Theory of relativity

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Theory of relativity

  1. 1. Theory of Relativity Albert Einstein Physics 100 Chapt 18
  2. 2. watching a light flash go by v c 2kThe man on earth sees c = √ κ (& agrees with Maxwell)
  3. 3. watching a light flash go by v cIf the man on the rocket sees c-v, he disagrees with Maxwell
  4. 4. Do Maxwell’s Eqns only work in one reference frame? If so, this would be the rest frame of the luminiferous Aether.
  5. 5. If so, the speed of light should change throughout the year upstream,downstream, light moves light moves slower faster “Aether wind”
  6. 6. Michelson-MorleyNo aether wind detected: 1907 Nobel Prize
  7. 7. Einstein’s hypotheses: 1. The laws of nature are equally valid in every inertial reference frame. Including Maxwell’s 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. 8. All observers see light flashes go by them with the same speed vNo matter how fastthe guy on the rocketis moving!! c Both guys see the light flash travel with velocity = c
  9. 9. Even when the light flash istraveling in an opposite direction v c Both guys see the light flash travel past with velocity = c
  10. 10. Gunfight viewed by observer at rest He sees both shots fired simultaneously Bang ! Bang !
  11. 11. Viewed by a moving observer
  12. 12. Viewed by a moving observer He sees cowboy shoot 1st & cowgirl shoot later Bang ! Bang !
  13. 13. Viewed by an observer in the opposite direction
  14. 14. Viewed by a moving observer He sees cowgirl shoot 1st & cowboy shoot later Bang Bang ! !
  15. 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. 16. Light clock
  17. 17. Seen from the ground
  18. 18. Eventsy (x2,t2) (x1,t1) x x x1 x2 x t
  19. 19. Prior to Einstein, everyone agreed the distance between events depends Same events, different observers upon the observer, but not the time. y’ y’y (x2,t2) (x1,t1) x x (x1’,t1’) (x2’,t2’) t’ t’ x1’ x1’ dist’ x2’ x’ x’ x1 x2 x t dist
  20. 20. Time is the 4th dimension Einstein discovered that there is no “absolute” time, it too depends upon the state of motion of the observer Einstein Newton Space-Timecompletely Space different & 2 different aspects concepts Time of the same thing
  21. 21. How are the times seen by 2 different observers related? We can figure this out with simple HS-level math ( + a little effort)
  22. 22. Catch ball on a rocket ship Event 2: girl catches the ball w v= =4m/s t w=4m t=1s Event 1: boy throws the ball
  23. 23. Seen from earth V0=3m/s V0=3m/s Location of the 2events is different = 5m Elapsed time is m 2 ) (4 the same m )2 + w=4m √ (3 The ball appears d= v0t=3m to travel faster d t=1s v= = 5m/s t
  24. 24. Flash a light on a rocket ship Event 2: light flash reaches the girl w c= t0 w t0 Event 1: boy flashes the light
  25. 25. Seen from earth V VSpeed has toBe the same 2 2+ w )Dist is longer √( vt w d=Time must be vt longer d =√ (vt) +w 2 2 c= t=? t t
  26. 26. How is t related to t0?t= time on Earth clock t0 = time on moving clock w c =√ (vt)2+w2 c = t0 t ct = √ (vt)2+w2 ct0 = w (ct)2 = (vt)2+w2 (ct)2 = (vt)2+(ct0)2  (ct)2-(vt)2= (ct0)2  (c2-v2)t2= c2t02 c2 1  t = 2 2 t 02  t2 = t2 2 c – v 2 1 – v /c 0 2 1  t= t0 √1 – v2/c2  t = γ t0this is called γ
  27. 27. Properties of γ = 1 √1 – v2/c2Suppose v = 0.01c (i.e. 1% of c) 1 1 γ = √1 – (0.01c)2/c2 = √1 – (0.01)2c2/c2 1 1 1 γ = √1 – (0.01)2 = = √1 – 0.0001 √0.9999 γ = 1.00005
  28. 28. Properties 1 of γ = √1 – v(cont’d) 2 /c2Suppose v = 0.1c (i.e. 10% of c) 1 1 γ = √1 – (0.1c)2/c2 = √1 – (0.1)2c2/c2 1 1 1 γ = √1 – (0.1)2 = = √1 – 0.01 √0.99 γ = 1.005
  29. 29. Let’s make a chart v γ =1/√(1-v2/c2)0.01 c 1.00005 0.1 c 1.005
  30. 30. Other values of 1 γ = √1 – v2/c2Suppose v = 0.5c (i.e. 50% of c) 1 1 γ = √1 – (0.5c)2/c2 = √1 – (0.5)2c2/c2 1 1 1 γ = √1 – (0.5)2 = = √1 – (0.25) √0.75 γ = 1.15
  31. 31. Enter into chart v γ =1/√(1-v2/c2)0.01 c 1.00005 0.1 c 1.005 0.5c 1.15
  32. 32. Other values of 1 γ = √1 – v2/c2Suppose v = 0.6c (i.e. 60% of c) 1 1 γ =√1 – (0.6c)2/c2 = √1 – (0.6)2c2/c2 1 1 1 γ = √1 – (0.6)2 = = √1 – (0.36) √0.64 γ = 1.25
  33. 33. Back to the chart v γ =1/√(1-v2/c2)0.01 c 1.00005 0.1 c 1.005 0.5c 1.15 0.6c 1.25
  34. 34. Other values of 1 γ = √1 – v2/c2Suppose v = 0.8c (i.e. 80% of c) 1 1 γ = √1 – (0.8c)2/c2 = √1 – (0.8)2c2/c2 1 1 1 γ = √1 – (0.8)2 = = √1 – (0.64) √0.36 γ = 1.67
  35. 35. Enter into the chart v γ =1/√(1-v2/c2)0.01 c 1.00005 0.1 c 1.005 0.5c 1.15 0.6c 1.25 0.8c 1.67
  36. 36. Other values of 1 γ = √1 – v2/c2Suppose v = 0.9c (i.e.90% of c) 1 1 γ = √1 – (0.9c)2/c2 = √1 – (0.9)2c2/c2 1 1 1 γ = √1 – (0.9)2 = = √1 – 0.81 √0.19 γ = 2.29
  37. 37. update chart v γ =1/√(1-v2/c2)0.01 c 1.00005 0.1 c 1.005 0.5c 1.15 0.6c 1.25 0.8c 1.67 0.9c 2.29
  38. 38. Other values of 1 γ = √1 – v2/c2Suppose v = 0.99c (i.e.99% of c) 1 1 γ =√1 – (0.99c)2/c2 = √1 – (0.99)2c2/c2 1 1 1 γ = √1 – (0.99)2 = = √1 – 0.98 √0.02 γ = 7.07
  39. 39. Enter into chart v γ =1/√(1-v2/c2)0.01 c 1.00005 0.1 c 1.005 0.5c 1.15 0.6c 1.25 0.8c 1.67 0.9c 2.290.99c 7.07
  40. 40. Other values of 1 γ = √1 – v2/c2Suppose v = c 1 1 γ = √1 – (c)2/c2 = √1 – c2/c2 1 1 1 γ = = = √1 – 12 √0 0 γ = ∞ Infinity!!!
  41. 41. update chart v γ =1/√(1-v2/c2)0.01 c 1.00005 0.1 c 1.005 0.5c 1.15 0.6c 1.25 0.8c 1.67 0.9c 2.290.99c 7.071.00c ∞
  42. 42. Other values of 1 γ = √1 – v2/c2Suppose v = 1.1c 1 1 γ = √1 – (1.1c)2/c2 = √1 – (1.1)2c2/c2 1 1 1 γ = √1 – (1.1)2 = = √1-1.21 √ -0.21 γ = ??? Imaginary number!!!
  43. 43. Complete the chart v γ =1/√(1-v2/c2) 0.01 c 1.00005 0.1 c 1.005 0.5c 1.15 0.6c 1.25 0.8c 1.67 0.9c 2.29 0.99c 7.07 1.00c ∞Larger than c Imaginary number
  44. 44. Plot results: ∞ Never-never land 1γ = √1 – v2/c2 x x x x x v=c
  45. 45. Moving clocks run slower v t0 t= 1 t 0 √1 – v2/c2 t t = γ t0 γ >1  t > t0
  46. 46. Length contraction v L0 time=t L0 = vt te r! or Shman on Time = t0 =t/γrocket Length = vt0 =vt/γ =L0/γ
  47. 47. Moving objects appear shorter Length measured when object is at rest L = L0/γ γ >1  L < L0 V=0.9999c V=0.86c V=0.1c V=0.99c
  48. 48. Length contraction
  49. 49. mass: change in v F=m0a = m0 time t0 a time=t0 m0 Ft0 change in v = m0 Ft0 m0 = change in v mass Ft γ Ft0m= = = γ m0 increases!! change in v change in v m = γ m0 t=γt0 by a factor γ
  50. 50. Relativistic mass increase m0 = mass of an object when it is at rest  “rest mass”mass of a moving γobject increases as vc, m∞ m = γ m0 as an object moves faster, it gets harder & harder to accelerate by the γ factor v=c
  51. 51. summary• Moving clocks run slow γ o f o r c t• Moving objects appear shorter f a a y• Moving object’s mass increases B
  52. 52. Plot results: ∞ Never-never land 1γ = √1 – v2/c2 x x x x x v=c
  53. 53. α- Twin paradox centauri rs y ea ht Twin brother lig & sister 4.3 She will travel to α -centauri (a near- by star on a specialHe will stay home rocket ship v = 0.9c& study Phys 100
  54. 54. Light yeardistance light travels in 1 year dist = v x time = c yr 1cyr = 3x108m/s x 3.2x107 s = 9.6 x 1015 m We will just use cyr units & not worry about meters
  55. 55. Time on the boy’s clock r cy 0.9c =4 . 3 v= d0 0.9c v= According to the boy & his clock on Earth: d0 4.3 cyr = 4.8 yrs tout = = 0.9c v d0 4.3 cyr = 4.8 yrs tback = = 0.9c v ttotal = tout+tback = 9.6yrs
  56. 56. What does the boy see on her clock? yr 0.9c 4. 3c v= d= 0.9c v= According to the boy her clock runs slower tout 4.8 yrs t = out = 2.3 = 2.1 yrs γ tback 4.8 yr tback = γ = = 2.1 yrs 2.3 ttotal = tout+tback = 4.2yrs
  57. 57. So, according to the boy: yr 0.9c 4. 3c v= d= 0.9c v= his clock her clock out: 4.8yrs 2.1yrs back: 4.8yrs 2.1yrs ges She a total: 9.6yrs 4.2yrs less
  58. 58. But, according to the girl, the boy’s clock ismoving &, so, it must be 0.9c running slower v= According to her, the boy’s clock on Earth says: tout 2.1 yrs tout = γ = = 0.9 yrs 2.3 tback 2.1 yrs = 0.9 yrs tback = = 2.3 .9c γ v=0 ttotal = tout+tback = 1.8yrs
  59. 59. Her clock advances 4.2 yrs& she sees his clock advance only 1.8 yrs, contradict ion??AShe should think he has aged less than her!!
  60. 60. Events in the boy’s life: As seen by him As seen by her She leaves 4.8 yrs 0.9 yrs She arrives & starts turn short time ???? Finishes turn & heads home 4.8 yrs 0.9 yrs She returns 9.6+ yrs 1.8 + ??? yrs
  61. 61. turning around as seen by her According to her, these 2 events occur very,very far apart from each other He sees herHe sees her finish turningstart to turn Time interval between 2 events depends on the state of motion of the observer
  62. 62. Gunfight viewed by observer at rest He sees both shots fired simultaneously Bang ! Bang !
  63. 63. Viewed by a moving observer
  64. 64. Viewed by a moving observer He sees cowboy shoot 1st & cowgirl shoot later Bang ! Bang !
  65. 65. In fact, ???? = 7.8+ years as seen by him as seen by her She leaves 4.8 yrs 0.9 yrs She arrives & starts turn short time 7.8+ yrs ??? Finishes turn & heads home 4.8 yrs 0.9 yrs She returns 9.6+ yrs 1.8 + ???yrs 9.6+ yrs
  66. 66. No paradox: both twins agree The twin that “turned around” is younger
  67. 67. Ladder & Barn Door paradoxStan & Ollie puzzle over howto get a 2m long ladder thrua 1m wide barn door ??? 1m 2m ladder
  68. 68. Ollie remembers Phys 100 & the theory of relativityStan, pick upthe ladder &run very fast 1m 2m tree ladder
  69. 69. View from Ollie’s ref. frame 1m Push, Stan! 2m/γ V=0.9cOllie Stan (γ=2.3)
  70. 70. View from Stan’s ref. frame But it 1m/ doesn’t fit, γ Ollie!!V=0.9c(γ=2.3) 2mOllie Stan
  71. 71. If Stan pushes both ends of theladder simultaneously, Ollie sees the two ends move at different times: Too late 1m Too Stan! soon nk Stan! clu clan k V=0.9c Ollie Stan Stan (γ=2.3)
  72. 72. Fermilab proton accelerator V=0.9999995c γ =1000 2km
  73. 73. Stanford electron acceleratorv=0.99999999995 c 3k mγ=100,000
  74. 74. statusEinstein’s theory of “special relativity” has been carefully tested in many very precise experiments and found to be valid.Time is truly the 4th dimension of space & time.
  75. 75. testγ=29.3

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