RELATIVITY Einstein published two theories of relativity In 1905 The Special Theory For uniform motion a 0 In 1916 The General Theory For non uniform motion a 0 First we will discuss The Special Theory At the beginning of the century most believed light was a wave Thus must have something that waves Sound has air Water has water etc 2 Physicists proposed that for waves of light something must wave They called it the ether for light This ether then must fill the universe The earth moves through the universe so How fast are we traveling through the ether To answer this we will start with a race between two boats that run at exactly the same speed in a river 3 The boat going downstream will have speed c v The boat going upstream will have speed c v 4 The calculated time for round trip is L L 2 Lc t 2 2 c v c v c v The speed for the boat going cross stream will have speed both ways c v 2 2 Therefore tcross 2L v c 2 2 Then the ratio of the two times will be 5 t t cross 1 v 1 c 2 1 2 Therefore we see that the boat that goes across and back wins the race even though both boats travel the same speed relative to the water The main point is the time to complete the race is different for the two boats Note 6 We can measure ratio of times for boats and calculate the speed of river if we know the speed of boats 7 The Michelson Morley Experiment Diagram of Apparatus 8 Drawing of actual apparatus We know the speed of light We want the speed of the earth through the ether 9 When light arrives at the eye it has traveled two paths to reach the observer There will be interference either constructive or destructive The resulting image will be a series of lines 10 Spectrometer lines Since the direction of the ether flow is not known the apparatus must be rotated 11 First one than the other path will be parallel to the flow of ether Therefore the interference lines should shift Michelson and Morley did the experiment very carefully and did not find a shift The conclusion has to be that The Ether does not exist or the earth travels along with it 12 Another experiment shows that we are not moving with it Stellar Aberration is that experiment While the light travels down the telescope the telescope moves with the earth 13 The telescope has to be tilted to keep the image in the center If the ether the substance that waves to cause the propagation of light moves with the scope there would be no need to tilt the it Therefore we conclude the ether does not exist Classical Relativity The transformation equations before Einstein 14 x x vt y y z z t t These are the Galilean Equations that allow observers to compare observations in two different frames moving relative to each other with constant velocity 15 Observer on ground and observer on railroad car moving in x direction The observer on the ground observes the birds separated by distance x 2 x1 The distances are equal 16 If an airplane flies over the railroad car traveling in the x direction at a speed u x measured by the observer on the ground what will be the speed u x of the airplane measured by the observer on the railroad car We can use the transformation equation for x x x vt and the equation for t t t Differentiate and divide to get 17 dx dx dv v t dt dt dt u x u x v 0 if the velocity of the railroad car is constant If the observer on the ground measures the velocity of the airplane as ux then the person on the railroad car will measure u x 18 What if the person on the ground points a flashlight in the x direction What will be the speed of light measured by the observer on the railroad car 19 We get u ux v x giving c c v We must keep this result in mind as we discuss Einstein s Theory 20 Einstein s postulates for the Special Theory of Relativity 1 Fundamental laws of physics are identical for any two observers in uniform relative motion 2 The speed of light is independent of the motion of the light source or observer 21 These postulates cannot be satisfied using the Galilean Equations as we will see However Einstein found that the following equations worked x x vt y y z z vx t t 2 c where 1 2 v 1 2 c 22 These are the Lorentz Transformation Equations Now consider the airplane flying over the railroad car in the x direction What is the speed of the airplane as measured by the observer on the ground What is the speed of the airplane as measured by the observer on the railroad car We need to answer these questions by using the Einstein Lorenze Equations 23 x x vt and vx t t 2 c differentiate dx dx vdt and vdx dt dt 2 c divide dx dx vdt dt dt vdx 2 24 divide by dt dx v dx dt dx dt v dt 1 2 c or ux v ux vu x 1 2 c Thus if an object an airplane flies over the railroad car the observer on the ground will 25 measure the speed in the x direction as observer on the car will find ux The u x What about the speed of light when a flashlight is pointed in the x direction The observer on the ground points a flashlight in the x direction What will be the speed of light measured by the observer on the car c v c v c v u c vc v c v c 1 2 1 c c x 26 Both observers even though they are moving relative to each other measure the same value for the speed of light This is in agreement with the Second Postulate LENGTH CONTRACTION Read the section on Length Contraction in the book We will do it a little differently 27 The observer on the moving railroad car has a rod moving with him He measures the length of the rod to be x x L0 2 1 Use the Lorentz equations to get x2 x2 vt 2 x1 x1 vt1 Then putting these in the equation 28 L0 x2 vt2 x1 vt1 L0 x2 x1 v t2 t1 If the observer on the ground measures the far end and near end of the rod at the same time t1 t2 Then L0 x 2 x1 L or L L0 and 1 29 So the observer on the ground with the rod moving past in the x direction measures the rod to be shorter than what is measured by the observer at rest relative to the rod and on the car Length Contraction is a prediction of the …