1April 21, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 48April 21, 2005 Physics for Scientists&Engineers 2 2EinsteinEinstein’’s Postulatess Postulates Postulate 1:• The laws of physics are the samein each reference frame,independent of the motion of thisreference frame. Postulate 2:• The speed of light, c, is the samein every reference frame.April 21, 2005 Physics for Scientists&Engineers 2 3Beta and GammaBeta and Gamma It is convenient to express speeds as a fraction of thespeed of light. Introduce a variable beta as the ratio of the speed to thespeed of light Another useful variable thatdepends on the speed and thespeed of light!=vc !=11 "#2=11 " (v / c)2April 21, 2005 Physics for Scientists&Engineers 2 4Time DilationTime Dilation Conventionally, we think of time as flowing at auniform rate in one direction (past to the future) But if you accept that the speed of light is thesame in any reference frame, then our oldunderstanding of time runs into conceptualproblems The time interval it takes for an event to happenas observed in a moving rest frame is dilated (=made bigger) as compared to the time interval forthe event to occur in the frame where it is at rest2April 21, 2005 Physics for Scientists&Engineers 2 5Time Dilation (2)Time Dilation (2) Let’s start with two parallel mirrors,distance h, and send a light beamfrom one to the other This takes a time Now observe the same event whilemoving with speed vh!t0=hchxvvApril 21, 2005 Physics for Scientists&Engineers 2 6Time Dilation (3)Time Dilation (3) Now the mirrors are moving adistance x = vΔt in horizontaldirection. Light beam has covered a distance(Pythagoras!) Light moves with c: L = cΔthhxvvL = h2+ x2April 21, 2005 Physics for Scientists&Engineers 2 7Time Dilation (4)Time Dilation (4) Put this together Solve for the time interval in themoving frame:hhxvvL = h2+ x2! c"t = (c"t0)2+ (v"t )2!t =!t01 " (v / c)2=#!t0April 21, 2005 Physics for Scientists&Engineers 2 8Time Dilation (5)Time Dilation (5) Once more, in words:The rate at which time flows is dependent on howfast the observer moves! Note: this is not just the subjective perception oftime, but the objective measurable length of time!!t =!t01 " (v / c)2=#!t0Simply the most astonishing formula derived in the entire course up to now!3April 21, 2005 Physics for Scientists&Engineers 2 9Example: Time DilationExample: Time Dilation Muon Decay Muons have a lifetime of 2.2 micro-seconds CERN experiment: muons produced with β = 0.9994 Lifetime of these moving muons should be 28.87 timeslonger, = 63.5 micro-secs, than those at rest. During this time, the muons can move a distance Without time dilation, this distance would only be 660 m => Direct observation of time dilation!!=11 "#2=11 " 0.99942= 28.87x = v!= v"!0= 0.9994 c # 28.87 # 2.200 µs=19 kmApril 21, 2005 Physics for Scientists&Engineers 2 10Length ContractionLength Contraction If you want to measure the length of a spaceship at rest, you can use alaser and measure the time it takes to move the length of the spaceship Now we redo the same experiment withthe shuttle and the clock to measure thetime inside it are moving by at speed v. This time the measurement then yields: Length contraction by a factor γ ! Note: length only changes in the direction of motion, not perpendicular L0= c!t0L =c!t0"=L0"April 21, 2005 Physics for Scientists&Engineers 2 11Twin ParadoxTwin Paradox Twin sisters Alice (astronaut) and Betty (stays home), both age 20 Alice takes a space trip with γ = 10 Alice flies to a nearby star and back (Note the direction of the velocity vector does not matter!) Alice flies for 5 years and is 25 when she returns home to Earth But in the meantime Betty has aged 5γ = 50 years and is now 70 whenher sister returns! But wait a minute! We can also put ourselves into Alice’s rest frame,and in this frame Betty (and the entire Earth) has moved with γ = 10,and so Alice would expect to be 25 on her return, but her sister shouldonly be 20.5 years old. Big difference! … in particular for Betty … Which answer is correct?April 21, 2005 Physics for Scientists&Engineers 2 12Twin Paradox - SolutionTwin Paradox - Solution Alice is not really in an inertial frame, because she has toaccelerate and slow down• True, but also not relevant• We can restrict the speeding up and slowing down phases to a verysmall fraction of the trip; so they will not change the answer Consider that the two endpoints of the trip are given in thecoordinate frame of Earth, and that they are fixed: Earthon one end and a star 25 light-years away at the other end. In Alice's traveling frame that distance appears lengthcontracted by a factor of 10, which is why she is able tomake the roundtrip in 5 years. Consequently Betty is 70 and Alice 25 years old when theymeet again.4April 21, 2005 Physics for Scientists&Engineers 2 13Frequency ShiftFrequency Shift Time dilation modifies all other observables that arerelated to time. Example: frequency Relativistic frequency shift acts similarly to DopplerEffect, but has origin in time dilation, not in movement ofthe wave medium relative to observer or source Upper signs for observer and emitted moving away fromeach other, lower for moving toward each other f = f0c ! vc ± vApril 21, 2005 Physics for Scientists&Engineers 2 14Wavelength ShiftWavelength Shift Since c = λ f, a shift in the frequency is also a shiftin wavelength Observed wavelength(again, upper signs for moving away from eachother, lower for towards each other) Red-shifted: object moves away from us Blue-shift: object moves towards us !=!0c ± vc ! vApril 21, 2005 Physics for Scientists&Engineers 2 15Red ShiftRed Shift Red-shift parameter z is defined as the ratio ofthe wave length shift of light divided by the wavelength of the same light when the source is at rest Red shift Observation: Practically all galaxies in theuniverse are red-shifted and thus move away fromus• The further away they are, the more red-shifted z =!""0="#"0"0=c ± vc ! v# 1April 21, 2005 Physics for Scientists&Engineers 2 16Lorentz Lorentz TransformationTransformationGalileix ' = x ! vty' = yz' = zt ' = tLorentzx ' =!(x " vt )y' = yz ' = zt ' =!(t " vx / c2)Inversex =!(x '+ vt)y = y'z = z 't =!(t '+ vx
View Full Document
Unlocking...