1Chapter 35Special Theory of Relativity22Motion is Relative• Motion is always described with respect to a reference frame• Therefore, one classically expects that the speed of light will change when measured against different reference frames• Is there a universal reference frame that all motion can be referenced against?33Michelson-Morley Experiment• Physicists speculated that there was a universal reference frame and light propagated on this frame– This frame permeated all space and was called the luminiferous aether• The Michelson-Morley experiment tried to measure the effects of this reference frame– There should be a travel time difference for light headed into the aether and light moving against the aether• There was no detected travel time difference– There is no universal reference frame– The speed of light is constant in every frame44The Luminiferous Aether55The Interferometer66Postulates of Special Relativity• “All of the laws of nature are the same in all uniformly moving frames of reference”• “The speed of light in free space has the same measured value for all observers, regardless of the motion of the observer; that is the speed of light is constant”77Simultaneity• Two events are simultaneous if they happen at the same time• A person inside a moving ship will see light hit the front end of his ship at the same time it hits the back• A person at rest outside the ship will the see back of the ship move toward the light and the front end away. This observer will see the light hit the back of the ship first!• “Two events that are simultaneous in one frame of reference need not be simultaneous in a frame moving relative to the first frame”88Space Time• We can define the location of an object with three numbers: x, y, and z• However, because space does not look the same in all reference frames, time will also appear different in different reference frames• Because of these differences, space and time are bound together so that the speed of light is always the same– We can treat time like a fourth dimension and together with space we call this space time99Time Dilation• The speed of light can only be constant if time in different reference frames is different• Consider a moving light clock and a light clock at rest– To an observer at rest, it takes light more time to go “tick-tock” when the clock is moving than when it is at rest:– But to an observer in the same frame as the moving clock, it does not take as long. Therefore, time is relative!vtocto2201cvtt−=1010Addition of Velocities• The rule for adding relativistic velocities:• Suppose we observe a rocket ship is moving at 0.5c away from us. The ship then boost is velocity 0.5c in its reference frame in the same direction. Adding the velocities together using the relation above gives us 0.8c instead of the expected c– No object can travel faster than the speed of light!• Now suppose the ship fires a laser in its direction of motion. The ship measures the speed of the light to be c. We will also measure the speed of light to be c!– The speed of light is c no matter what!221211cvvvvV−+=1111Length Contraction• Time dilation and length contraction are two sides of the same coin:• Consider travelers leaving Earth for a star 4 light years away– Someone on Earth will see travelers going at relativistic speeds have their clocks run slower; their clock will not have ticked off 4 years when they arrive– The travelers will see their clock ticking perfectly normal; they arrive before four years because the length between them and the star contractedRest lengthLength measured as it passes by 2201cvLL −=Length will only contract on the axis of the direction of motion.1212Length Contraction0.1c 0.865c0.99c 0.9999cHere is a nice series of animations that shows an observer at rest seeing the increasing contraction of passing rocket ship as the speed of the ship increase. The closer and closer you get to the speed of light, the more and more contracted you are. Also, it is noted here that only the length in the direction of motion is contracted.1313Relativistic Momentum• Classical momentum says p=mv but v=d/t contains two factors that are affected at relativistic speeds. Therefore, we find relativistic momentum to be:• At the speed of light:– Length decreases to zero– Time increases to infinity (all of eternity would pass in an instant)– Momentum increases to infinity• So, if you were to ride on a photon (and have no mass) time would not exist and space would disappear!221cvmvp−=1414Problem Solving: Relativity• If a person at rest observes a ship traveling at 0.865c, calculate the time dilatation, length contraction and relativistic momentummvmvcvmvpLcvLLttcvttcc2211211 221121865.01)865.0(122022000220222==−==−===−==−=−1515Correspondence Principle• We must be able to reproduce Newtonian physics with any new theory• Velocities on our scales are much, much less than the speed of light, so v2/c2~0• Using velocities we commonly deal with, we do indeed reproduce classical physics:mvmvpLLLttt =−==−==−=01 01