PH 301 Dr. Cecilia Vogel Lecture 2
Review Outline length contraction Doppler Lorentz Consequences of Einstein’s postulates Constancy of speed of light time dilation
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.
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
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
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
Recall Classical Relativity recall
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.
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
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
Finding and Plug x ’=0, t = t, and x = vt into x=0 x’=0 vt So the relationship becomes
Symmetry We found The symmetric equation, just changing the sign of v is:
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’
Finding Solve for : Plug x = ct and x’ = ct’ into
Lorentz Transformation Eqns Can also combine x’= (x-vt) and x= (x’+vt’), and solve for t’: where
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
More Example According to Bolthians, our class started at t=3.75 (Sep 2004), and x= 53 c-yr
More Example The Vulcan’s say our class started 70.5 light-years from them, 46.7 years before they passed Bolth!
Vulcan Timeline x=0 is Vulcan ship timeplaceEvent yr70.5 c-yrOur class begins Pass planet Bolth yr Light from our class beginning arrives yr + 1day Vulcans calculate that our class began 70.5 yrs ago, i.e. t = 23.8 – 70.5 = yr