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Cosmic Adventure 5.6 Time Dilation in Relativity


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Time Dilation in Relativity

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Cosmic Adventure 5.6 Time Dilation in Relativity

  1. 1. © ABCC Australia 2015 TIME DILATION IN RELATIVITY Cosmic Adventure 5.6
  2. 2. © ABCC Australia 2015 Temporal Transform Not only frame transformation changes the length of an object, but also the timing on the object.
  3. 3. © ABCC Australia 2015 Proper Time An observer at the rest frame with his single clock made a measurement of the time on another clock. This time interval, 𝑡′′ − 𝑡′ = ∆𝑡0, is called the proper time interval between the events. 𝑡′′𝑡′ Proper time: ∆𝑡0= 𝑡′′ − 𝑡′ Rest Frame
  4. 4. © ABCC Australia 2015 Proper Time Equation To find the relationship between the time separations as measured by O and O’ , the relativistic way is to subtract two of the Lorentz time transformations: ∆𝑡0= 𝑡′′ − 𝑡′ → 𝑡′′ + 𝑣𝑥′′ 𝑐2 1 − 𝑣2 𝑐2 − 𝑡′ + 𝑣𝑥′ 𝑐2 1 − 𝑣2 𝑐2 = 𝑡′′ + 𝑣𝑥′′ 𝑐2 − 𝑡′ − 𝑣𝑥′ 𝑐2 1 − 𝑣2 𝑐2
  5. 5. © ABCC Australia 2015 Resulting Difference ∆𝑡0= 𝑡′′ + 𝑣𝑥′′ 𝑐2 − 𝑡′ − 𝑣𝑥′ 𝑐2 1 − 𝑣2 𝑐2 These time measurements are made in the moving frame. When they are made at the same location, the expression will be reduced to: ∆𝑡 = 𝑡′′ − 𝑡′ 1 − 𝑣2 𝑐2 = ∆𝑡0 1 − 𝑣2 𝑐2 = 𝛾∆𝑡0 Where 𝛾 = 1 1− 𝑣2 𝑐2
  6. 6. © ABCC Australia 2015 Time Dilation Since: 1 − 𝑣2 𝑐2 < 1, ∆𝑡 > ∆𝑡0 Being the time interval between the two events measured by O’ is considered to be dilated (enlarged), thereby giving rise to the phenomenon of “time dilation”.
  7. 7. © ABCC Australia 2015 Equation of Time Dilation ∆𝑡 = ∆𝑡0 1 − 𝑣2 𝑐2 Observed time Proper time Lorentz Factor
  8. 8. © ABCC Australia 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 10.0 8.0 6.0 4.0 2.0 1.0 Timedilation ∆𝑡 = ∆𝑡0 1 − 𝑣2 𝑐2 Velocity as a fraction of the speed of light 𝑣/𝑐
  9. 9. © ABCC Australia 2015 Time Changes due to Frame Transfer For example, one can place cameras at the location of clock B and at the location of the clock C, and a picture is taken by each camera when the clock C passes clock B. Each picture will show that the clock C has advanced through ∆𝑡0 while clock B has advanced through ∆𝑡. ∆𝑡 and ∆𝑡0 are related by the time dilation equation. Clock A Clock BClock B Clock C Clock C 𝑣 𝑣 Clock C advances through ∆𝑡 𝑜 Clock A & B advances through ∆𝑡
  10. 10. © ABCC Australia 2015 Slower Clocks For the observer at rest on the ground, the object’s clock is running at a lower rate. But for the clock carrier, there is no difference at all. His clock is running at the normal rate as when he is at rest. No! It is not! Your clock is slowing down!
  11. 11. © ABCC Australia 2015 Time Lapse by Slower Clock Once the moving clock re- unites with the ground clock, it will again run at the same rate as the ground clock. It is only when the clock is in motion with respect to the other clock that the phenomenon of time dilation takes place. Naturally, the slower clock will remain behind by the amount of time it spent.
  12. 12. © ABCC Australia 2015 Time Dilation is Retained by Slower Clock Relativists believe that if you take two very accurate and well synchronized atomic clocks and put one on a high-speed trip on an airplane. When the plane returned, the clock that took the plane ride was slower.
  13. 13. © ABCC Australia 2015 The Twin Paradox The wonderful thing is that the returned clock keeps the time it gained in the trip. Anyone who travels with the clock will consequently live longer. This leads to the phenomenon of “Twin Paradox”. It is a paradox because nobody has verified it by trying to live longer this way.
  14. 14. © ABCC Australia 2015 Dr. Einstein will return to earth nearly as young, while his twin on earth will have aged terribly.
  15. 15. © ABCC Australia 2015 Example of Calculation The earth bound Einstein calculate the time dilation with the equation: ∆𝑡 = ∆𝑡0 1 − 𝑣2 𝑐2 where: ∆𝑡 = time observed in the other reference frame ∆𝑡0 = time in observers own frame of reference (rest time) 𝑣 = the speed of the moving rocket 𝑐 = the speed of light in a vacuum
  16. 16. © ABCC Australia 2015 Calculating the Dilation So in the situation we will let: 𝑣 = .95𝑐 𝑡0 = 10 𝑦𝑒𝑎𝑟𝑠 and we will solve for t which is the time that the earth bound Einstein brother measures. 𝑡 = 10 1 − 0.95𝑐 2 𝑐2 = 10 1 − 0.9025 = 10 0.0975 = 10 0.3122 = 32 𝑇𝑖𝑚𝑒 = 32 𝑦𝑒𝑎𝑟𝑠
  17. 17. © ABCC Australia 2015 Muon Life Time The extended lifetime of the muon provides another proof of time dilation. But so far the results are not absolutely precise and the relativistic explanation is not too clear. There are other ways of achieving the results and in a more logical way. We will be glad to talk about it in a different session.
  18. 18. © ABCC Australia 2015 Both Answers Real and Correct Objectively speaking, the observer is seeing two different times. So which time is correct? According to the relativists, they both are. The reason is that time is not absolute but is relative, it depends on the relative relationship of the reference frames. So why then is the phenomenon called a paradox?
  19. 19. © ABCC Australia 2015 VISONIC TIME DILATION – CLOCKS AT REST To be continued on: Cosmic Adventure 5.7