# Theory of relativity

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12-Jun-2015Category

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- 1. Theory of Relativity Albert Einstein Physics 100Chapt 18

2. watching a light flash go byvc2kThe man on earth sees c = (& agrees with Maxwell) 3. watching a light flash go by vcIf the man on the rocket sees c-v,he disagrees with Maxwell 4. Do Maxwells Eqns only work inone reference frame? If so, this would be the rest frameof the luminiferous Aether. 5. If so, the speed of light shouldchange throughout the year upstream,downstream, light moves light moves slowerfaster Aether wind 6. Michelson-MorleyNo aether wind detected: 1907 Nobel Prize 7. Einsteins hypotheses:1. The laws of nature are equally validin every inertial reference frame.IncludingMaxwells eqns2. The speed of light in empty space is same for all inertial observers, regard- less of their velocity or the velocity of the source of light. 8. All observers see light flashes goby them with the same speedvNo matter how fastthe guy on the rocketis moving!!cBoth guys see the light flashtravel with velocity = c 9. Even when the light flash istraveling in an opposite direction vc Both guys see the light flash travel past with velocity = c 10. Gunfight viewed by observer at rest He sees both shotsfired simultaneously Bang !Bang! 11. Viewed by a moving observer 12. Viewed by a moving observerHe sees cowboy shoot 1st & cowgirl shoot later Bang !Bang! 13. Viewed by an observer in the opposite direction 14. Viewed by a moving observerHe sees cowgirl shoot 1st & cowboy shoot later BangBang !! 15. Time depends of state of motion of the observer!!Events that occur simultaneously according to one observer can occur at different times for other observers 16. Light clock 17. Seen from the ground 18. Eventsy(x2,t2)(x1,t1)x x x1x2xt 19. Prior to Einstein, everyone agreedthe distance between events dependsSame events, different observers upon the observer, but not the time.y yy (x2,t2)(x1,t1)xx(x1,t1) (x2,t2) tt x1 x1 distx2 xxx1 x2 x tdist 20. Time is the 4th dimensionEinstein discovered that there is noabsolute time, it too depends uponthe state of motion of the observer Einstein Newton Space-Timecompletely Space different &2 different aspects conceptsTimeof the same thing 21. How are the times seenby 2 differentobservers related?We can figure this out withsimple HS-level math ( + a little effort) 22. Catch ball on a rocket ship Event 2: girl catches the ballw v= =4m/stw=4mt=1sEvent 1: boy throws the ball 23. Seen from earth V0=3m/s V0=3m/s Location of the 2events is different= 5m Elapsed time ism2) (4the same m )2 +w=4m (3 The ball appears d= v0t=3m to travel fasterd t=1sv= = 5m/st 24. Flash a light on a rocket shipEvent 2: light flash reaches the girl wc= t0wt0Event 1: boy flashes the light 25. Seen from earthV VSpeed has toBe the same 22+ w)Dist is longer ( vtwd=Time must bevtlongerd= (vt) +w 2 2 c= t=?tt 26. How is t related to t0?t= time on Earth clock t0 = time on moving clockwc =(vt)2+w2c =t0t ct = (vt)2+w2ct0 = w (ct)2 = (vt)2+w2 (ct)2 = (vt)2+(ct0)2 (ct)2-(vt)2= (ct0)2 (c2-v2)t2= c2t02c2 1 t = 22t 02 t2 =t2 2c v 21 v /c 0 21 t=t01 v2/c2 t = t0this is called 27. Properties of = 1 1 v2/c2Suppose v = 0.01c(i.e. 1% of c) 11 = 1 (0.01c)2/c2=1 (0.01)2c2/c2 11 1 = 1 (0.01)2 = = 1 0.0001 0.9999 = 1.00005 28. Properties 1of = 1 v(contd) 2/c2Suppose v = 0.1c(i.e. 10% of c) 1 1 = 1 (0.1c)2/c2 =1 (0.1)2c2/c2 11 1 = 1 (0.1)2 = =1 0.01 0.99 = 1.005 29. Lets make a chart v =1/(1-v2/c2)0.01 c1.00005 0.1 c 1.005 30. Other values of 1 = 1 v2/c2Suppose v = 0.5c(i.e. 50% of c) 1 1 = 1 (0.5c)2/c2 =1 (0.5)2c2/c2 1 11 = 1 (0.5)2 = =1 (0.25) 0.75 = 1.15 31. Enter into chart v =1/(1-v2/c2)0.01 c 1.00005 0.1 c1.005 0.5c1.15 32. Other values of 1 = 1 v2/c2Suppose v = 0.6c(i.e. 60% of c)11 =1 (0.6c)2/c2=1 (0.6)2c2/c2 11 1 = 1 (0.6)2= = 1 (0.36)0.64 = 1.25 33. Back to the chart v =1/(1-v2/c2)0.01 c 1.00005 0.1 c1.005 0.5c1.15 0.6c1.25 34. Other values of 1 = 1 v2/c2Suppose v = 0.8c(i.e. 80% of c) 1 1 = 1 (0.8c)2/c2 =1 (0.8)2c2/c2 1 11 = 1 (0.8)2 = =1 (0.64) 0.36 = 1.67 35. Enter into the chart v =1/(1-v2/c2)0.01 c1.00005 0.1 c 1.005 0.5c 1.15 0.6c 1.25 0.8c 1.67 36. Other values of 1 = 1 v2/c2Suppose v = 0.9c(i.e.90% of c) 1 1 = 1 (0.9c)2/c2 =1 (0.9)2c2/c2 11 1 = 1 (0.9)2 = =1 0.81 0.19 = 2.29 37. update chart v =1/(1-v2/c2)0.01 c 1.00005 0.1 c1.005 0.5c1.15 0.6c1.25 0.8c1.67 0.9c2.29 38. Other values of 1 = 1 v2/c2Suppose v = 0.99c(i.e.99% of c)1 1 =1 (0.99c)2/c2 =1 (0.99)2c2/c2 1 11 = 1 (0.99)2 == 1 0.980.02 = 7.07 39. Enter into chart v =1/(1-v2/c2)0.01 c 1.00005 0.1 c1.005 0.5c1.15 0.6c1.25 0.8c1.67 0.9c2.290.99c7.07 40. Other values of 1 = 1 v2/c2Suppose v = c 1 1 = 1 (c)2/c2 = 1 c2/c21 1 1 ===1 120 0 = Infinity!!! 41. update chart v =1/(1-v2/c2)0.01 c 1.00005 0.1 c1.005 0.5c1.15 0.6c1.25 0.8c1.67 0.9c2.290.99c7.071.00c 42. Other values of 1 = 1 v2/c2Suppose v = 1.1c 1 1 = 1 (1.1c)2/c2 =1 (1.1)2c2/c2 1 11 = 1 (1.1)2 ==1-1.21 -0.21 =???Imaginary number!!! 43. Complete the chartv =1/(1-v2/c2) 0.01 c 1.000050.1 c1.0050.5c1.150.6c1.250.8c1.670.9c2.29 0.99c7.07 1.00c Larger than c Imaginary number 44. Plot results: Never-never land1 =1 v2/c2 x x x x x v=c 45. Moving clocks run slowervt0 t= 1 t01 v2/c2tt = t0 >1 t > t0 46. Length contractionv L0time=tL0 = vtte r! orShman on Time = t0 =t/rocket Length = vt0 =vt/ =L0/ 47. Moving objects appear shorterLength measured when object is at rest L = L0/ >1 L < L0 V=0.9999cV=0.86cV=0.1cV=0.99c 48. Length contraction 49. mass: change in v F=m0a = m0timet0 a time=t0m0 Ft0 change in v =m0Ft0m0 =change in v mass Ft Ft0m= = = m0 increases!! change in v change in vm = m0 t=t0by a factor 50. Relativistic mass increasem0 = mass of an object when it is at rest rest massmass of a movingobject increasesas vc, m m = m0 as an object moves faster, it gets harder & harderto accelerateby the factor v=c 51. summary Moving clocks run slow o f o rc t Moving objects appear shorter f a a y Moving objects mass increasesB 52. Plot results: Never-never land1 =1 v2/c2 x x x x x v=c 53. -Twin paradox centauri rs y eaht Twin brother lig & sister 4.3 She will travel to -centauri (a near-by star on a specialHe will stay home rocket ship v = 0.9c& study Phys 100 54. Light yeardistance light travels in 1 year dist = v x time = c yr1cyr = 3x108m/s x 3.2x107 s = 9.6 x 1015 m We will just use cyr units & not worry about meters 55. Time on the boys clock rcy0.9c =4. 3v= d00.9cv=According to the boy& his clock on Earth: d0 4.3 cyr = 4.8 yrstout = = 0.9cvd04.3 cyr = 4.8 yrs tback = = 0.9cvttotal = tout+tback = 9.6yrs 56. What does the boy see on her clock? yr0.9c4. 3cv= d=0.9cv=According to the boyher clock runs slower tout 4.8 yrs t =out= 2.3 = 2.1 yrs tback 4.8 yrtback = = = 2.1 yrs2.3ttotal = tout+tback = 4.2yrs 57. So, according to the boy:yr 0.9c4. 3c v= d= 0.9c v=his clock her clockout: 4.8yrs2.1yrs back: 4.8yrs2.1yrsges She a total: 9.6yrs 4.2yrs less 58. But, according to the girl, the boys clock ismoving &, so, it must be 0.9c running slowerv= According to her, theboys clock on Earth says:tout 2.1 yrstout = = = 0.9 yrs 2.3tback 2.1 yrs = 0.9 yrs tback = = 2.3.9c v=0ttotal = tout+tback = 1.8yrs 59. Her clock advances 4.2 yrs& she sees his clock advanceonly 1.8 yrs,contradict ion??AShe should think he has aged less than her!! 60. Events in the boys life:As seen by him As seen by her She leaves4.8 yrs 0.9 yrsShe arrives & starts turnshort time ????Finishes turn& heads home4.8 yrs0.9 yrs She returns9.6+ yrs 1.8 + ??? yrs 61. turning around as seen by her According to her, these 2 events occur very,veryfar apart from each otherHe sees herHe sees herfinish turningstart to turnTime interval between 2 events dependson the state of motion of the observer 62. Gunfight viewed by observer at rest He sees both shotsfired simultaneously Bang !Bang! 63. Viewed by a moving observer 64. Viewed by a moving observerHe sees cowboy shoot 1st & cowgirl shoot later Bang !Bang! 65. In fact, ???? = 7.8+ years as seen by him as seen by her She leaves4.8 yrs0.9 yrsShe arrives & starts turn short time 7.8+ yrs???Finishes turn& heads home4.8 yrs0.9 yrsShe returns9.6+ yrs 1.8 + ???yrs9.6+ yrs 66. No paradox: both twins agreeThe twin that turned aroundis younger 67. Ladder & Barn Door paradoxStan & Ollie puzzle over howto get a 2m long ladder thrua 1m wide barn door ??? 1m 2mladder 68. Ollie remembers Phys 100 & thetheory of relativityStan, pick upthe ladder &run very fast 1m 2mtreeladder 69. View from Ollies ref. frame1m Push, Stan!2m/ V=0.9cOllieStan(=2.3) 70. View from Stans ref. frame But it1m/doesnt fit,Ollie!!V=0.9c(=2.3)2mOllie Stan 71. If Stan pushes both ends of theladder simultaneously, Ollie sees thetwo ends move at different times: Too late1m TooStan! soonnkStan!clu clankV=0.9c Ollie StanStan(=2.3) 72. Fermilab proton acceleratorV=0.9999995c =10002km 73. Stanford electron acceleratorv=0.99999999995 c3km=100,000 74. statusEinsteins theory of special relativity hasbeen carefully tested in many very preciseexperiments and found to be valid.Time is truly the 4th dimension of space &time. 75. test=29.3

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