From Special Relativity to Feynman Diagrams.pdf

# Under galilean transformations therefore the

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mass, acceleration, have exactly the same values in the two frames. Under Galilean transformations therefore the conservation of linear momentum is an example of a covariant law, while the Newtonian law of motion is invariant . As a last point we observe that the independence of the laws of classical mechanics from the inertial frame is easily verified in our everyday life. Everybody traveling by car, train, or ship and moving with uniform rectilinear motion with respect to the earth (considered as an inertial frame) can observe that the oscillation of a pendulum, the bouncing of a ball, the collisions of billiard-balls, etc., occur exactly in the same way as in the earth frame. On the other hand if the moving frame is accelerated the laws of mechanics are violated, since the new frame is no longer inertial. 1.2 The Speed of Light and Electromagnetism It is well known that many mechanical phenomena, such as vibrating strings, acoustic waves in a gas, ordinary waves on a liquid, can be described in terms of propagating waves. These mechanical waves describe the propagation through a given material medium of a perturbation originating from a source located in a point or a region in its interior (like for instance the impact of a stone on the surface of a pond), the propagation being due to the interactions among the molecules of the medium. If the medium is homogeneous and isotropic (which we shall always assume to be the case) the speed of propagation of a wave has the same constant value in every direction with respect to the medium itself. For example, in the case of acoustic waves propagating through the atmosphere, the speed of sound is v ( s ) 330 m / s with respect to the air (supposed still). Let S be a frame at rest with respect to the air and S another frame in relative uniform motion with velocity V with respect to the former (we assume the stan- dard configuration between the two coordinate systems). By means of ( 1.6 ) we may compute the velocity v ( s ) of a sound signal with respect to S . v ( s ) = v ( s ) V . (1.25)

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1.2 The Speed of Light and Electromagnetism 11 If the sound is emitted along the x -direction, that is in the same direction as the relative motion, v ( s ) = (v ( s ) , 0 , 0 ) and one obtains: v ( s ) x = v ( s ) V , v ( s ) y = 0 , v ( s ) z = 0 , (1.26) so that the velocity of the sound measured in S along the x -direction will be lower than in S , v ( s ) x = v ( s ) V < v ( s ) . Vice versa, if the sound is emitted in the negative x -direction, that is v s = ( v ( s ) , 0 , 0 ) , then the modulus of the velocity measured in S will be greater than in S ; indeed v ( s ) x = − v ( s ) V , v ( s ) y = 0 , v ( s ) z = 0 , (1.27) implying | v ( s ) x | = | v ( s ) + V | > v ( s ) . Let us now consider a sound wave propagating in S along a direction perpendicular to that of the relative motion, say along the negative y -axis, v ( s ) = ( 0 , v ( s ) , 0 ) (see Fig. 1.3 ). In S the velocity of the sound signal is: v ( s ) = v ( s ) V = ( V , v ( s ) , 0 ) (1.28) It follows that for the observer in S , the sound wave will propagate along a direction forming an angle α with respect to the y -axis given by (Fig. 1.4 ): tan α = V v ( s ) , (1.29) while the modulus of the velocity v ( s ) ≡ | v ( s ) | turns out to be: v ( s ) = v 2 ( s ) + V 2 > v ( s ) . (1.30) Note that, if V v ( s )
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