1Phy107, Lecture 17 1Mon. Feb 27, 2006From last time…• Galilean Relativity– Laws of mechanics identical in all inertial ref. frames• Einstein’s Relativity– All laws of physics identical in inertial ref. frames– Speed of light=c in all inertial ref. frames• Consequences– Simultaneity: events simultaneous in one frame willnot be simultaneous in another.– Time dilation: time interval between events appeardifferent to different observersPhy107, Lecture 17 2Mon. Feb 27, 2006Einstein’s principle of relativity• Principle of relativity:– All the laws of physics are identical in all inertialreference frames.• Constancy of speed of light:– Speed of light is same in all inertial frames(e.g. independent of velocity of observer, velocityof source emitting light)(These two postulates are the basis of thespecial theory of relativity)Phy107, Lecture 17 3Mon. Feb 27, 2006Consequences of Einstein’s relativity• Many ‘common sense’ results break down:– Events that seem to be simultaneous are notsimultaneous in different inertial frames– The time interval between events is not absolute.it will be different in different inertial frames– The distance between two objects is not absolute.it is different in different inertial frames– Velocities don’t always add directlyPhy107, Lecture 17 4Mon. Feb 27, 2006Time dilation• Laser bounces up and down from mirror on train.• Joe on ground measures time interval w/ his clock.• Joe watches Jane’s clock on train as she measuresthe time interval.• Joe sees that these two time intervals are different.Reference frame of Jane on trainReference frameof Joe on groundPhy107, Lecture 17 5Mon. Feb 27, 2006Why is this?• Jane on train: light pulse travels distance 2d.• Joe on ground: light pulse travels farther• Relativity: both Joe and Jane say light travels at c– Joe measures longer travel time of light pulse• This is time dilationReference frame ofJane on trainReference frame ofJoe on groundPhy107, Lecture 17 6Mon. Feb 27, 2006Time dilation, continued• Observer Jane on train: light pulse travels distance 2d.• Time = distance divided by velocity = 2d/c• Time in the frame the events occurred at same locationcalled the proper time ΔtpReference frame ofJane on trainReference frame ofJoe on ground2Phy107, Lecture 17 7Mon. Feb 27, 2006Joe measures a longer timeTime dilationTime interval in Jane’s frame! "tJane=round trip distan cespeed of light=2dc! "tJoe=2 d2+ v"tJoe/2( )2c! "=11# (v /c)2> 1! "tJoe=11# (v /c)22dc$ % & ' ( ) =*"tpd(vΔt)/2Phy107, Lecture 17 8Mon. Feb 27, 2006The ‘proper time’• We are concerned with two time intervals.Intervals between two events.– A single observer compares time intervals measured indifferent reference frames.• If the events are at the same spatial location in oneof the frames…– The time interval measured in this frame is called the ‘proper time’.– The time interval measured in a frame moving withrespect to this one will be longer by a factor of γ! "tother frame=#"tproper, #> 1Phy107, Lecture 17 9Mon. Feb 27, 2006QuestionI am on jet traveling at 500 mph.I throw a ball in the air, catch it in my hand, andtime the round trip with a clock on my jet.You are on the ground watching my clock, and you alsomeasure the time interval with your own clock.How do these time intervals compare?A. Δtjet=ΔtEarthB. Δ tjet>ΔtEarthC. Δtjet<ΔtEarthProper time is measured in the jet frame(events occur at same spatial location).Times measured in other frames are longer(time dilation).Phy107, Lecture 17 10Mon. Feb 27, 2006Atomic clocks and relativity• In 1971, four atomic clocks were flown around the worldon commercial jets.• 2 went east, 2 went west -> a relative speed ~ 1000 mi/hr.• On return, average time difference was 0.15 microseconds,consistent with relativity.First atomic clock: 1949Miniature atomic clock: 2003Phy107, Lecture 17 11Mon. Feb 27, 2006Global Positioning System (GPS)Network of 24 satellites orbiting earth at14,000 km/hr, each carrying atomic clocks,At least three visible from any location.GPS receiver on ground comparestime signals from several satellites.Distance from each satellite givenby travel time.Position on ground determined bydistance from satellites.Relativity: clocks run slow by 7 microseconds!Phy107, Lecture 17 12Mon. Feb 27, 2006Traveling to the starsSpaceship leaves Earth, travels at 0.95cSpaceship later arrives at star0.95c0.95cd=4.3 light-years3Phy107, Lecture 17 13Mon. Feb 27, 2006The ship observer’s frame..then star arrives0.95c0.95cd=4.3 light-yearsEarth leaves…Phy107, Lecture 17 14Mon. Feb 27, 2006QuestionBoth the astronaut and the earthling measure the timeinterval between heatbeats of the astronaut?Who measures the proper timeA. The astronautB. The EarthlingC. BothD. NeitherPhy107, Lecture 17 15Mon. Feb 27, 2006• The ship observer measures ‘proper time’– Heartbeats occur at the same spatial location(in the astronaut’s chest).• On his own clock, astronaut measures his normal heart-rate of1 second between each beat.• Earth observer measures, with his earth clock, a time muchlonger than the astronaut’s ( Δtearth = γ Δtastronaut )Comparing the measurements! "tearth=#"tastronaut="tastronaut1$ v2/c2= 3.2 % "tastronaut= 3.2 secEarth observer sees astronaut’s heart beating slow,and the astronaut’s clock running slow.Earth observer measures 3.2 sec between heartbeats ofastronaut.Phy107, Lecture 17 16Mon. Feb 27, 2006The twin ‘paradox’The Earth observer sees the astronautage more slowly than himself.– On returning, the astronaut would beyounger than the earthling.– And the effect gets more dramatic withincreasing speed!– All this has been verified - the‘paradox’ arises when we take theastronaut’s point of view.Phy107, Lecture 17 17Mon. Feb 27, 2006• Special relativity predicts that astronaut woulddisagree, saying earthling is younger!• Why?0.95cd=4.3 light-yearsApparently a direct contradiction.If both measure the time interval between heartbeats of theearthling, the earthling measures the proper time.Any other measurement of the time interval is longer!The astronaut says the earthling’s heart beats more slowly.Phy107, Lecture 17 18Mon. Feb 27, 2006Resolution• Special relativity applies only toreference frames moving at constant speed.• To turn around and come back, the astronaut mustaccelerate over a short interval.• Only the Earthling’s determination of the
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