1Phy107 Fall 2006 1From 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 Fall 2006 2Einstein’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 Fall 2006 3Consequences 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 Fall 2006 4Time 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 Fall 2006 5Why 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 Fall 2006 6Time 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 tpReference frame ofJane on trainReference frame ofJoe on ground2Phy107 Fall 2006 7Joe measures a longer timeTime dilationTime interval in Jane’s frametJane=round trip distancespeed of light=2dctJoe=2 d2+ vtJoe/2()2c=11 (v /c)2> 1tJoe=11 (v /c)22dc = tpd(vt)/2Phy107 Fall 2006 8The ‘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 tother frame=tproper, > 1Phy107 Fall 2006 9Atomic 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 Fall 2006 10Traveling to the starsSpaceship leaves Earth, travels at 0.95cSpaceship later arrives at star0.95c0.95cd=4.3 light-yearsPhy107 Fall 2006 11The ship observer’s frame..then star arrives0.95c0.95cd=4.3 light-yearsEarth leaves…Phy107 Fall 2006 12• 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 ( tearth = tastronaut )Comparing the measurementstearth=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.3Phy107 Fall 2006 13The 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 Fall 2006 14• 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 Fall 2006 15Resolution• 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 timeintervals using special relativity are correct.• General relativity applies to accelerating referenceframes, and will make the measurements agree.Phy107 Fall 2006 16Total trip timeSpaceship leaves Earth, travels at 0.95c0.95cd=4.3 light-yearstearth=dvc 95.0years-light 3.4=years 5.4=Time for astronaut passes more slowly by a factor gamma.Trip time for astronaut is 4.5 yrs/3.2 = 1.4 yearsPhy107 Fall 2006 17Relative velocity ofreference frames=11 ( v / c )2=11 ( 0.95 )2= 3.203Speed of lightvvEarth frameRocket frameBoth observers agree on relative speed, hence also gamma.Phy107 Fall 2006 18Are there other ‘paradoxes’?• Both observer’s agree on the speed (0.95c)– Earth observer: ship moving– Ship observer: earth and star moving– They both agree on the speed• But they disagree about the total trip time.• If the time intervals are different, and speed isthe same, how can distances be the same?• The distances are not the same!Length contraction4Phy107 Fall 2006 19Length Contraction• People on ship and on earth agree on relative velocity v = 0.95 c.• But they disagree on the time (4.5 vs 1.4 years).• What about the distance between the planets?Earth frame dearth = v tEarth= .95 (3x108 m/s) (4.5 years)= 4x1016m (4.3 light years)Ship frame dship = v tship= .95 (3x108 m/s) (1.4 years)= 1.25x1016m (1.3 light
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