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Lecture 1

PHYS 242 Lecture 1: PHYS242_Lecture_10-12

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PHYS 242
Wolfgang Rau

31 ENPHPHYS 242 Fall 2014 L 10 1.3.4 Velocity Transformation, FourVelocity After we have derived the relativistically correct transformation law for the space and time coordinates we go to the next step and discuss the velocity transformation. This means we consider an object moving with a certain velocity in one frame and ask: what is the velocity an observer in another reference frame would measure for this object? Since is usually used for the relative velocity of two frames, and in an attempt to avoid too many subscripts it is common to use (or ) for the velocity of an object observed within a given frame. The velocity is defined as change of position with time: We will separate the three spatial components starting with the component paallel to (as usual we assume that is parallel to the axis) and apply what we learned in the previous section: The first step is the definition of the velocity; in the second step we replace and according to the Lorentz transformation. In the third step we pull the out and use . Then we cancel and divide numerator and denominator by . In the last step we use again the definition of the velocity (now in the primed frame). In classical physics we would just add the velocities (Galilean velocity transformation). Here we also add the velocities (note the numerator in the above equation), but we have to apply a correction factor which is similar in structure to the square of the relativistic factor, except that instead of it contains the term . If we apply the same procedure to the direction we find: In all transformations that we have seen so far (classical transformation, but also the Lorentz transformation for the space and time variables in Special Relativity), the transverse components were not affected. This here is the first time that this changes. The situation is different for the velocity since the velocity involves the time, and the time changes when going from one frame to another. The complete velocity transformation for along the axis is given by: ; ; (26) (note that in all three equations we have in the denominator!) W. Rau
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