1. PH 301 Dr. Cecilia Vogel Lecture 2

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

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. Just How Proper is it? 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. slow Astronaut you

6. Time Dilation Plus  Light source with frequency f o (in its own frame)  Emits N cycles of EM waves  in time  t o.  N = f o  t o.   t o is the proper time to emit N cycles,  since in source’s reference frame all cycles are emitted at same place, “right here”

7. Additional Effect  In another reference frame, the light source is moving toward the observer.  Time to emit N cycles is given by time dilation equation  t =  t o.  There is a second effect due to the fact that the light takes time to arrive  And in that time, the source has moved ctct vtvtN ’

8. Doppler Effect Geometry With this geometry ctct vtvtN ’

9. Doppler Effect ― Approaching Now plug in Since ’ =c/ ’, 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 receding Holds if source and observer approaching Higher frequency ― blue shift 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. Lorentz Transformations  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

14. Recall Classical Relativity  recall

15. Postulates and Assumptions  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.

16. Definition  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 x=0 x’=0 t=0 x’=0 vt

17. More Assumptions  The relationships between x, t and x’, t’ are linear.  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. Finding  and   Plug x ’=0, t = t, and x = vt into x=0 x’=0 vt  So the relationship becomes

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

20. Timing a Flash of Light  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=0 x’=0 light’s wavefront x=ct x’=ct’

21. Finding   Solve for  :  Plug x = ct and x’ = ct’ into

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

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. B E V

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

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 timeplaceEvent -46.7 yr70.5 c-yrOur class begins 00 0 0 Pass planet Bolth +23.8 yr Light from our class beginning arrives +23.8 yr + 1day Vulcans calculate that our class began 70.5 yrs ago, i.e. t = 23.8 – 70.5 = -46.7 yr