Special Relativity and Quantum Physics Special Relativity and Quantum Physics Albert Einstein s theory of special relativity describes this phenomenon correctly It is based on two postulates 1 The Principle of Relativity Although Newtonian mechanics works very well at low speeds it fails when applied to particles whose speeds approach that of light 1 The laws of physics are the same in all inertial reference frames Experimentally the predictions of Newtonian mechanics at high speeds can be tested by accelerating an electron through a large electric potential difference For example an electron can be accelerated to v 0 99c by using a potential difference of several million volts 2 The speed of light in vacuum has always the same value of c and is independent of the motion of the observer or of the light source According to Newtonian mechanics the speed should be doubled to v 1 98c if the potential difference is increased by a factor 4 To describe a physical event it is necessary to specify a frame of reference i e a coordinate system and a clock Reference frames in which Newton s first law law of inertia is valid are called inertial frames However experiments show that the speed of the electron remains lower than the speed of light no matter how large the potential difference is Since the laws of physics are the same in all inertial reference frames there is no preferred frame and there is no experiment that can distinguish between different inertial frames Dr D Wackeroth Spring 2005 PHY102A Dr D Wackeroth Special Relativity and Quantum Physics Spring 2005 PHY102A Special Relativity and Quantum Physics 2 The Speed of Light 2 the laws of electricity and magnetism are not the same in all inertial frames There is a serious contradiction between the Newtonian addition law for velocities and the fact that the speed of light is the same for all observers If 1 is correct notions of absolute time and length are incorrect c is sent out by an observer Suppose a light pulse with speed v relative to the ground According in a truck moving with speed v to Newtonian addition of velocities the light pulse has a speed v c relative to a stationary observer v If 2 is correct a preferred reference frame must exist In the 19th century it was thought that electromagnetic waves require a medium to propagate the so called ether which was to be present everywhere even in empty space If the ether actually existed a preferred absolute frame would exist If the Sun is assumed to be at rest in the ether and v is the velocity of the Earth with respect to the ether the speed of light would then be c v downwind and c v upwind The Michelson Morley experiment was designed to detect these small changes in the speed of light Therefore either 1 the law for addition of velocities is wrong or Dr D Wackeroth Spring 2005 Its result was negative thus contradicting the ether hypothesis PHY102A Dr D Wackeroth Spring 2005 PHY102A Special Relativity and Quantum Physics Special Relativity and Quantum Physics Using the Pythagorean theorem gives 2 2 v t c t d2 2 2 3 Consequences of Special Relativity The Relativity of Time Consider a vehicle moving to the right with a speed v A mirror is fixed to the ceiling of the vehicle and an observer in the vehicle holds a flash gun a distance d below the mirror The time it takes a light flash to travel from the moving observer to the mirror and back again measured by the moving observer is Solving for t and substituting d c t0 2 one finds t p Now consider the same set of events as viewed by an observer at rest For him mirror and flash gun move with v to the right When the light strikes the mirror it has moved a distance v t 2 horizontally and d vertically Here t is the time difference as measured by the observer at rest Spring 2005 PHY102A Dr D Wackeroth Special Relativity and Quantum Physics PHY102A 4 Relativistic Momentum In order to properly describe the motion of relativistic particles Newton s 2nd law and the definitions of momentum and energy need to be modified Consider a spaceship traveling with speed v between two stars The proper distance between the stars is L0 For an observer at rest with respect to the stars the time it takes for the trip is t L0 v The space traveler measures t0 t and the distance between the stars is t L v t0 v Because L0 v t p L0 L0 1 v 2 c2 Spring 2005 Special Relativity and Quantum Physics Length Contraction The measured distance between two points depends on the frame of reference The proper length of an object is the length measured by an observer at rest with respect to the object L 1 p where 1 1 v 2 c2 In words The time interval measured by the observer in the stationary frame is longer than that measured by the observer in the moving frame i e a moving clock runs more slowly than an identical stationary clock time dilation t0 is called the proper time The proper time is always the time measured by an observer moving along with the clock Example for time dilation Lifetime of cosmic muons 2d t0 c Dr D Wackeroth t0 t0 1 v 2 c2 Requiring momentum conservation and that the relativistic momentum approaches the classical value m0 v for v c 0 one can show that the relativistic momentum is given by p p m0 v m0 v 1 v 2 c2 3 where m0 is the rest mass of the particle The rest mass is the mass measured by an observer at rest with respect to the particle 2 In words length contraction takes place along the direction of motion Dr D Wackeroth Spring 2005 PHY102A Dr D Wackeroth Spring 2005 PHY102A Special Relativity and Quantum Physics Special Relativity and Quantum Physics 6 Relativistic Addition of Velocities Imagine a motorcycle rider moving with v 0 8c past a stationary observer If the rider tosses a ball with a speed of v 0 7c in the forward direction relative to himself what is the speed of the ball recorded by the stationary observer v 5 Mass and the Ultimate Speed Einstein also showed that the mass of an object depends on the frame of reference It increases with speed according to Common sense tells us the answer is v v v 0 8c 0 7c 1 5c This must be wrong because no material object can go faster than the speed of light 4 Thus for v c objects become infinitely massive This means that an infinite amount of energy is required to accelerate an object to the speed of light m0 1 v 2 c2 m p Einstein resolved this dilemma by deriving a corrected formula for adding velocities v v v 5 BM M G 1 2 Thus the speed of light is the ultimate speed for any material object