Loreptz Transformations: (Purpose is to be Considtent with the Special Theory of Relativity) (x — vt) P (event) @2004 Thomson - Brooks/Cole x'—x—vt
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Frame S Frame S' Those also satisfy (2). Obviously this will be the case when the relation - ctl) -cl) For negative direction 2
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c/' acl bit How to find out a and b? For the origin of S' we have permanently x' = O, and hence according to the first of the equations (5) bc If we call v the velocity with which the origin of S' is moving relative to S, we then have bc
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Furtherrllore, the principle of relativity teaches us that, as judged from S, the length of a unit measuring-rod which is at rest with reference to S' must be exactly the same as the length, as judged from S', of a unit measuring-rod which is at rest relative to S. In order to see how the points of the x'-axis appear as viewed from S, we only require to take a snapshot" of S' from S; this means that we have to insert a particular value of t (time of S), e.g. t = O. For this value of t we then obtain from the first of the equations (5) x' = ax. Two points of the x'-axis which are separated by the distance x'=l when measured in the S' system are thus separated in our instantaneous photograph by the distance 1
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But if the snapshot be taken from S'(t' = O), and if we from the equations (5), taking into account the eliminat expressio (6), we obtain From this we conclude that two points on the x-axis and separated by the distance 1 (relative to S) will be represented on our snapshot by the distance v x in (7) must be equal tovx' in (7a), so that we obtain (70). c2
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The equations (6) and (7b) determine the constants a and b. By insergng the values of these constants in (5), we obtain the first and the fourth of the eauations given in Thus we have obtained the Lorentz transformation for events on the x-axis. It satisfies the condition
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The extension of this result, to include events which take place ou Ride the x-axis, is obtained by retaining equations (8) and s pplementing them by the relations In this way we satisfy the postulate of the constancy of the velocity of light in vacuo for rays of light of arbitrary direction, both for the system S and for the system S' 1 V C
Discussion
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