RELATIVITY Special Relativity
FROM WARMUP It was all interesting! How important is it for us to know the Galilean transformation equations and the math of the Michelson-Morley experiment? Know the Galilean transformations. The math for the MM experiment is less important. I am not sure I understand it and if I did I am not sure I believe it.
GALILEAN RELATIVITY v 2 = 100 mph v 1 = 80 mph Credit: this slide and next one from Dr. Durfee
GALILEAN RELATIVITY Reference frame moving with car 1 v 2 = 20 mph v 1 = 0 mph
POSTULATES OF SPECIAL RELATIVITY 1. The laws of physics apply in all inertial reference frames. 2. The speed of light is the same for all inertial observers, regardless of the motion of the source or the observer.
FROM WARMUP Explain in your own words (in language that your non-physics-major roommates could understand) how the Galilean velocity transformation equation and the two postulates of special relativity are at odds with one another What if a car is racing a light beam. The car manages to travel at half the speed of light. How fast does the driver observe the light beam to travel according to: Galilean relativity? Einstein s relativity?
EXAMPLE: LIGHT RAY ON A TRAIN If height of train car inside is h, how long did that take (to me, inside the train)? Credit: animations from Dr. Durfee Answer: t = 2h/c
AS SEEN FROM GROUND How long did it take, really? If height of train car inside is h, how long did that take (to you, on the ground)? Train is traveling at speed v Answer: t = 2h/c (1-v 2 /c 2 ) -1/2
NOTATION Answer 1 (measured on train): t = 2h/c Answer 2 (measured on ground): t = 2h/c (1-v 2 /c 2 ) -1/2 Time measured on train: Dt Time measured on ground: Dt v c 1 1 2 Dt v/c Dt For v = 0.9c: = 0.9 = 2.3
THINK ABOUT THIS Suppose Alice (in the train), measure a time interval to be 1 second. If the train is moving at 0.9c, Bob (on the ground) measure that time interval to be 2.3 s. To Bob, it looks like things in the train are running in slow motion. However, what if Bob on the ground is the one that is bouncing light rays back and forth. If Bob measure a time interval to be 1 s, how long will that interval look like, to Alice on the train? A. 1 s. That To is, Bob, to Alice Alice s it looks time like things appears on the to ground be slowed. are running normally To Alice, Bob s time appears to be slowed. Who is right? B. (1/2.3) s. That is, to Alice it looks like things on the ground are sped up C. 2.3 s. That is, to Alice it looks like things on the ground are running in slow motion.
CONSIDER SPACE TRAVEL A space traveler takes his rocket (0.9 c, gamma = 2.29) to a planet 1 light year away. Earth frame: how long does it take? Traveler frame: how long does it take? Traveler frame: How much does the traveler age? Earth frame: how much does the traveler age? Traveler frame: how fast is the planet approaching? Do you see a contradiction? x = vt: t = 1.11 year 0.485 year 0.485 year 0.485 year 0.9 c x = vt
LENGTH CONTRACTION Which is correct? The space traveler aged so little because. A. Time was slowed down (Earth frame of reference) B. The distance shrank (Traveler frame of reference)
MUON DECAY Muons are unstable particles (heavier than electrons and neutrinos, but lighter than all other matter particles). When cosmic rays strike the upper atmosphere, a series of reactions produce muons (gamma rays + atomic nuclei -> pions -> muons). Muons have an average lifetime of about 2.2 micro seconds. Ignoring relativistic effects, how far can muons travel on average, if they move at the speed of light? On earth, about 10,000 muons originating from cosmic ray reactions reach ever square meter of the earth s surface. How does relativity explain this? Including relativistic effects, how far could muons potentially travel?
FROM WARMUP What is the rest length of an object? The proper, or rest, length of an object is the length measured by some observer who is stationary, or at rest, with respect to the object. The rest length is the length of the object from its reference frame. Note that the book calls this proper length.
FROM WARMUP What is the proper time of a journey? Proper time is the time measured by an observer for whom the two events take place in the same location. Proper time of a journey is measured by someone who remains in the same position when witnessing the beginning and end points of a journey.
TIME DILATION AND LENGTH CONTRACTION Two sides of the same coin The space traveler aged so little because A. Time was slowed down for him (Earth view) B. The distance shrank for him (Traveler view) Who measure s the trip s rest length? Who measures the traveler s proper time?
FROM WARMUP The book discusses the famous "twin paradox". What is the seeming paradox, and how is it resolved? The twin paradox is to figure out if less time has elapsed for one twin, the space traveler, than has the "stationary" twin on earth. Basically, is the space traveling twin moving away from earth, or is earth moving with respect to the space traveler. Thus the paradox is which twin is younger upon the return of the space traveler. Perhaps it is solved by realizing that the space traveling twin is not in an inertial reference frame for his whole journey. SR assumes that all inertial reference frames are the same, and the space traveler cannot accelerate nor turn around without leaving his inertial frame. This implies that the traveling twin will be younger upon return.
SPACE-TIME DIAGRAMS What is an event on this graph? What do we mean by a worldline? More on space-time diagrams: Monday Next Friday (group problem)