43 the twin paradox ua twin making a round trip to a

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Unformatted text preview: is in the rest frame of the object defining the events. o  for example a clock Cck in its rest frame o  or a parCcle which is created then decays u༇ The Cme in the rest frame is called the proper Cme. o  The proper Cme can someCmes be used like the universal Cme was before relaCvity. 30 Calculate Time DilaCon u༇  A clock, at the origin in its rest frame K Ccks twice. o  first Cck at t1 and x1=0. o  second Cck at t2 and x2=0. o  Cme between Ccks is t2- t1 u༇  In frame K’ moving with velocity v, o  First Cck is at t1’=γ(t1- βx1)=γt1 o  Second Cck is at t2’=γ(t2- βx2)=γt2 o  Cme between Ccks is γ (t2- t1) u༇  Time is longer by a factor of γ in the moving system. u༇  What about a clock at rest in the K frame? What is Cck Cme in K’ frame? 31 (Fitzgerald) Length ContracCon u༇  Consider a meter sCck at rest in the K’ system. o  the ends are at x1’=0 and x2’=1 at any t’. o  the proper length is 1 meter u༇  Use the same LT to write x2’ and x1’ in terms of the unprimed quanCCes. o  x1’=γ(x1- βt1) o  x2’=γ(x2- βt2) u༇  We must measure the length in K at the same Cme at both ends, pick t1=t2=0. o  x2’- x1’ =γ(x2- x1) o  (x2- x1)=(x2’- x1’)/ γ o  Length ContracCon by factor of γ Proper length is the longest. u༇  Lengths perpendicular to boost direcCon are unchanged. u༇  32 Try Meter SCck in K System (to do length contracCon the hard way) u༇  We can use the normal LT, but we must measure the length in the K’ system with t2’=t1’ o  let t1=0 u༇  This gives the same answer, but is a liole more work. 33 Physical Measurements u༇ The Cme dilaCon and length contracCon are real effects. o  for example muons with a lifeCme of about a microsecond that are produced in the upper atmosphere travel to the earth’s surface because their lifeCme in a moving frame is much larger. u༇ But many physical effects should be measured by an “observer in an inerCal frame”. o  not at the same Cme but rather when the light from an event reaches the observer. o  Doppler effect is an example. 34 Velocity AddiCon u༇  Consider an object moving with a velocity u in an arbitrary direcCon. o  but lets say the x component of u is...
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