From Special Relativity to Feynman Diagrams.pdf

8 relativity of simultaneity another consequence of

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8 Relativity of Simultaneity : Another consequence of the transformation law for time is the relativity of simultaneity. Indeed, let us consider again the inertial frames S and S , and suppose that A and B are two events which are simultaneous in S , namely t A = t B ( t = 0). When observed by S the two events will be separated by a time interval t = γ ( V ) V c 2 x = 0 . (1.69) This implies that two events which are simultaneous in S, but occur at different points, are not simultaneous with respect to a frame S in motion with respect to S. 8 Note that the time dilation is a relative effect , that is if we have a clock at rest in S , from Eq. (1.60) it follows that t = γ ( V ) t , that is time in S is dilated with respect to S . The same observation applies to the length contraction to be discussed in the following.
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1.4 Kinematic Consequences of the Lorentz Transformations 25 Length Contraction : The fact that simultaneity between events is not an absolute concept implies that the distance between two points depends on the particular ref- erence frame in which it is measured. Let us consider the situation described in Sect.1.1 , in which a rod is placed at rest along the x -axis of a frame S moving with respect to S at velocity V = ( V , 0 , 0 ) . Let the endpoints A and B of the rod be located in the points x B and x A . We can repeat one by one the arguments given in Sect.1.1 , from formula ( 1.10 ) to formula ( 1.12 ), using now the Lorentz transformations instead of the Galilean ones. In S the length is defined as: x L = x B x A , while the same length is measured in S as the difference between the coordinates of the endpoints taken at the same time, that is simultaneously : x = L = x B ( t B ) x A ( t A ) x B ( t ) x A ( t ), where we have set t = t B = t A . From ( 1.52 ) we then find: L = x = γ ( V )( x V t ) = γ ( V ) x = γ ( V ) L , that is: L = γ ( V ) 1 L . Since γ ( V ) 1 = ( 1 V 2 c 2 ) 1 / 2 < 1, the observer S in motion with respect to the rod will measure a length L contracted by the factor γ ( V ) 1 with respect to L , which is the length of the object at rest. The conclusion is that: The length L of an object in motion is contracted with respect its length L at rest: L = 1 V 2 c 2 L < L . (1.70) We note, instead, that lengths along the directions perpendicular to that of the relative motion are not affected by the motion itself y = y , z = z . This in particular implies that a volume V = x y z transforms like the length of a rod parallel to the motion, namely: V = 1 γ V < V . (1.71) This in turn has the important consequence that the concept of rigid body , so useful un classical mechanics, looses its meaning in the framework of a relativistic theory.
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26 1 Special Relativity 1.5 Proper Time and Space–Time Diagrams We have learned, in the previous section, that both time and spatial intervals depend on the reference frame, that is, space and time are not absolute as they were in Newtonian mechanics, rather their transformation laws are combined in such a way that only the velocity of light is absolute . Note that, in the standard configuration , the transformation properties ( 1.57 ), ( 1.60 ) of x and t under Lorentz transformations
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