Transcript Document
Relativity PHY232 Remco Zegers [email protected] Room W109 – cyclotron building http://www.nscl.msu.edu/~zegers/phy232.html An argument between Newton and Einstein Newton is standing along a road. Einstein is passing by in a car driving with a speed of 10 m/s. They later meet in a pub, and Newton congratulates Einstein with achieving a speed of 10 m/s. Modest Einstein answers that he was not moving with 10 m/s at all: he thought Newton was moving with a speed of 10 m/s, in the opposite direction and congratulates him. A long argument follows, which is only settled after many beers. What did they agree on in the end? a) Newton was right that the car was moving at a speed of 10 m/s b) Einstein was right that Newton was moving at a speed of 10 m/s c) neither were right d) both were right e) that they had too many beers For the record: Newton: 1643 – 1727 Einstein: 1879-1955; the above encounter is entirely fictional…unless…see later PHY232 - Remco Zegers - Relativity 2 relative motion 70 mph 20 mph Which train is having what speed, relative to what? In what ways can one distinguish between the 2? PHY232 - Remco Zegers - 3 If light is a wave… …then what is oscillating? waves in water: water molecules move up and down Transverse waves sound: molecules in air move to and fro (no sound in vacuum!) Longitudinal waves light waves: ??? let’s call it the ‘ether’ Transverse or Longitudinal?? PHY232 - Remco Zegers - 4 properties of ‘ether’ • It must be a very tenuous gas (doesn’t interfere with our movements) • It must be very rigid (speed of light is very large) … possible??? against common sense, but… we need something to measure motion against. Something must be the absolute frame of reference to measure motion against: • earth? • sun? • Galaxy? • ETHER! (mid 19th century) PHY232 - Remco Zegers - 5 Measuring the ether wind… Our postulates: • So now we have ether as our fixed reference frame • Light is a transverse oscillation of the ether material light vearth vlight,ether ether vlight,measured on earth=vlight,ether+vearth PHY232 - Remco Zegers - 6 Michelson Morley experiment PHY232 - Remco Zegers - 7 ether & Michelson-Morley ether wind c: speed of light v: ether wind speed v tf=L/(c+v) + L/(c-v) = 2Lc/(c2-v2) tm=L/vup + L/vdown vup=vdown= (c2-v2) c vup tm=2L/ (c2-v2) PHY232 - Remco Zegers - 8 ether wind ether wind? c: speed of light v: ether wind speed 1 tf/tm=c/(c2-v2)= 2 v 1 2 c if v<c then tf>tm corresponding virtual path length ratio (use x=vt) 1 df/dm= v2 1 2 c Path length difference creates interference pattern! PHY232 - Remco Zegers - 9 ether wind? If ether would exist… tf>tm tf=tm The interference pattern would change when setup is rotated! DOES NOT HAPPEN!!! so ether does not exist PHY232 - Remco Zegers - 10 Wow!! There is no ether, so there is no way to determine absolute motion All motion must be considered relative to some object arbitrarily taken at rest. Any reference object (reference frame) can be taken: no object/frame is more at rest than other Einstein PHY232 - Remco Zegers - 11 Einstein’s postulates 1. All laws of physics are the same in all frames of reference moving with constant velocity relative to each other (whatever you try to measure, the result is the same independent on the frame of reference) 2. The speed of light is constant (2.9979245x108 m/s) in all inertial reference frames. Postulates of Special relativity: no acceleration involved. A massive object cannot move faster than the speed of light This has some important (and weird) consequences… PHY232 - Remco Zegers - 12 train paradox lightning strikes two ends of a fast moving train simultaneously, when seen by an observer standing along the railroad. However, the person in the train sees the light come in at different times and thus thinks the times were different. Who is right? a) observer along the railroad b) observer in the train c) neither d) both PHY232 - Remco Zegers - 13 simultaneous? Newton: ‘Absolute, true and mathematical time, of itself and from nature, flows equably without relation to anything external’ (time is absolute) Einstein: Simultaneity is not an absolute concept but one that depends on the state of motion of the observer. (time is relative) PHY232 - Remco Zegers - 14 Time dilation d vtB A is sitting in a train moving with velocity v. She shines a light toward a mirror on the roof (height d). The reflected light takes tA=2d/c to get back to A. An observer (B) outside the train sees A moving with v. The distance traveled by the train until the reflected light returns is vtB PHY232 - Remco Zegers - 15 ctB/2 d vtB/2 Time dilation ct B vt B 2 d 2 2 2d 2d t B t A c v2 c 1 2 c t A 1 t B t A 2 2 v v 1 2 1 2 c c 2 PHY232 - Remco Zegers 2 - 16 Time dilatation A clock moving past an observer at speed v runs more slowly than an identical clock at rest with respect to the observer by a factor of 1/. v 1 c 1 2 tp: proper time is the time interval between two events as measured by an observer who sees the two events occur at the same positions. Or in other words, is in the same frame of reference Note: this equation can only be applied in a frame of reference that is not accelerating (else Einstein’s postulates of special relativity do not hold) PHY232 - Remco Zegers - 17 question A person on a far-away planet X takes 5 hours to read a book. If earth moves with a velocity of 0.9 times the speed of light relative to the planet, how long does the reader take to finish the book when viewed by an observer on earth? Step 1) In which frame is the proper time measured? a) in the frame of the reader on planet X b) in the frame of the person on earth The ‘events’, starting and finishing the book occur at the same place for the person on planet X, so he measured the proper time Step 2) Use the time dilation equation: t=(5 hours)/(1-[0.9c/c]2)=11.47 hours PHY232 - Remco Zegers - 18 twin paradox Two identical twins, Newton and Einstein, live on earth and are both 20 years old. Einstein decides to settle on planet Y. He travels there by spaceship with an average speed of 0.95 times the speed of light. As measured by a clock in his spaceship it takes 5 years. Upon arrival he feels homesick and returns immediately at the same average speed. Which of the following is correct? a) Upon his return, Einstein has aged 10 years and Newton has aged x10 years=10 years/(1-[0.95c/c]2)=32 years b) Upon his return, Einstein has aged 10 years and Newton has aged 10/ years=10 years/(1-[0.95c/c]2)=3.1 years c) both are 20+32=52 years old d) both are 20+10=30 years old The proper time is in the frame of Einstein. Note that since Einstein is at certain times accelerating/decelerating, the time dilation equation cannot be applied by Einstein, even though from his point of view Newton is traveling at 0.95c. PHY232 - Remco Zegers - 19 Lorentz-Fitzgerald length contraction A person A measures the length of a rope to be 10 m Lp=10 m (we call this the proper length Lp, proper meaning that the rope is in the same reference frame as the person). A space ship passes by with v=0.9c. How long is the rope according to a person B in the space ship? 1. According to A, it takes the ship t=Lp/v to get from one end of the rope to the other 2. For B, this time is reduced to tp= t/ (dilation) proper time because in frame of B PHY232 - Remco Zegers - 20 Lorentz-Fitzgerald length contraction L=4.4 m Lp=10 m 3. So, according to B, the length of the rope is: L=vtp= vt/=Lp/=Lp(1-v2/c2) L=Lp x 0.44 = 4.4 meter The length of an object measured in a frame moving with respect to the object is less than the proper length. L=Lp/ Length contraction only takes place along direction of motion PHY232 - Remco Zegers - 21 slow question L=Lp/ When passing by very slowly (v<<c) the length of a space ship is 10 m as observed from the ground. What is its length (as observed from the ground) if the ship has a velocity of 0.9c (=2.27) ? a) L=Lp/=10/2.27=4.4 meter c) L=Lp=10m b) L=Lpx=10x2.27=22.7 meter L=Lp/=10/2.27=4.4 meter The proper length is now in the frame of the ship! PHY232 - Remco Zegers - 22 loncapa now do questions 1,2,3,4,5 of set 10 note: a light year is the distance traveled by light in one year. It is ‘just’ another measure of distance like miles or kilometers. PHY232 - Remco Zegers - 23 Relativistic addition of velocities A person is walking in a moving train. The train moves with a speed of 10 m/s to the right, and the person walks with a speed of 2 m/s to the right, relative to the moving train. You are standing on a platform in the station. The speed of the person, from your point of view is: a) 8 m/s b) 10 m/s c) 12 m/s • A spaceship is passing by you with a velocity of 0.8c. It shoots a rocket in the same direction as the moving ship, which according to the pilot of the ship, has a velocity of 0.6c. What is the velocity of the rocket from your point of view? a) v=0.2c b) v=0.8c c) v>0.8c but v<c d) v=c e) v=1.4c if we simply add 0.6c+0.8c, the velocity is larger than c, which is impossible. PHY232 - Remco Zegers - 24 Relativistic addition of velocities frame d is moving in the +x direction relative to frame b with a velocity vdb. The velocity of an object a is measured in frame d to be vad. Then the above equation gives the velocity vab of a in the frame b. Note that if vad and vdb are small, vab=vad+vdb which is the common equation for relative motion. PHY232 - Remco Zegers - 25 Relativistic addition of velocities • A spaceship is passing by you with a velocity of 0.8c. It shoots a rocket in the same direction as the moving ship, which according to the pilot of the ship, has a velocity of 0.6c. What is the velocity of the rocket from your point of view? a: rocket d: frame of reference of spaceship b: your frame of reference vad=0.6c vdb=0.8c vab=(0.6c+0.8c)/(1+0.6cx0.8c/c2)=0.946c PHY232 - Remco Zegers - 26 Doppler effect: a non-moving source v source f=v/ you The velocity v (say of light or sound) is fixed PHY232 - Remco Zegers - 27 doppler effect: a source moving towards you the distance between the wave front is shortened vsource vsound vsource f f f vsource source v f you vsound f vsound vsource prime’: received observable The frequency becomes larger: wavelength smaller PHY232 - Remco Zegers - 28 doppler effect: a source moving away from you the distance between the wave front becomes longer vsource vsound vsource f f f vsource you v f source vsound f vsound vsource prime’: received observable vsource negative The frequency becomes lower: wavelength higher PHY232 - Remco Zegers - 29 applications of the doppler effect: speed radar v vobserver f f v vsource v f vv approachin g car PHY232 - Remco Zegers - 30 lon-capa now do question 6,7,10 from set 10. PHY232 - Remco Zegers - 31 relativistic energy and momentum We have seen that Newtonian laws for motion do not hold at relativistic energies. The equations for momentum, energy and kinetic energy must also be modified. momentum: an important conclusion by Einstein was that energy and mass are equivalent: The total energy of an object is given by: The kinetic energy is the total energy of an object minus its rest mass (energy): (use in question 12) By combining: PHY232 - Remco Zegers - 32 relativistic protons v c 1 1 2 a proton (rest mass of 938 MeV/c2) is accelerated over a potential difference of 2x107 V. what are: a) the kinetic energy of the proton b) the velocity of the proton c) the total energy of the proton. a) the energy given by the potential is qV=1.6x10-19 x 2x107 J=3.2x10-12 J Note: 1eV = 1.6x10-19 J so kinetic energy is also: 2x107 eV=20 MeV b) so =1+Ekin/(mc2)=1+Ekin/Erest=1+20/938=1.021 =(1-1/2)=0.2 so v=0.2c c) E=Ekin+Erest=20 MeV+938 MeV=958 MeV = 1.53x10-10 J PHY232 - Remco Zegers - 33 lon-capa now do questions 8, 9, 11, 12 of loncapa 10. Note: for question 9: you need a calculator that can handle many digits. If yours doesn’t do that (like mine) one option is the standard windows calculator, but be sure to set it to ‘scientific view’. PHY232 - Remco Zegers - 34 Einstein’s General relativity person in a free-falling elevator feels similar (weightless) to a person in a rocket far away from any planet (gravitational field) person in an accelerating rocket feels similar (same weight) as a person standing on a planet The force of gravity is the acceleration you feel when you move through space-time… PHY232 - Remco Zegers - 35 space-time This has as a consequence that a ray of light would bend in a gravitational field (observed!!). PHY232 - Remco Zegers - 36 postulates of general relativity 1. All the laws in nature have the same form for observers in any frame of reference (accelerated or not). 2. In the vicinity of any given point, a gravitational field is equivalent to an accelerated frame of reference without a gravitational field (principle of equivalence). The gravitational effect at a certain point is given by the so-called curvature of space time… PHY232 - Remco Zegers - 37 curvature of space time… Masses produce a curvature in space-time (which would otherwise be flat). Smaller masses (earth) follow the curvature of larger masses (sun). PHY232 - Remco Zegers - 38 really?? Strong gravity makes time run slower… boulder (Colorado) 1 mile Washington Atomic clocks have an error of 1 in 1014 (1 s in 3 million year) After 1 day, an atomic clock in Boulder runs faster by 15 ns (15x10-9s) than an atomic clock in Washington. This difference is 17 times larger than the error!! Gravity is slightly different: time is different! Important for satellites (GPS systems!!) PHY232 - Remco Zegers - 39