Special Theory of Relativity

1. Special Theory of Relativity

2. 1900 : Science Has Explained Nature • Physics at the end of the Nineteenth Century was presumed to have explained most of the major aspects of nature.

3. Newtonian Mechanics • Newton’s Laws explained the motions of objects on earth and in the heavens. They also formed a basis for explaining fluids, wave motion and sound.

4. Kinetic Theory • Kinetic theory explained the behaviour of gases and other materials.

5. Maxwell’s Theory of Electromagnetism • Maxwell’s Electromagnetic Theory explained magnetism and electricity. It also explained how electromagnetic waves would behave as light does, so light was thought of as electromagnetic waves.

6. The Crystal Palace : Nineteenth Century Pride • The Great Exhibition of 1851 featured all the latest inventions, new technologies produced by Science, new art works along with copies of ancient art masterworks.

7. The Crystal Palace : Nineteenth Century Pride • The Great Exhibition was a giant comparison of the work of Ancient man and Modern man. Modern man felt that civilization was progressing – witness the huge cast iron and glass building that no ancient had ever conceived or built.

8. Classical Physics • The ideas of physics up to 1900 is called Classical Physics and the new ideas (relativity and quantum theory) to come after 1900 are called Modern Physics.

9. Relativity: Inertial Reference Frames • An inertial reference frame is something in which Newton’s First Law (Law of Inertia) is valid. In an inertial reference frame, objects remain at rest or in uniform motion if the force on the object are balanced. Inertial reference frames are standing still or moving steadily, NOT accelerating. • A pickup is carrying a bowling ball and slows down for a stop sign. Newton’s first law appears to be violated using the truck as a reference frame but not if earth is reference

10. The Relativity Principle • The relativity principle is that the basic laws of physics are the same in all inertial reference frames. • If a coin is dropped in a steadily moving car it will appear to drop down with the car as a reference. But to a stationary observer on the earth, the coin would appear to move in a parabolic path. Yet the laws of motion in the car reference frame and the earth reference frame are the same laws of motion.

11. Absolute Quantities • Absolute quantities are those which remain the same in all frames of reference. In classical mechanics, space and time are viewed as absolute. But an object’s position and velocity varies from one reference frame to another. An object’s acceleration, however, is the same (absolute) from one reference frame to another.

12. Comparing Inertial Reference Frames • Since all the laws of physics are the same in inertial reference frames, it follows that all inertial reference frames are equally valid. This is to say that no one inertial frame of reference is special in any sense. When travelling in a steadily moving car it is just as valid to say that the car is standing still and the earth is moving beneath the car as it is to say that the car is moving and the earth is standing still.

13. Which Reference Frame is Really “At Rest”? • There is no experiment which can show which reference frame is really at rest so no single inertial reference frame is more valid than any other.

14. The Speed of Light : What Reference Frame? • When Maxwell presented his theory of electromagnetism, he showed that light could be considered an electromagnetic wave that he predicted would travel at 3 x 108 m/s. Actual experiments showed nearly the same speed, within experimental error. But the question was, What should be the reference frame for the speed of light?

15. Ether, The Medium For Light Waves • Waves require a medium to propagate so if light was electromagnetic waves, what was its medium? Scientists hypothesized that the medium for light waves was a transparent substance that they called ether. The speed of light was hypothesized to be relative to the ether (an inertial reference frame).

16. Luminiferous Aether’s Extraordinary Properties • Based on Maxwell’s theory of light, the luminiferous ether (aether) had to have the following properties: It was massless, is was nondispersive, it was incompressible, it was totally transparent, and it was continuous everywhere.

17. Problems With Maxwell’s Equations • Maxwell’s equations did NOT appear to satisfy the relativity principle. They changed for different reference frames. They were the simplest in the frame where the speed of light, c, was 3.00 x 108 m/s, in the reference frame of the ether at rest. The ether reference frame seemed special.

18. Experiments Devised to Detect Motion in the Ether • A number of experiments were devised to detect earth’s speed relative to the ether. A team of experimenters, Albert A. Michelson and Edward W. Morley, conducted a number of experiments in the 1880’s to measure the speed of the earth relative to the ether.

19. A Boat Analogy for the Michelson-Morley Expt. • A boat has a velocity of 5 m/s relative to still water. A 500 m wide river is flowing with a velocity of 3 m/s [E]. A boat makes a trip straight across the river and back (To do this the boat must angle into the current both ways). Then the boat makes a trip downstream 500 m and back 500m upstream to its starting point. How much time is required for each trip?

20. The Boat Velocity [N] When Crossing the River • To determine the boat velocity north, the vectors for the river and boat need to be connected so that the same objects coincide. The boat is on the tip of an arrow 5 in length in any direction with water on its tail. The water vector has water on its tip and earth on its tail, pointing in the east direction. The boat wants to go north relative to the earth which leads to the following diagram. boat earth is where 4 water and 3 intersect Thus the velocity of the boat north is 4 m/s

21. Calculating Times for Both Trips • From the velocity formula, v = d/t, t = d/v. To cross the river takes t = 500m/4m/s = 125 s. To return also takes 125 s for a total of 250 s. • To go downstream the v = 5 + 3 m/s (8 m/s) and to go upsteam the v = 5 – 3 m/s (2 m/s). Thus the time to go downstream is 500m/8m/s (62.5 s) and the time to go upstream is 500m/2m/s (250 s). So, the total time to go downstream and back upstream to the start point is 312.5 s

22. Summary of the Boat Analogy • If two boats, each traveling at the same speed relative to the water, were traveling perpendicular to each other – one across the current and the other with and against the current, the boats would arrive back at different times.

23. The Michelson Morley Experiment (1887) • As the earth moves through the ether in space, the effect is to create an “ether wind” in some direction (much like a river current in a river). At some points on the earth’s orbit, the current would be in different directions and at different velocities.

24. The Michelson Morley Experiment • If beams of light were sent out at right angles, then reflected back, they should arrive back at different times at some point in the earth’s orbit.

25. The Michelson Morley Experiment • Light beams arriving back at different times should be out of phase. This would cause a shift in the interference pattern that could be measured and used to determine the velocity of the ether wind or of earth relative to the ether (The larger the ether wind velocity, the greater the time delay).

26. Michelson-Morley Experiment : A Null Result • Michelson and Morley’s device (called an inferometer) made of a solid block of granite floating on a pool of mercury in a basement of a building virtually eliminated the effect of vibrations and allowed the device to be turned in various directions. Running their experiment hundreds of times, they found absolutely no evidence of any shift – light always came back at exactly the same time. This non-result is called a Null result.

27. How to Explain the Null Result • In the 1890s, H. Lorentz and G. Fitzgerald proposed that the arm of the interferometer pointing into the ether wind contracted by a factor of the square root of (1 - v2/c2). Since the arm parallel to the ether wind shortened, it allowed the light beam moving parallel to the ether wind to return at the same time as the light beam traveling perpendicular to the ether wind. • ether wind direction

28. Another Possible Explanation for the Null Result • If there was no evidence of an ether wind, perhaps the ether was still relative to the earth, suggesting that maybe the earth and the ether were special, perhaps at the centre of the universe like the Greeks thought! This explanation seemed to fly in the face of Newton and Science since his time. The idea that the earth was special in the universe in that for all the objects in the universe, only the earth was still relative to the ether, seemed a remote possibility.

29. Einstein Gave Today’s Explanation for the Null Result • Albert Einstein used many “gedanken” (thought experiments) to reason what must be true. He asked himself “What would I see if I rode a light beam?” The answer would be that he would see alternating electric and magnetic fields at rest whose magnitude changed in space, but did not change in time.

30. The Speed of Light Can Not Be Reduced to Zero • Since oscillating magnetic and electric fields at rest have never been detected and were not consistent with Maxwell’s theory, Einstein concluded it was unreasonable to think of light’s speed being reduced to zero or for that matter to be reduced at all. It must always (in space) be travelling at 3.00 x 108 m/s. This explained the Null Result of the Michelson Morley Experiment.

31. Einstein’s Special Theory of Relativity (1905) • In his 1905 paper, Einstein proposed two postulates (Things to be taken as self-evidently true, not requiring proof) • 1. The Laws of Physics have the same form in all inertial reference frames (the relativity principle) • 2. Light propagates through space with speed c (3.00 x 108 m/s), independent of the speed of the source or the observer.

32. Special and General Relativity Einstein produced two theories of relativity. His Special Theory of Relativity dealt with inertial reference frames (non-accelerating) while his General Theory of Relativity dealt with relativity effects for accelerating reference frames.

33. The Relativity of Simultaneity • In one reference frame, two simultaneous events (happening at the same time) may be seen to happen at different times in another reference frame (moving with respect to the first reference frame), especially if the events happen at great distances from each other and the moving reference frame is travelling at a high speed.

34. The Relativity of Simultaneity

35. The Relativity of Simultaneity • A light flash is sent from A to mirrors at p and q. For observer A in a reference frame moving relative to observer B, p and q reflect the light back at the same time. But for observer B, the light hits mirror p before it hits q. One observer sees the events as simultaneous but the other does not.

36. Relativity Time Dilation • Relativity predicts (and experiments confirm) that time runs slower for objects moving at high speeds, especially speeds close to the speed of light.

37. A Thought Experiment (Gedanken) For Time Dilation • An astronaut in a moving space ship observes the time it takes a beam of light to be sent to a mirror and bounce back. An observer on earth sees the same light beam travelling the longer red path.

38. A Thought Experiment (Gedanken) For Time Dilation • Since the speed of light remains the same for both reference frames and since the distance that the light travels for the earth observer is about 6X the distance for the astronaut, the time that the earth observer records for the light travel is about 6X what the astronaut records. Note: vlight for astr. = v light for earth obs. . Since v = d/t, then 1d/1t = 6d/6t .

39. Time Dilation Summary and Equation • Clocks moving relative to an observer are measured by the observer to run slowly (Time is measured to pass more slowly in any moving reference frame compared to your own). The time dilation equation, adjusting for relativity, is: • The to in the equation is called the proper time. Proper time is the time recorded in the reference frame in which the two timed events occur at the same point in space.

40. A Sample Calculation of Time Dilation • If a person travelling at .98 c records a time of 1 year, what time would an earth observer record? • t = 1 y/√(1 - [.98c/c]2) • t = 1 y/√(.0396) • t = 5.025 y

41. The Twin Paradox • One person of a set of twins leaves on a rocket travelling close to the speed of light, returning many years later to find his earthbound twin much older. But from the perspective of the rocket twin the earth twin was moving away from his stationary rocket and so the rocket twin expects to find the earth twin much younger and himself older. But, both of these can not be true so which twin is older?

42. The Twin Paradox • Only the observer in the inertial reference frame (non-accelerating) will make accurate observations. The rocket twin accelerates to start his journey and accelerates to turn around so his reference frame is not an inertial reference frame where the earth twin is in a close-to-inertial reference frame and thus makes accurate observations.

43. The Earth Twin Makes True Observations • The rocket twin comes back younger while the earth twin is older due to time dilation. In the dilation equation, place the age of the earth twin in t and solve for to . This will produce a smaller value than t, the correct result.

44. Sample Problem • What will be the mean lifetime (average lifetime) of a muon as measured in the laboratory if it is travelling at v = .60 c with respect to the laboratory? Its mean life (birth to death) at rest is 2.2 x 10-6 s.

45. Length Contraction • The length of an object is measured to be shorter when it is moving relative to the observer than when it is at rest. This shortening or contraction of length is in the direction of the motion and greater as the speed approaches c. No vertical shortening is observed.

46. Why Length Contraction is Observed • For an earth observer, a spaceship is moving with velocity v between two planets separated by a distance of Lo while from the spaceship’s frame of reference it is standing still while the planets are moving with the same velocity v. The earth observer measures a shorter time for the moving spaceship so the distance must be measured shorter by the same factor since the velocity remains the same. From the spaceship’s reference, time is shortened in the planet system so its length must contract to L. Since v = d/t and v ship = v planets ,so .5d/.5t ship = v (planet’s reference frame) and .5d/.5t planets = v (ship’s reference frame)

47. Relativistic Length Contraction Formula • In the formula, d (NOT d‘ ) is the proper distance, the distance endpoints measured at the same point in space (in the moving object’s reference frame).

48. Sample Problem for Length Contraction • A rectangular painting measuring 1.00 m tall and 1.5 m wide is hanging on a side wall of a spaceship moving past earth at a speed of ,90 c. What are the dimensions of the picture according to the spaceship captain? What are the dimensions of the picture as seen by an earth observer?

49. The Appearance of Objects When Travelling Near c • Due to the high speed, buildings will look narrower but the same height. The side of the building will be visible even when viewing the building from the front.

50. Four-Dimensional Space-Time • Imagine a person in a train moving at .65c. He eats his meal in 15 minutes but an outside observer records that it takes 20 minutes. The meal is served on a 20 cm plate but an outside observer records the plate as 15 cm. For the outside observer, the plate has shortened while the time has lengthened. Going from the train reference frame to the outside reference frame, its as if some space (one of its dimensions-length) has been transformed into time.