1Lecture 6Reminders:Reading prior to this lecture TZD 1.7-1.10Reading for next lecture TZD 1.11-1.13 and 2.1-2.3Homework #2 posted and due next WednesdayExam #1 will be Friday, September 25 in class.Reminder – no class or problem solving session on MondayToday’s topic: Time Dilation andLength ContractionClicker questionWhere was Leia actually going when she and her brother were first separated?A) SiriusB) NabooC) AlderaanD) TantooineE) The Death Star“The daughter of Jedi General Anakin Skywalker and SenatorPadmé Amidala of Naboo, Leia was the younger twin sister of Luke Skywalker, and shortly after her birth she became the adopted daughter of Bail Prestor Organaand Queen Breha of Alderaan, making her a Princess of Alderaan”http://starwars.wikia.com/wiki/Leia_Organa_SoloLuke is on Earth watching Leia’s spaceship fly by at a speed v.Leia’s spaceship flies by Earth at speed vLuke has observers set up at the location of the two events. Using their synchronized clocks he finds that a time of ΔtR(rest frame of earth) passes between the light flash and when the reflected light reaches the detector.LukeAny observer moving with respect to this clock sees it run slowly (i.e., time intervals are longer). This is time dilation:If two events occur at the same location, then the time between them can be measured by a single observerwith a single clock (the “Leia time” in our example). The time measured between these types of events is called the proper time,Proper Time and Time Dilation0tΔ0ttγΔ=ΔExample: any given clock never moves with respect to itself. It keeps proper time in its own frame. 211βγ−=cv=βwhere and How come we never notice the effect of time dilation?Time DilationHow much time will your watch lose on a non-stop flight from Los Angeles to Sydney (11 hours at 300 m/s)?Not usually noticeable but in fact measured in 1971 by carrying atomic clocks onboard commercial airplanes around the world.()0050000000000.11011m/s 103m/s 1031111122 8222=−=××−=−=−cvγIf our watch measures 11 hours, the time recorded on the ground will be longer: = 11 hours + 20 nanoseconds0 tt Δ=ΔγTime dilation is0 tt Δ=ΔγQ. Luke and Leia are twins. While Luke stays on Earth, Leia departs for the star Sirius, 8 light-years away, traveling at a speed v = 0.8c (Note γ = 5/3). According to observers in Luke’s frame, how long does the trip take?A. 6 years B. 8 years C. 10 years D. 16.67 yearsvLukeLeiaClicker questionThe distance is 8 light-years and the speed is 0.8 c so we getyears 10c 80.0/years)c 8(/=⋅==vdt2Solving for the proper time givesQ. Luke stays on Earth and Leia departs for the star Sirius, 8 light-years away, traveling at a speed v = 0.8 c (γ = 5/3). According to Leia, how long does the trip take?A. 6 years B. 8 years C. 10 years D. 16.67 yearsvLukeLeiaClicker questionLeia’s clock is the only one present at both events (leaving Earth and arriving at Sirius). Therefore, she is measuring the proper time! This tells us which time is which in the time dilation formula.years 63/5years 100==Δ=Δγtt0ttγΔ=ΔFollow the proper time!We found that only 6 years pass in Leia’s reference frame as she travels 8 light-years to Sirius at 0.8 c.We say that every inertial frame is valid so won’t she observe the space ship going faster than the speed of light (8 light years in 6 years!)?This discrepancy is resolved by length contraction – moving objects are shorter (in the direction of motion) γ/0LL =Length ContractionvLukeLeiaLength ContractionLeia sees a star approaching at 0.80c which givesyrsc 8.4)3/5( / yrsc 8/0⋅=⋅==γLL3/56.0136.0164.118.011)/(1122===−=−=−=cvγand measures the length between Earth and star asSo in Leia’s frame the ship covers 4.8 light years in 6 years which works out to a velocity of 4.8/6 = 0.8c, same as the velocity seen on earth by Luke.Length ContractionLength of an object... -3 -2 -1 0 1 2 3 ...This stick is 3m long. I measure both ends at the same time in my frame of reference. Or not. It doesn’t matter, because the stick isn’t going anywhere.But as we know, “at the same time” is relative – it depends on how you’re moving.This length, measured in the stick’s rest frame, is its proper length.Length of an object... -3 -2 -1 0 1 2 3 ...Another observer comes whizzing by at speed v. This observer measures the length of the stick, and keeps track of the time.S0 vS′Event 1 – Origin of S′ passes left end of stick.Length of an object... -3 -2 -1 0 1 2 3 ...S0 vS′Event 1 – Origin of S′ passes left end of stick.Event 2 – Origin of S′ passes right end of stick.How many observers are required to measure the time between these two events in reference frame S and S′?For S: 2 observers at 0 and 3. For S′: just one at the origin3In frame S:length of stick = L (this is the proper length)time between measurements = Δtspeed of frame S′ is v = L/ΔtIn frame S′:length of stick = L′ (this is what we’re looking for)time between measurements = Δt′speed of frame S is v = L′/Δt′Q. Which is the correct time dilation formula for this setup?A. Δt= γΔt′ B. Δt′ = γΔtFollow the proper time!Clicker questionLength ContractionSpeeds are the same (both refer to the relativespeed)tLtLv′Δ′=Δ=γLL =′Length in moving frameLength in stick’s rest frame (proper length)Length contraction is a consequence of time dilation (and vice-versa).tt′Δ=ΔγtLtL′Δ=Δ′which givesto replace Δt′ we arrive atUsingcontraction formulaand identifying L as the proper length we get the lengthγ0LL =The detectors were then moved to sea level to see what fraction of the muons survived.Muons are elementary particles similar to electrons but they decay (into other particles) in about 2.2 μs (at rest).Muons can be produced from cosmic rays interacting in the upper atmosphere.Detectors to count muons with v=0.99c were placed on top of Mt. Washington (2000 m above sea level) and 563 muons/hour were observed.Experimental tests of time dilation & length contractionDown the mountain at 0.99c takes Δt=d/v=2000m/(0.99•3•108m/s)=6.7μs.Muons live ~2.2μs so might expect very few at sea level (~25 per hr).However, from special relativity, the muon clock is slow and the 6.7μs observed on Earth (Δt) will be