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UW-Madison PHYSICS 107 - Lecture 18 Notes

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1Mon. Oct 18 Phy107 Lecture 18From Last Time…•Events observed to be simultaneous in oneframe may not be simultaneous in another.•Measured interval between events different fordifferent observers.— Time dilation. Proper time is that measured inframe where events occur at same spatial location— All other measured times are longer by factor γ•Measured distance between events different fordifferent observers.— Length contraction. Proper length is that measuredin frame where events are simultaneous.— All other lengths are shorter by factor γMon. Oct 18 Phy107 Lecture 18Hour Exam 2• Wednesday, Oct. 27• In-class (1300 Sterling Hall)• Twenty multiple-choice questions• Will cover 8.1-8.6 (Light and E&M) 9.1-9.5 (E&M waves and color) 10, 11 (Relativity)• You should bring– 1 page notes, written double sided– Calculator– Pencil for marking answer sheetMon. Oct 18 Phy107 Lecture 18Time dilationI am on jet traveling at 500 mph and throw a ballup and catch it in my hand.You are on the ground and watch me.How do the time intervals compare for you and I?A. tjet=tEarthB. tjet>tEarthC. tjet<tEarthProper time is measured in the jet frame(events occur at same spatial location).Times measured in other frames are longer(time dilation).Mon. Oct 18 Phy107 Lecture 18Geometrical space-time diagrams• We came to these conclusionsvia particular thought experiments.• Were able to make quantitative determinations,using sequential logical arguments.(e.g. to get from time dilation to length contraction)• Shortly after relativity, Minkowski developed thespace-time diagram.• A geometrical way to display observations indifferent reference frames.Mon. Oct 18 Phy107 Lecture 18Events and relativity• In relativity, it is helpful to think of ‘events’– E.g. a lightning strike,– a flashbulb going off,– a spaceship arriving at a planet• Describe event by spatial position and time as (x, t)• Can think of this as a point in two-dimensional space,but axes would have different units (meters, sec)• Use event ‘coordinates’ of (x, ct).– Units of ct = (m/s)(s) = meters.• Can now represent events graphicallyMon. Oct 18 Phy107 Lecture 18The space-time continuum• An event is indicated bya point in this graph.xct• The time-dependentmotion of a particle wouldbe a string of these points.– This is called theparticle’s ‘world-line’x1ct12Mon. Oct 18 Phy107 Lecture 18Constant velocity motion• Worldline of an objectmoving at constantvelocity is a linexctWorld line low ofvelocity objectWorld line of highervelocity objectCoordinate axesMon. Oct 18 Phy107 Lecture 18Scale of the space-time graph• The axes ct and x both have thesame units (meters).• We scale them so that theworld line of a light beamis at a 45˚ angle• For instance,x tick-marks 1 meter apart,ct tick-marks 3x108 meter apartxctWorld line oflight beamMon. Oct 18 Phy107 Lecture 18Observing from a new frame• The point of relativity is thatthese events will look differentin reference frame moving atsome velocity.• Can do this by measuring sameevent along different coordinateaxes.• Coordinates determined byprojecting parallelto x and ct axes (parallel tox’ and ct’ in new coord system).Coordinates inoriginal frameCoordinates innew framect’x’xctNew frame movingrelative to originalMon. Oct 18 Phy107 Lecture 18Different inertial frames• Relativistic effects are apparentwhen we compare inertialframes moving at differentvelocities.• Speed of light constant in allreference frames says tick-marksscale same way on space andtime axes.• Coordinate axes tilt in towardlight worldline as relativevelocity increases.• Tick-marks get farther apart asrelative velocity increases.ct’x’ct’’x’’xctLightMon. Oct 18 Phy107 Lecture 18The train again‘Track’ observer‘Train’ observerScorch marks on tracksLightning strikeson trainMon. Oct 18 Phy107 Lecture 18Flashesreach trainobserver atdiff timesxct‘Train’ observerA’B’World lines of objectsstationary on the train(moving at velocity v in track frame)Low-velocityworldlinesFrame of observer on trackxct‘Track’ observerABWorld lines of objectsstationary on the trackZero-velocityworldlinesSimultaneousflashes3Mon. Oct 18 Phy107 Lecture 18Frame of Observer on TrainTrain observer has newcoordinate axes.• A’, B’,& observer remain at fixed x’• But world lines same in both frames.•On these new coordinate axes,the original lightning strikes atA’ and B’ areNOT simultaneous.Event A’ occurs atEvent B’ occurs at€ ′ t A= 0€ ′ t B< 0Coordinate axes for refframe of observer on trainxctA’B’ct’x’€ ′ t = 0€ ′ t < 0Mon. Oct 18 Phy107 Lecture 18Events• The view here is that the events themselvestranscend the observer in the sense that theyare fixed on the space-time diagram.• Different observers record the time and spacecoordinates differently,but the events themselves have not changed.Mon. Oct 18 Phy107 Lecture 18Is any measurement the same for all observers?• Relativity seems to say that there are nomore absolutes.– Distance between objects depends on observer.– Time between events depends on observer.• But this occurs because we weren’tconsidering the situation as a whole.• Immutable character of events suggest thatthere might be invariant quantities.• Analogy: 2D view of 3D object.Mon. Oct 18 Phy107 Lecture 18The big picture• Views of the samecube from twodifferent angles.• Distance betweencorners (length of redline drawn on the flatpage) seems to bedifferent dependingon how we look at it.• But clearly this is just because we are not considering thefull three-dimensional distance between the points.• The 3D distance does not change with viewpoint.Mon. Oct 18 Phy107 Lecture 18The real ‘distance’ between events• Can do something similar for the space-time continuum• Need a quantity that is the same for all observers• Hint: Speed of light is constant in all frames.– World line for light propagation is x = ct ( or x = -ct )• A quantity all observers agree on is• Can think of this as the full ‘distance’ between events.• Looking at position x or time t separatelyjust confuses things. € x2− c2t2≡ separation( )2− c2interval( )2Mon. Oct 18 Phy107 Lecture 18The 4D space-time continuum• Should be considering time as another‘coordinate’ which contributes to the 4D‘distance’ along with 3 dimensional spatialdistance• Relativisitic


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UW-Madison PHYSICS 107 - Lecture 18 Notes

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