Relativity. Physics April 2002 Lecture 8. Einstein at 112 Mercer St. 11 Apr 02 Physics 102 Lecture 8 1

Transcription

1 Relativity Physics April 2002 Lecture 8 Einstein at 112 Mercer St. 11 Apr 02 Physics 102 Lecture 8 1

2 Physics around 1900 Newtonian Mechanics Kinetic theory and thermodynamics Maxwell s equations There were a few problems but many thought these would be resolved using known principles. There is nothing new to be discovered in physics now. All that remains is more and more precise measurement Lord Kelvin 11 Apr 02 Physics 102 Lecture 8 2

3 Measuring the speed of light The first measurement of c was made by Rømer (1676) by studying the moons of Jupiter. Fizeau made first terrestrial measurement in Foucault, then Michelson, measured with rotating mirror. Small displacement due to mirror motion d Fixed mirror Measurement arm of length D lens Storage arm of length L Rotating mirror θ = ω t During time light bounces between mirrors, the rotating mirror turns slightly 11 Apr 02 Physics 102 Lecture 8 3 m storage

4 The Luminiferous Æther If light is a wave, in what does it propagate? What properties must ether have? Fills all space (even vacuum and intermolecular space) Doesn t hinder motion Perfectly elastic Infinitely diffuse The existence of such a medium is now universally assumed by physicists. Millikan and Gale, A First Course in Physics, Apr 02 Physics 102 Lecture 8 4

5 The Michelson-Morley Experiment A. A. Michelson (later with E. W. Morley) set out to measure the aether wind in Imagine a plane flying 1000 miles round trip at 500 miles/hr With no wind, time is given by: 500 miles 500 MPH 500 miles + = 1+ 1= 2hrs 500 MPH With 100 mile/hr wind, time is given by: 500 miles 600 MPH 500 miles + = = 208. hrs 400 MPH If you knew the distance, you could time the flight and find whether or not there was a wind. 11 Apr 02 Physics 102 Lecture 8 5

6 Michelson Interferometer The Earth s motion through the aether is analogous to traveling in a wind. We measure the path length difference with an interferometer. Earth s v A In B Beam splitter Out Mirror With an aether, the time delay in the two arms is different and thus the interference condition is different. If initially there were no output due to destructive interference we might get constructive interference. Result of experiment: Output is independent of Earth s motion thus there is no aether. 11 Apr 02 Physics 102 Lecture 8 6

7 The Postulates of Special Relativity The laws of physics are the same in every inertial reference frame. The speed of light (in a vacuum), measured in any inertial reference frame, always has the same value c no matter how fast the source and observer are moving relative to each other. 11 Apr 02 Physics 102 Lecture 8 7

8 LIGHT PULSE LAB FRAME 11 Apr 02 Physics 102 Lecture 8 8

9 LIGHT PULSE ROCKET FRAME 11 Apr 02 Physics 102 Lecture 8 9

10 Speed of light 11 Apr 02 Physics 102 Lecture 8 10

11 Implications of Special Relativity Space and time are not absolute: measurements depend on the observer s reference frame. Moving clocks run slow Moving rulers are foreshortened in direction of motion Moving masses increase Nothing can travel faster than the speed of light in a vacuum, and only massless particles can travel that fast. Mass and energy can be transformed, one into the other. 11 Apr 02 Physics 102 Lecture 8 11

12 A gedanken experiment. Imagine we construct a clock in which we time the interval between when we send out a flash and when it is received. D f The time for a tick would be: A. B. C. D. E. t D c 0 = / t c D 0 = / t = D/ c 0 2 t = D/ c 0 2 t = 2c/ D 0 11 Apr 02 Physics 102 Lecture 8 12

13 now imagine our clock is moving with velocity v such that it covers a distance 2L in time t, that is: D 2L = v t E. The time for a tick, as measured by us, is: A.The same as before because the speed of light is the same in all frames. B. C. D. t = 2 L/ c 2 2 / t = 2 D + L c t = 2 D/ v t = 2 ( D/ c) + ( L/ v) Apr 02 Physics 102 Lecture 8 13

14 To summarize: Clock at rest with respect to us: t t = D/ c 0 2 = t 0 1 ( v/ c) 2 Clock moving with velocity v with respect to us: tc = 4D + 4L t 2 2 t = 2 D + L / c = 4D + v t 2D/ c = 1 ( v/ c) Apr 02 Physics 102 Lecture 8 14

15 Moving clocks As 1-v 2 /c 2 is always less than 1 in the expression t = t v c the time between ticks for the moving clock,, is always longer than the time between ticks for the clock that is at rest with respect to us,. Moving clocks run slower. This has been measured with atomic clocks on planes. It holds for biological clocks. It s important for the GPS system. is called the proper time. 11 Apr 02 Physics 102 Lecture

16 Atmospheric Muons Muons (electron-like particles) have a typical life of only 2.2 microseconds in the lab. At the speed of light, a particle moves 2/3 km in 2.2 µsec Muons produced at the top of the atmosphere (10 km) nevertheless reach the ground. 11 Apr 02 Physics 102 Lecture 8 16

17 A. Muons decay into different particles that can make it to the Earth s surface. B. When moving near the speed of light the distance you cover is no longer given by d C. In the muon s frame, our clock runs faster and so the distance is shorter. D. In our frame, the muon s clock runs slower and so it has longer to traverse the atmosphere. = v t 11 Apr 02 Physics 102 Lecture 8 17

18 .more muons They are moving at 0.999c and moving clocks run slower! lifetime in Earth bound frame is 22. µ s = 49. 2µ s Thus they can travel 15 km! Note: muon decay is statistical: Nt () t / τ = N0 e τ = lifetime. 11 Apr 02 Physics 102 Lecture 8 18

19 We can define a proper length in the same manner as we defined proper time. It is the distance one measures when the object is at rest. Let s say we have a stick that is meters long and we assess its length by timing how long it takes a rocket traveling at velocity v to go by it. We find: A., where is the time interval for the moving clock. B., where is the time interval on a stationary clock C. D. l v t 0 = l = v t 0 0 t t 0 l0 = c t0 l = v t/ 1 ( v/ c) 0 l Apr 02 Physics 102 Lecture 8 19

20 Now imagine that you are on a rocket ship traveling with velocity v and moving by the same stick. The length you measure, call it, will be: A. B. C. D. l = v t l = v t 1 ( v/ c) 0 2 l = v t 0 l = v t/ 1 ( v/ c) l 2 11 Apr 02 Physics 102 Lecture 8 20

21 Combine these two ways of looking at the stick... Standing on Earth timing the passage of a rocket with the moving clock: l = v t 0 l = v t 1 ( v/ c) = l 1 ( v/ c) 0 Standing in the rocket timing how long it takes you to go by the stick. 11 Apr 02 Physics 102 Lecture l = v t 0 Moving objects appear shorter! 2 Plug in time relation from before

22 Relativistic momentum In special relativity, the conservation laws which we have covered still hold true as long as we redefine momentum and energy. The relativistic momentum (which is conserved) is mv p = mv (for v << v c c) Note that if v exceeds c the momentum is not defined. We can think of this as an increase in the mass of the particle when it moves fast. 11 Apr 02 Physics 102 Lecture 8 22

23 Relativistic energy Mass and energy are equivalent and are not independently conserved. The total energy of an object (not including potential, electrical. etc.) is: 2 mc E = = E v c 0 E K At v=0, an object has its rest mass energy: This means that m kg of mass is equivalent to mc 2 Joules of energy. 11 Apr 02 Physics 102 Lecture 8 23

24 Newton was wrong Some misconceptions Classical mechanics concerned itself with only the low velocity limit. Newtonian mechanics is a subset of special relativity. Nothing travels faster than light Material particles and information cannot travel faster than light. However, some types of waves do and objects can appear to travel faster than light. A cosmic reference frame cannot exist The distant stars define a cosmic frame. However, the laws of physics are still the same in all inertial frames. 11 Apr 02 Physics 102 Lecture 8 24

25 Cosmic Microwave Background Dipole This shows that we are moving with respect to a cosmic reference frame at 200 km/sec. 11 Apr 02 Physics 102 Lecture 8 25

26 General Relativity: Gravitation In special relativity, space and time are the (four dimensional) scaffolding upon which events occur. In the general theory of relativity, space and time become an active part of physics, and the curvature of spacetime is identified with gravity. Matter tells space how to curve, space tells matter how to move. Because energy and mass are equivalent, a hotter object-- with lots of thermal energy-- has greater gravitational pull than a cold one. Predictions: planetary orbits not quite elliptical; bending of light near the Sun; redshifting of light by gravity, 11 Apr 02 Physics 102 Lecture 8 26

27 First tested prediction of Einstein s theory: 1919 Bending of light An Einstein ring 11 Apr 02 Physics 102 Lecture 8 27

28 Cluster Gravitational Lens 11 Apr 02 Physics 102 Lecture 8 28

29 A black hole in a distant galaxy (scale of picture: half a million light years across!) 11 Apr 02 Physics 102 Lecture 8 29

30 Gravitational Waves Just as accelerating electric charges produce electromagnetic waves, accelerating masses produce gravitational waves (spacetime ripples) Hulse and Taylor proved the existence of these waves The Laser Interferometer Gravitational-wave Observatory (LIGO) will try to detect them directly. 11 Apr 02 Physics 102 Lecture 8 30