From last time…Einstein’s principle of relativityConsequences of Einstein’s relativityTime dilationWhy is this?Time dilation, continuedSlide 7The ‘proper time’Atomic clocks and relativityTraveling to the starsThe ship observer’s frameComparing the measurementsThe twin ‘paradox’PowerPoint PresentationResolutionTotal trip timeSlide 17Are there other ‘paradoxes’?Length ContractionLength contraction and proper lengthThe real ‘distance’ between eventsEvents in the Earth FrameA relativistic invariant quantityTime dilation, length contractionAddition of Velocities (Non-relativistic)Addition of Velocities (Relativistic)Relativistic Addition of VelocitiesSlide 28Slide 29Phy107 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 will not be simultaneous in another.–Time dilation: time interval between events appear different to different observersPhy107 Fall 2006 2Einstein’s principle of relativity•Principle of relativity:–All the laws of physics are identical in all inertial reference frames.•Constancy of speed of light:–Speed of light is same in all inertial frames(e.g. independent of velocity of observer, velocity of source emitting light)(These two postulates are the basis of the special theory of relativity)Phy107 Fall 2006 3Consequences of Einstein’s relativity•Many ‘common sense’ results break down:–Events that seem to be simultaneous are not simultaneous 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 measures the time interval.•Joe sees that these two time intervals are different.Reference frame of Jane on trainReference frame of 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 dilation Reference frame of Jane on trainReference frame of Joe 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 location called the proper time tpReference frame of Jane on trainReference frame of Joe on groundPhy107 Fall 2006 7Joe measures a longer time Time dilationTime interval in Jane’s frame € ΔtJane=round trip distancespeed of light=2dc€ ΔtJoe=2 d2+ vΔtJoe/2( )2c€ γ =11− (v /c)2>1€ ΔtJoe=11− (v /c)22dc ⎛ ⎝ ⎜ ⎞ ⎠ ⎟= γΔtpd(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 in different reference frames.•If the events are at the same spatial location in one of the frames…–The time interval measured in this frame is called the ‘proper time’.–The time interval measured in a frame moving with respect to this one will be longer by a factor of € Δtother frame= γΔtproper, γ >1Phy107 Fall 2006 9Atomic clocks and relativity•In 1971, four atomic clocks were flown around the world on 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 0.95c Spaceship 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 of 1 second between each beat. •Earth observer measures, with his earth clock, a time much longer than the astronaut’s ( tearth = tastronaut )Comparing the measurements € Δtearth= γΔtastronaut=Δtastronaut1− v2/c2= 3.2 × Δtastronaut= 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 of astronaut.Phy107 Fall 2006 13The twin ‘paradox’The Earth observer sees the astronaut age more slowly than himself. –On returning, the astronaut would be younger than the earthling.–And the effect gets more dramatic with increasing speed!–All this has been verified - the ‘paradox’ arises when we take the astronaut’s point of view.Phy107 Fall 2006 14•Special relativity predicts that astronaut would disagree, saying earthling is younger!•Why?0.95cd=4.3 light-yearsApparently a direct contradiction.If both measure the time interval between heartbeats of the earthling, 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 to reference frames moving at constant speed.•To turn around and come back, the astronaut must accelerate over a short interval.•Only the Earthling’s determination of the time intervals using special relativity are correct.•General relativity applies to accelerating reference frames, and will make the measurements agree.Phy107 Fall 2006 16Total trip timeSpaceship leaves Earth, travels at 0.95c 0.95cd=4.3 light-years€ Δtearth=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 of reference frames€ γ =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
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