SRnotes - PHY3101 Special Relativity Notes Harrison B....

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PHY3101 — Special Relativity Notes Harrison B. Prosper September, 2009 Notation u — speed of an object relative to a frame of reference v — relative speed between two frames of reference c — speed of light in vacuum β = v/c t,x,y,z — time and space coordinates in a frame of reference, S , with respect to which the observer is stationary t 0 ,x 0 ,y 0 ,x 0 — time and space coordinates in a frame of reference, S 0 , moving at speed v relative to frame S . I. POSTULATES AND IMMEDIATE CONSEQUENCES 1. Principle of Relativity : The laws of physics are the same in all inertial frames of reference. (Inertial: non accelerating.) 2. Constancy of the Speed of Light : The speed of light in vacuum is independent of the motion of the light source. These two postulates imply that the speed of light is also independent of the motion of the observer. Therefore, one could replace postulate 2 with something like: The speed of light in vacuum is always the same. A. Time dilation and length contraction For simplicity, let’s assume that the two frames of reference S and S 0 share one event, labeled O in Fig. 1, that serves as a common origin; that is, the values t = 0 ,x = 0 in frame 1
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D O E C x ct x ct ! A B FIG. 1: Shown are two inertial reference frames S and S 0 with a relative velocity between them of magnitude v . In the S frame events are labeled by t and x , while in the S 0 the same events are labeled by t 0 and x 0 . (Reference frame S could be, for example, the frame of the Earth, while S 0 could be the reference frame of a starship.) We have multiplied the time coordinates by the speed of light c so that the units for time and space are the same. A few points to note . If the elapsed time along worldline OD is ct (in space units) then the elapsed time along the wordline OC is ct 0 = ct/γ . The events O and A are simultaneous in frame S , as are events D and C . However, it is events such as O and B and E and C that are simultaneous in frame S 0 . Since the time dilation law is true in both reference frames, the elapsed time along OE is the elapsed time along OC reduced by the factor γ ; that is, the elapsed time along OE is ct 0 = ct/γ 2 . This implies that the elapsed time along ED , that is, the temporal discrepancy (which is the same as that along AB ) is c Δ t = ct - ct/γ 2 = vx/c . The parallel worldlines OD and AC can be thought of as the ends of a rod that is stationary in the S frame. In the frame S 0 , however, the same rod is moving relative to that frame. Consequently, in that frame the length of the rod is not OA , but rather OB . If we let OA equal x then the length of OB is given by the length contraction law x/γ . 2
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S and t 0 = 0 ,x 0 = 0 in frame S 0 , label the same event O . The postulates imply the following laws of physics: t 0 = t/γ, (1.1) x 0 = x/γ, (1.2) where γ = 1 / 1 - β 2 and β = v/c . According to postulate 1, these laws hold in every inertial frame. Consequently, if you are moving relative to me, from my point of view your
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SRnotes - PHY3101 Special Relativity Notes Harrison B....

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