From Special Relativity to Feynman Diagrams.pdf

We readily obtain y β v y β v β v y β 2 v y β 2

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) we readily obtain: y = β( V ) y = β( V )β( V ) y = β 2 ( V ) y β 2 ( V ) = 1 , which implies β( V ) = ± 1. On the other hand, since we have orientated y and y in the same direction, we must have β( V ) 1. By the same token we also find z = z . Thus the first three equations of the transformations ( 1.33 ) take the simple form: x = α( V )( x V t ), (1.40) y = y , (1.41) z = z . (1.42) Let us now consider the fourth equation involving the time variable t . Solving the first of ( 1.40 ) with respect to t we find: t = 1 V x x α( V ) , (1.43) Using the same argument which led to ( 1.37 ), if we consider S in motion with velocity V with respect to S , the equation obtained from ( 1.43 ) by replacing t with t , x with x and V with V must also be true: t = − 1 V x x α( V ) = − 1 V α( V )( x V t ) x α( V ) = α( V ) t + 1 V 1 α( V ) α( V ) x . (1.44)

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1.3 Lorentz Transformations 19 Fig. 1.8 Light signal as seen by S and S We may then rewrite the transformation ( 1.44 ) as follows: t = α( V ) t + δ( V ) x , (1.45) where we have set δ( V ) = 1 V 1 α( V ) α( V ) . (1.46) By simple considerations we have reduced the problem of determining all the coef- ficients in ( 1.33 ), to that of computing a single function α( V ) . This coefficient will be now determined by implementing the principle of con- stancy and isotropy of the speed of light. Let us suppose that at t = t = 0, when O O , a light (or electromagnetic wave) source emits a signal isotropically, see Fig. 1.8 . According to this principle, the signal propagates isotropically with the same constant speed c for both observers S and S . Thus with respect to the two frames the wave front of the electromagnetic signal will be described by spheres of radii r = ct and r = ct respectively. The equations for the wave front of the spherical wave are thus given by: x 2 + y 2 + z 2 c 2 t 2 = 0 , (1.47) for the observer S , and x 2 + y 2 + z 2 c 2 t 2 = 0 , (1.48) for the observer S . Since the four coordinates ( x , y , z , t ) and ( x , y , z , t ) refer to the same physical events, that is the locus of points reached by the signal at a fixed time, they must hold simultaneously. We must then have: x 2 + y 2 + z 2 c 2 t 2 = κ x 2 + y 2 + z 2 c 2 t 2 , (1.49) where κ is a constant. If we now substitute the expression of x , y , z , t in terms of x, y, z, t as given by ( 1.40 ), ( 1.41 ), ( 1.42 ) and ( 1.45 ), in ( 1.49 ), we obtain:
20 1 Special Relativity x 2 + y 2 + z 2 c 2 t 2 = κ α 2 ( x V t ) 2 + y 2 + z 2 c 2 t + δ x ) 2 , (1.50) and this relation must be an identity in ( x , y , z , t ) . Conparing the coefficients of z and y on both sides, we immediately find κ = 1 . Next, equating the coefficients of t 2 , one finds: α 2 ( V ) V 2 = c 2 1 α 2 ( V ) α( V ) = ± 1 1 V 2 c 2 . Since at t = t = 0, x and x have the same orientation, we conclude that: α( V ) = α( V ) = 1 1 V 2 c 2 . (1.51) One can easily verify that, with the above value of α( V ) , also the coefficients of x 2 and xt are equal. The transformation laws ( 1.40 ), ( 1.41 ), ( 1.42 ) and ( 1.44 ) now take the following final form: x = γ ( V )( x V t ), (1.52) y = y , (1.53) z = z , (1.54) t = γ ( V ) t V c 2 x , (1.55) where γ ( V ) 1 1 V 2 c 2 > 1 . (1.56) Equations ( 1.52 ) are the Lorentz transformations . They represent the correct trans- formation laws connecting two inertial frame, which allow to extend the principle of relativity to electromagnetism.

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