Notes on Special Relativity Physics 252, Michael Fowler, University of Virginia March 21, 2008 FRAMES OF REFERENCE AND NEWTON’S LAWS .......................................................................... 3 THE SPEED OF LIGHT ............................................................................................................................. 6 Early Ideas about Light Propagation ................................................................................................... 6 Measuring the Speed of Light with Jupiter’s Moons............................................................................. 6 Starlight and Rain................................................................................................................................. 7 Fast Flickering Lanterns....................................................................................................................... 8 Albert Abraham Michelson ................................................................................................................... 8 Sailing the Silent Seas: Galilean Relativity .......................................................................................... 9 Michelson Measures the Speed of Light.............................................................................................. 10 THE MICHELSON-MORLEY EXPERIMENT..................................................................................... 10 The Nature of Light............................................................................................................................. 10 The Wavelike Nature of Sound............................................................................................................ 11 Is Light a Wave? ................................................................................................................................. 11 If Light is a Wave, What is Waving?................................................................................................... 12 Detecting the Aether Wind: the Michelson-Morley Experiment ......................................................... 13 Einstein’s Answer................................................................................................................................18 SPECIAL RELATIVITY .......................................................................................................................... 19 Galilean Relativity again .................................................................................................................... 19 Generalizing Galilean Relativity to Include Light: Special Relativity................................................ 20 You Really Can’t Tell You’re Moving!................................................................................................ 21 Truth and Consequences..................................................................................................................... 21 SPECIAL RELATIVITY: WHAT TIME IS IT? .................................................................................... 23 Special Relativity in a Nutshell ........................................................................................................... 23 A Simple but Reliable Clock................................................................................................................ 23 Looking at Somebody Else’s Clock..................................................................................................... 24 Fitzgerald Contraction........................................................................................................................ 27 Experimental Evidence for Time Dilation: Dying Muons................................................................... 27 SPECIAL RELATIVITY: SYNCHRONIZING CLOCKS.................................................................... 28 THE LORENTZ TRANSFORMATIONS ............................................................................................... 30 Problems with the Galilean Transformations..................................................................................... 30 Deriving the Lorentz Transformations................................................................................................ 32 Spheres of Light .................................................................................................................................. 33 Lorentz Invariants............................................................................................................................... 34 The Light Cone.................................................................................................................................... 36 TIME DILATION: A WORKED EXAMPLE......................................................................................... 37 “Moving Clocks Run Slow” plus “Moving Clocks Lose Synchronization” plus “Length Contraction” leads to consistency! ........................................................................................................................... 37 MORE RELATIVITY: THE TRAIN AND THE TWINS...................................................................... 42 Einstein’s Definition of Common Sense.............................................................................................. 42 Trapping a Train in a Tunnel.............................................................................................................. 42 2 The Tunnel Doors are Closed Simultaneously.................................................................................... 43 Or are they?........................................................................................................................................ 43 Does the Fitzgerald Contraction Work Sideways? ............................................................................. 43 How to Give Twins Very Different Birthdays...................................................................................... 44 The Twins Stay in Touch..................................................................................................................... 44 Figuring the Observed Time between Flashes.................................................................................... 45 What does she see? ............................................................................................................................. 46 What does he see?............................................................................................................................... 46 The Doppler Effect.............................................................................................................................. 47 ADDING VELOCITIES: A WALK ON THE TRAIN ........................................................................... 48 The Formula ....................................................................................................................................... 48 Walking Across the Train.................................................................................................................... 50 Testing the Addition of Velocities Formula ........................................................................................ 51 RELATIVISTIC DYNAMICS .................................................................................................................. 52 The Story So Far: A Brief Review....................................................................................................... 52 Newton’s Laws Revisited .................................................................................................................... 53 Conservation Laws.............................................................................................................................. 53 Momentum Conservation on the Pool Table....................................................................................... 54 A Symmetrical Spaceship Collision .................................................................................................... 55 Just how symmetrical is it? ................................................................................................................. 55 Einstein rescues Momentum Conservation ......................................................................................... 56 Mass Really Does Increase with Speed............................................................................................... 56 Mass and Energy Conservation: Kinetic Energy and Mass for Very Fast Particles .......................... 57 Kinetic Energy and Mass for Slow Particles ...................................................................................... 58 Kinetic Energy and Mass for Particles of Arbitrary Speed................................................................. 58 Notation: m and m0 ............................................................................................................................. 59 MASS AND ENERGY ............................................................................................................................... 60 Rest Energy......................................................................................................................................... 60 Einstein’s Box ..................................................................................................................................... 60 Mass and Potential Energy................................................................................................................