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Understanding Time Contraction & Doppler Effect in Relativity | PH.301 Dr. Cecilia Vogel Lecture

Explore Einstein's postulates, time dilation, Lorentz transformations, and Doppler effects. Learn how speed impacts time and distance measurements in different frames of reference. Discover the fascinating connections between light, motion, and perception.

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Understanding Time Contraction & Doppler Effect in Relativity | PH.301 Dr. Cecilia Vogel Lecture

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  1. PH 301 Dr. Cecilia Vogel Lecture 2

  2. length contraction Doppler Lorentz Review • Consequences of Einstein’s postulates • Constancy of speed of light • time dilation Outline

  3. Recall Trip Example John saw Nick’s 12 light-year trip to take 20 years. Nick saw the trip to take 16 years … saw the Earth receding and the other planet approaching for 16 yrs.

  4. Nick’s Frame • In Nick’s frame • Earth is at x=vt = -0.6c*16yr = -9.6 c-yr • How far apart are the Earth and the planet in Nick’s frame? 9.6 c-yr • Recall the distance from Earth to planet is 12c-yr according to John! • Lengths and distances are not the same for all observers! • Thus length contraction • John measures proper length, because he is at rest relative to Earth & planet • Nick measures length-contracted distance

  5. If there is a proper time and a proper length, is there a proper reference frame? NO!!!! Proper time of trip in example: Nick Proper length of trip in example: John Proper time of astronaut’s heartbeat: Astronaut’s heartbeat looks ____ to you. Proper time of your heartbeat: Your heartbeat looks _____ to astronaut. Just How Proper is it? Astronaut slow you slow

  6. Light source with frequency fo (in its own frame) Emits N cycles of EM waves in time Dto. N = fo Dto. Dto is the proper time to emit N cycles, since in source’s reference frame all cycles are emitted at same place, “right here” Time Dilation Plus

  7. In another reference frame, the light source is moving toward the observer. Time to emit N cycles is given by time dilation equation Dt = gDto. There is a second effect due to the fact that the light takes time to arrive And in that time, the source has moved Additional Effect cDt vDt Nl’

  8. With this geometry Doppler Effect Geometry vDt Nl’ cDt

  9. Doppler Effect ― Approaching Now plug in Since l’ =c/n’, Holds if source and observer approaching

  10. Doppler Effect ― Receding Can repeat the previous derivation for receding source or observer Holds if source and observer approaching Higher frequency ― blue shift Holds if source and observer receding Lower frequency ― red shift

  11. Doppler Effect ― Evidence Hydrogen absorption spectrum: moving H-atoms absorb different frequencies than H-atoms at rest in lab. Because they “see” a Doppler-shifted freq.

  12. Application Laser cooling Aim a laser with a slight lower freq than an (at-rest) absorption line. Atoms at rest won’t absorb the laser light. Approaching atoms will “see” a slightly higher freq such atoms can absorb the laser light this will slow the atoms (head-on) At-rest atoms unaffected, moving atoms slowed (on average) Overall effect – slower atoms -- COOLER

  13. Relates time and position of an event in one frame to those in another frame Event is something that happens at one place at one time. our class is not an event, because it lasts 75 min New Years day is not an event, because it happens all over Earth x and t from x’ and t’ Can be used generally for any event Lorentz Transformations

  14. recall Recall Classical Relativity

  15. Postulate: Both the primed and unprimed observer measure the speed of light to be c. Assumption: The primed and unprimed frames have agreed on an origin, so that x=0, t=0 corresponds to x’=0, t’=0. Postulates and Assumptions

  16. The primed frame moves at velocity v relative to the unprimed frame (defines v). Which means that the origin of the x’-axis separates from the origin of the x-axis at velocity v in the positive-x direction t=0 vt x=0 x’=0 x’=0 Definition

  17. The relationships between x, t and x’, t’ are linear. More Assumptions • So that a constant velocity in one frame will be a constant velocity in the other (Law of inertia). • The relationships between x, t and x’, t’ are symmetric. • since neither frame is special • the only difference is the sign of v

  18. vt x=0 x’=0 Finding g and a • Plug x’=0, t=t, and x=vt into • So the relationship becomes

  19. We found Symmetry • The symmetric equation, just changing the sign of v is:

  20. If a flash of light occurs at t=t’=x=x’=0, it will travel away from the origin at speed c. according to both observers. x=ct x=0 Timing a Flash of Light light’s wavefront x’=ct’ x’=0

  21. Finding g • Plug x = ct and x’ = ct’ into • Solve for g:

  22. Lorentz Transformation Eqns where • Can also combine x’=g(x-vt) and x=g(x’+vt’), and solve for t’:

  23. Example • As a Vulcan spacecraft passes planet Bolth in Jan 2001 at a speed of 0.7 c, it synchronizes its origin of time and position with the Bolthians. Both agree that time is zero and that place is zero. Where and when did we begin lecture #1, relative to the Vulcans on the ship? Earth and Bolth are at rest relative to each other, 53 light-years apart. E B V

  24. According to Bolthians, our class started at t=3.75 (Sep 2004), and x= 53 c-yr More Example

  25. More Example • The Vulcan’s say our class started 70.5 light-years from them, 46.7 years before they passed Bolth!

  26. Vulcan Timeline • x=0 is Vulcan ship -46.7 yr 70.5 c-yr Our class begins 0 0 Pass planet Bolth Light from our class beginning arrives +23.8 yr 0 Vulcans calculate that our class began 70.5 yrs ago, i.e. t = 23.8 – 70.5 = -46.7 yr +23.8 yr + 1day 0

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