PH 301 Dr. Cecilia Vogel Lecture 2
Review Outline Relativity classical relativity Einstein’s postulates Constancy of speed of light consequence: time dilation consequence: Doppler effect
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
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.
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.
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
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
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.
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
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=?
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
Time Dilation Equation Eliminating d between these equations: d d v t/2
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?
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”
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?
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:
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 = ?
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
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”
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 ctct vtvtN ’
Doppler Effect Geometry With this geometry ctct vtvtN ’
Doppler Effect ― Approaching Now plug in Since ’ =c/ ’, Holds if source and observer approaching
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
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.
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