1. PH 301 Dr. Cecilia Vogel Lecture 2

2. Review Outline  Relativity  classical relativity  Einstein’s postulates  Constancy of speed of light  consequence: time dilation  consequence: Doppler effect

3. Recall Classical Relativity  very close to true when v<<c:  Different observers measure same time  Different observers measure same distance between objects  Different observers measure different position and velocity  of each other. Pattern:  of another object. Pattern:  Different observers conclude the same laws of mechanics apply

4. Postulates  Classical relativity not quite right  Einstein's relativity right (so far)  Einstein’s postulates  Laws of physics are the same for all inertial (constant velocity) observers  Speed of light is the same, independent of the motion of source or observer.

5. Postulates  Classical relativity not quite right  Einstein's relativity right (so far)  Einstein’s postulates  Laws of physics are the same for all inertial (constant velocity) observers  Speed of light is the same, independent of the motion of source or observer.

6. You Can Hide But You Can’t Run  Speed of light is measured to be c = 3X10 8 m/s by all.  Can you catch up? NO! If you chase a light beam, it will still recede from you at 3X10 8 m/s  Can you run away? NO! If you fly away from a light beam, it will still catch up to you at 3X10 8 m/s  What if the source moves? Light from a moving bulb still moves at 3X10 8 m/s relative to you

7. Some Consequences  Can be derived from constancy of speed of light:  Time interval between events depends on observers state of motion  Length of object or length of a trip depends on observers state of motion

8. Recall Classical Relativity  Suppose two observers time the pretzel you throw and catch. One observer on airplane, one on Earth. Same pretzel. Go-stop. t’=5 s Go-------------------------stop. t=? Classical relativity says this is also 5 s.

9. Recall Classical Relativity  At any point, let the velocity of the pretzel measured by the plane observer be u’.  Then the velocity measured by Earth observer is u = u’ + v, therefore u is faster than u’.  Pretzel goes farther, faster in Earth frame.  Same time Compared to this frame, in this frame, the pretzel goes… farther

10. Now Einstein’s Relativity  That worked for pretzels, what about light?  Person on super-plane shines light at mirror.  Suppose two observers time the light that shines and reflects. One observer on plane, one on Earth. Same light. Go-stop. t’=5  s Go----------- stop. t=?

11. Now Einstein’s Relativity Compared to this frame, In this frame, light goes farther  At any point, the velocity of the light measured by the plane observer is c.  And the velocity measured by Earth observer is  also c.  Light goes further at the same speed in Earth frame  t is longer than t’!! d d v  t/2

12. Time Dilation Equation  Eliminating d between these equations: d d v  t/2

13. Time Dilation  General result:   t and  t o are both time interval  between same two events!  measured by two different observers  v is relative velocity of two observers  Notice that if v<<c, the times are approximately the same  Hard part: which time is which?

14. Proper time  What’s the difference between  t and  t o ?   t o is the “proper time”  Proper time is always shortest.  Def: measured in frame in which the two events happen at the same place.  For example  Person who shines light, since light comes right back.  You measure proper time between your heartbeats.  Person who takes a trip measures proper time of trip, since departure and arrival both happen “right here”

15. Example  Nick travels to a planet 12 light-years away at a speed of 0.6 c. John stays on Earth. Each measures the trip to take a different amount of time. Note:  A light-year is distance light goes in a year  d = (3X10 8 m/s)(1 yr) = 9.46X10 15 m  d = ( c )( 1 yr) = 1 c-yr  The values in problem are relative to Earth.  Question: How long does the trip take according to each?

16. Solution  In John’s (Earth’s) frame (in any one frame), the laws of physics hold,  including d = vt, or t = d/v  John measures time = (12 c-yr)/(0.6c) = 20 yr  To find time in another frame (Nick’s), we need to use time dilation:

17. Solution  Who measures proper time?  Nick – departure and arrival both “right here”  John does not – departure is “right here,” but arrival is way away on another planet.   t = 20 yr,  t o = ?

18. 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

19. 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”

20. 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 ’

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

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

23. 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

24. 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.

25. 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