lecture_3_07

# lecture_3_07 - Special Relativity Einstein(1905 required...

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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 1 Special Relativity Einstein (1905) required Laws of Physics are independent of inertial frame or one can not detect absolute motion Speed of light (c) is independent of the motion of the source or c is velocity of EM wave in any inertial frame How can these 2 innocent requirements be compatible? Construct a light clock

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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 2 Light clock - accurate time keeper Bouncing light beam mirror S´: rest frame of clock Time for 1 tick is time for one round trip of light [ or 2 × distance/speed] ∆t´ = 2L 0 /c´ for 1 tick letting c´ be speed in S´ L 0 source & detector
05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 3 Light clock on fast moving train  seen  from platform L 0 0 1 2 3 4 5 6 7 8 time units (unit = t/8) Train speed × time u t Light clock moves right at train speed u while light pulse travels up & down speed, covers more distance than 2 × L 0 , c t >2 L 0 with c the speed in S 0| t/8| t/4|3 t/8| t/2|

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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 4 Moving light clock S: moving clock frame of reference - one round trip is one time unit. ∆t=2 L 0 /[c√(1-u 2 /c 2 )] from PythagorianTheorem (see next slide for algebra) Compare with synchronized clocks in stationary frame. L 0 u =Speed of clock More than L 0 Elapsed time c t/2 u t/2
05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 5 Time in S frame c t/2 u t/2 L 0 In time t/2 upper mirror has moved distance u t/2 and light beam has met the mirror, so follows red, slanted path. Pythagorian Theorem: c Dt 2 2 = L 0 2 + u Dt 2 2 so 1 4 c 2 Dt 2 = L 0 2 + 1 4 u 2 Dt 2 or c 2 Dt 2 = 4L 0 2 + u 2 Dt 2 Collecting Dt terms: 4L 0 2 = c 2 Dt 2 - u 2 Dt 2 = (c 2 - u 2 )Dt 2 = c 2 1- u 2 c 2 Dt 2 Take square root: 2L 0 = c 1- u 2 c 2 Dt Hence Dt = 2L 0 c 1- u 2 c 2 is time elapsed in S frame

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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 6 Compare 2 views of light clock For sound wave or water wave  c (1-u 2 /c 2 )=c´ (1-u 2 /c 2 ) is ≤ 1 so c ≥ c ´ (S´is moving with current, S is shore observer) and  t= t  of course. But Einstein says
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lecture_3_07 - Special Relativity Einstein(1905 required...

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