Physics 334 Modern Physics - Faculty Websites

Physics 334Modern PhysicsCredits: Material for this PowerPoint was adopted from Rick Trebino’s lectures from Georgia Tech which werebased on the textbook “Modern Physics” by Thornton and Rex. I have replaced some images from the adoptedtext “Modern Physics” by Tipler and Llewellyn. Others images are from a variety of sources (PowerPoint clip art,Wikipedia encyclopedia etc) and were part of original lectures. Contributions are noted wherever possible in thePowerPoint file. The PDF handouts are intended for my Modern Physics class.1

CHAPTER 1Special Theory of Relativity 11-11-21-31-41-51-6The Experimental Basis of RelativityEinstein’s PostulatesThe Lorentz TransformationTime Dilation and Length ContractionThe Doppler EffectThe Twin Paradox and Other SurprisesAlbert Michelson(1852-1931)It was found that there wasno displacement of theinterference fringes, so thatthe result of the experimentwas negative and would,therefore, show that there isstill a difficulty in the theoryitself - Albert Michelson, 19072

1–1: Newtonian (Classical) RelativityNewton’s laws of motion must be implemented with respect to (relative to)some reference frame.yy’z’x’zxA reference frame is called an inertial frame if Newton’s laws arevalid in that frame.Such a frame is established when a body, not subjected to netexternal forces, moves in rectilinear motion at constant velocity.3

Difference Between Inertial and Non-InertialReference Frame4

Newtonian Principle of RelativityIf Newton’s laws are valid in one reference frame, then they are alsovalid in another reference frame moving at a uniform velocity relative tothe first system.This is referred to as the Newtonian principle of relativity or Galileaninvariance.If the axes are also parallel, these frames are said to be InertialCoordinate Systems5

The Galilean TransformationFor a point P:In one frame S: P (x, y, z, t)In another frame S’: P (x’, y’, z’, t’)The Inverse Relationsx′ x vty′ yz′ zt′ t1. Parallel axes2. S’ has a constant relativevelocity (here in the x-direction)with respect to S.3. Time (t) for all observers is aFundamental invariant, i.e.,it’s the same for all inertialobservers.x x′ vt ′y y′z z′t t′6

A need for etherIn Maxwell’s theory, the speed oflight, in terms of the permeabilityand permittivity of free space, wasgiven by:vThus the velocity of light is constantAether was proposed as anabsolute reference system in whichthe speed of light was this constantand from which othermeasurements could be made.Maxwell’s equations are not invariant underGalilean transformations.The Michelson-Morley experiment wasan attempt to show the existence ofaether.Properties of Aether:Low densityElasticityTransverse wavesGalilean transformation7

Michelson-Morley experimentMichelson and Morley realized thatthe earth could not always bestationary with respect to theaether. And light would have adifferent path length and phase shiftdepending on whether itpropagated parallel and antiparallel or perpendicular to theaether.PerpendicularpropagationSupposedvelocity of earththrough theaetherParallel andanti-parallelpropagation8

Michelson-MorleyExperimental AnalysisExercise 1: Show that the time difference between pathdifferences after 90 rotation is given by:2 ( t t ) 2 LRecall that the phase shift isω times this relative delay:2ω Lv2c3or:v2c3L v24πλ c2The Earth’s orbital speed is: v 3 104 m/s , and the interferometersize is: L 1.2 m, So the time difference becomes: 8 10 17 s, which,for visible light, is a phase shift of: 0.2 rad 0.03 periodsThe Michelson interferometer should’verevealed a fringe shift as it was rotated withrespect to the aether velocity. MM expected0.4 of the width of a fringe, and could onlysee 0.01 equal to the uncertainty in themeasurement.Interference fringes showedno change as theinterferometer was rotated.Thus, aether seems not to exist!9

1-2: Einstein’s PostulatesAlbert Einstein was only two years oldwhen Michelson and Morley reportedtheir results.At age 16 Einstein began thinkingabout Maxwell’s equations in movinginertial systems.In 1905, at the age of 26, hepublished his startling proposal:the Principle of Relativity.It nicely resolved the Michelson andMorley experiment (although thiswasn’t his intention and hemaintained that in 1905 he wasn’taware of Michelson and Morley’swork )Albert Einstein (1879-1955)It involved a fundamentalnew connection betweenspace and time and thatNewton’s laws are only anapproximation.10

Einstein’s Two PostulatesWith the belief that Maxwell’s equations must be valid inall inertial frames, Einstein proposed the followingpostulates:The principle of relativity: The laws of physics are the same in allinertial reference frames.The constancy of the speed of light: The speed of light in avacuum is equal to the value c, independent of the motion ofthe source.11

Relativity of SimultaneityIn Newtonian physics, wepreviously assumed that t’ t.With synchronized clocks, eventsin S and S’ can be consideredsimultaneous.Einstein realized that each systemmust have its own observers with theirown synchronized clocks and metersticks.Events considered simultaneous in Smay not be in S’.Also, time may pass more slowly insome systems than in others.12

The constancy of the speed of light13

1-3 Lorentz TransformationExercise 2: The equations for a spherical wavefronts in S isx2 y2 z2 c2t2 , Show that the equation for the spherical wavefronts inS’ cannot be x’2 y’2 z’2 c2t’2 in the Galilean transformation.Exercise 3: Show that x’ γ (x – vt) so that x γ’ (x’ vt’) , yieldsthe γ factoid1γ 1 v2c2and that for small velocities1 v2γ 1 22c14

Lorentz TransformationExercise 4: Use x’ γ (x – vt) and x γ’ (x’ vt’) , to find t’ γ (t – v x/c2)Exercise 5: Use x’ γ (x – vt) and t’ γ (t – v x /c2) to show that theequations for spherical wave fronts in S and S’ are the same.15

Lorentz Transformation EquationsA more symmetrical form:β v/cγ 11 v2 / c 216

Properties of γRecall that β v / c 1 for all observers.γ equals 1 only when v 0.In general:Graph of γ vs. β:(note v c)17

The complete Lorentz Transformationx′ x vt1 v2 / c2y′ yx z′ zt′ 1 v / c1 v2 / c 2y y′z z′t vx / c 22x′ vt ′2t t ′ vx′ / c 21 v2 / c 2If v c, i.e., β 0 and γ 1, yielding the familiar Galilean transformation.Space and time are now linked, and the frame velocity cannot exceed c.18

The complete Lorentz Transformationx′ x vt1 v2 / c2y′ yz′ zt′ 1 v / cLengthcontraction2t Timedilationx′ vt ′1 v2 / c 2y y′z z′Simultaneityproblemst vx / c 22x t ′ vx′ / c 21 v2 / c 2If v c, i.e., β 0 and γ 1, yielding the familiar Galilean transformation.Space and time are now linked, and the frame velocity cannot exceed c.19

Relativistic Velocity TransformationExercise 6: Suppose a shuttle takes off quickly from a space shipalready traveling very fast (both in the x direction). Imagine that thespace ship’s speed is v, and the shuttle’s speed relative to thespace ship is u’. What will the shuttle’s velocity (u) be in the restframe?ux u ′x vdxγ (dx′ v dt ′) dt γ [dt ′ (v/c 2 ) dx′] 1 u x′ v/c 2uy u′ydydy′ dt γ [ dt ′ (v/c 2 ) dx′] γ (1 u′x v/c 2 )uz dz ′dzu′z dt γ [dt ′ (v/c 2 ) dx′] γ (1 u ′x v/c 2 )20

The Inverse Lorentz Velocity TransformationsIf we know the shuttle’s velocity in the rest frame, we can calculate itwith respect to the space ship. This is the Lorentz velocitytransformation for u’x, u’y , and u’z. This is done by switching primedand unprimed and changing v to –v:u ′x ux vdx′ dt 1 u x v/c 2u ′y uydy′ dt γ (1 u x v/c 2 )u ′z dz ′uz dt γ (1 u x v/c 2 )21

Gedanken(Thought)experimentsIt was impossible to achievethe kinds of speedsnecessary to test his ideas(especially while working inthe patent office ), soEinstein used Gedankenexperiments or Thoughtexperiments.Young EinsteinEinstein pic: cle id 6050Lightning and space ship: 22

Space-timeWhen describing events inrelativity, it’s convenient torepresent events with aspace-time diagram.In this diagram, one spatialcoordinate x, specifiesposition, and instead of time t,ct is used as the othercoordinate so that bothcoordinates will havedimensions of length.Space-time diagrams werefirst used by H. Minkowski in1908 and are often calledMinkowski diagrams. Pathsin Minkowski space-time arecalled world-lines.23

Particular WorldlinesStationaryobservers liveon vertical lines.A light wave hasa 45º slope.Worldline is the record of the particle’s travelthrough spacetime, giving its speed (1/slope) andacceleration ( 1/rate of change of slope).24

1-4: Time Dilation and Length ContractionMore very interesting consequences of the Lorentz Transformation:Time Dilation:Clocks in S’ run slowly with respect to stationary clocks in S.Length Contraction:Lengths in S’ contract with respect to the same lengths instationary S.25

We must think about how wemeasure space and time.In order to measure an object’s length in space,we must measure its leftmost and rightmostpoints at the same time if it’s not at rest.If it’s not at rest, we must ask someoneelse to stop by and be there to help out.In order to measure an event’s duration in time,the start and stop measurements can occur atdifferent positions, as long as the clocks aresynchronized.If the positions are different, we must ask someoneelse to stop by and be there to help out.Ruler: x1024.jpgClock:http://www.istockphoto.com/file thumbview approve/155322/2/istockphoto 155322ticking clock.jpg26

Proper TimeTo measure a duration, it’s best to usewhat’s called Proper Time.The Proper Time, τ, is the time betweentwo events (here two explosions) occurringat the same position (i.e., at rest) in a system as measured by aclock at that position.Same locationProper time measurements are in some sense the most fundamentalmeasurements of a duration. But observers in moving systems, wherethe explosions’ positions differ, will also make such measurements.What will they measure?27

Time Dilation and Proper TimeFrank’s clock is stationary in S where two explosions occur.Mary, in moving S’, is there for the first, but not the second.Fortunately, Melinda, also in S’, is there for the second.MelindaMaryS’Mary and Melindaare doing the bestmeasurement thatcan be done.Each is at the rightplace at the righttime.SFrankIf Mary andMelinda arecareful to time andcompare theirmeasurements,what duration willthey observe?28

Time DilationMary and Melinda measure the times for the two explosions in system S’ as t’1and t’2 . By the Lorentz transformation: t ' t ' t ' (t2 t1 ) (v c)( x2 x1 )2 11 v2 c2This is the time interval as measured in the frame S’.This is not proper time due to the motion of S’: x1′ x2′.Frank, on the other hand, records x2 – x1 0 in S with a (proper)time: τ t2 – t1, so we have: t ' τ21 v c2 γ t γτ29

Time Dilation1) t ’ t:(γ 1) the time measuredbetween two events at differentpositions is greater than the timebetween the same events atone position: this is time dilation.2) The events do not occur at the same space and timecoordinates in the two systems.3) System S requires 1 clock and S’ requires 2 clocks for themeasurement.4) Because the Lorentz transformation is symmetrical, timedilation is reciprocal: observers in S see time travelfaster than for those in S’. And vice versa!http://images.google.com/imgres?imgurl g&imgrefurl http://www.juliantrubin.com/einsteinjokes.html&h 84&w 125&sz 3&tbnid xQW4G0ngLFYbqM:&tbnh 56&tbnw 84&hl en&start 119&prev GGLD:en%26sa%3DN30

Time Dilation Example: ReflectionMirrorLc t/2Mirrorv t/2vS’MarySFrankFredExercise 7: Show that the event in its rest frame (S’) occurs fasterthan in the frame that’s moving compared to it (S).31