From Special Relativity to Feynman Diagrams.pdf

Are reminiscent of the way in which the components of

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are reminiscent of the way in which the components of a vector on the plane transform under a rotation of the corresponding coordinate axes. It is then natural to describe the effect of a change in the inertial frame as a kind of “rotation” of the space and time axes x, t . Considering also the other two coordinates y, z , which do not transform if the two inertial frames are in the standard configuration , one may regard a Lorentz transformationasakindof“rotation”inafour-dimensionalspace,thefourthdirection being spanned by the time variable. A more precise definition of this kind of rotation will be given in Chap. 4 ; for the time being we call this four-dimensional space of points space–time or Minkowski space . Every point in space–time defines an event which occurs at a point in space of coordinates x, y, z , at a time t , and is labeled by the four coordinates t, x, y, z . In three-dimensional Euclidean space R 3 a rotation of the coordinate axes im- ply a transformation in the components x , y , z of the relative position vec- tor between two points, which however does not affect their squared distance | x | 2 = x 2 + y 2 + z 2 . In analogy to ordinary rotations in Euclidean space, a Lorentz transformation preserves a generalized “squared distance” between events in Minkowski space which generalizes the notion of distance between two points in space. To show this let us recall that, in determining the Lorentz transformations, we required the equality: x 2 + y 2 + z 2 c 2 t 2 = x 2 + y 2 + z 2 c 2 t 2 . (1.72) the left- and right-hand sides of this equation being separately zero, in accordance to the constancy of the speed of light in every inertial frame. The two events in A and B , in that case, were the emission of a spherical light wave in O = O at the time t = t = 0 and the passage of the spherical wave-front through a generic point of coordinates x, y, z, t and x , y , z , t , respectively. Consider now a light wave which is emitted in a generic point, instead of the origin, at a generic time, and let the spherical wave propagate for a time t in S . Equation 1.72 can be written as: | x | 2 c 2 t 2 x 2 + y 2 + z 2 c 2 t 2 = x 2 + y 2 + z 2 c 2 t 2 = | x | 2 c 2 t 2 . (1.73) It is now a simple exercise to verify that equality ( 1.73 ) holds even if the two events do not refer to the propagation of a light ray. It is sufficient to express the primed quanti- ties on the right-hand side in terms of the unprimed ones by using the Lorentz trans- formations ( 1.57 )–( 1.60 ). One then finds that ( 1.73 ) is identically satisfied. Defining
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1.5 Proper Time and Space–Time Diagrams 27 the four-dimensional distance , also called proper distance between two events, as 2 = | x | 2 c 2 t 2 , (1.74) Equation ( 1.73 ) then implies that: The proper distance between two events in space–time is invariant under Lorentz transformations. In particular, if there exists a frame where the two events are simul- taneous , t = 0, the proper distance reduces in that frame to the ordinary distance = | x | .
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