic waves changes under a Galilean transformation, but, as one can easily ascertain, also the Maxwell equations themselves are not left invariant by such transformations. Since we cannot
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1.2 The Speed of Light and Electromagnetism 13 give up the principle of relativity for electromagnetic phenomena, there are only two possibilities: • either the Maxwell equations and their consequences are valid only in a particular frame, and thus should change their form by a change in reference frame; • or the Maxwell equations are valid in every inertial frame, but the principle of relativity should not be implemented by the Galilean transformations. Instead, the right transformation laws should be chosen in such a way as to keep the validity of this principle also for electromagnetism as expressed by the Maxwell equations. Let us first discuss the former hypothesis. If there existed a privileged reference frame in which the Maxwell equations hold, and thus with respect to which light has velocity c , we should be able to experimentally detect it. In this respect, over the course of the nineteenth century, physicists made various hypotheses, among which we quote the following two.
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