From Special Relativity to Feynman Diagrams.pdf

On time due to the relative motion of s and s x a t a

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on time, due to the relative motion of S and S : x A ( t A ) = ( x A ( t A ), y A ( t A ), z A ( t A )) and x B ( t B ) = ( x B ( t B ), y B ( t B ), z B ( t B )) . Their expression in terms of the coordinates of A and B in S are given by ( 1.4 ): x A ( t A ) = x A + V t A , x B ( t B ) = x B + V t B , y A ( t A ) = 0 , y B ( t B ) = 0 , z A ( t A ) = 0 , z B ( t B ) = 0 . (1.11) In order to compute the length L of the rod in S we must consider the coordinates of the endpoints A and B at the same instant , since evaluating them at different times would lead to a meaningless result. Setting t = t A = t B we find:
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6 1 Special Relativity x B x A = ( x B ( t B ) V t B ) ( x A ( t A ) V t A ) = x B ( t ) x A ( t ). (1.12) Equation ( 1.12 ) then implies: L = L , (1.13) that is, the length of the rod is the same for both observers. Note that, in defining the measure of the length L of the moving rod, we have used the notion of simultaneity of two events, t B = t A . This concept is, however, independent of the reference frame since, having assumed from the beginning the equality of time durations, that is t = t in different frames, simultaneity in S ( t = 0 ) implies simultaneity in S ( t = 0 ) for any two inertial frames S and S . We have thus proven that invariance of the lengths ( absolute space ) is a consequence of invariance of the time intervals ( absolute time ). In the previous discussion we have considered the rod lying along the x -axis, which is the direction of the relative motion. It is obvious that the distances along the y -or z -axes are also invariant since y = y and z = z . This means that the vector describing the relative position of any two points in space is invariant under Galilean transformations. More specifically, if A and B are two points at rest in S (not necessarily along the x -axis) with position vectors x A , x B and relative position vector x x B x A , and if x A ( t A ) , x B ( t B ) are the position vectors of the two points relative to S at different times, we define the relative position vector in S as the difference between the position vectors taken at the same instant t : x ( t ) x B ( t ) x A ( t ) = ( x B + V t ) ( x A + V t ) = x B x A = x . (1.14) We conclude that not only the spatial distance between A and B , but also the direction from A to B , i.e. thedirectionandorientationof therelativepositionvector, is invariant under Galilean transformations. So far we have examined the change of inertial frames due to a relative motion with a constant velocity V . The change of an inertial frame due to a rotation or to a rigid translation of the coordinate axes are in a sense trivial. They correspond to the congruence transformations of the Euclidean geometry leaving invariant the space relations between figures and objects. They have the form x = Rx + b , where R denotes a 3 × 3 matrix which implements a generic rotation or reflection. Another trivial transformation is the change of the time origin, or time translation namely t = t + β.
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