2. IB Physics – Relativity Relativity Lesson 1 1.Galilean Transformations (one frame moving relative to another) Michelson Morley experiment– ether. 2.Speed of light constant 3.Simultaneity. 4.Inertial Observers. Frames of Reference. 5.2 Postulates of Special relativity – c is constant and all laws are the same for all inertial observers. 6.Light Clocks and how they work; Pythagorean treatment Click for good background site

3. IB Physics – Relativity Inertial frames of reference An inertial frame of reference moves at a constant velocity. It is not accelerated 0 39 6 Lightning strikes at x = 60 m and t = 3 s x = 0x = 10x = 20x = 30x = 40x = 50x = 60x = 70 y x Tsokos, 2005, p553 An Event

4. IB Physics – Relativity Absolute Rest There is no such thing as absolute rest. In an inertial frame of reference there are no experiments you can do which prove you are moving. (Unless you look outside!) Newton’s Laws consider zero and constant velocity to be identical

5. IB Physics – Relativity Galilean Transformations Relatively easy? Consider a stationary frame and a frame moving at velocity v in the x direction.

6. IB Physics – Relativity Use of the Galilean Transforms Tsokos, 2005, p553 y x velocity, v with respect to ground origins coincide when clocks at both origins show zero 0 39 6 0 39 6 vt y x 0 39 6 0 39 6 the train has moved away; when the clocks show 3 s, lightning strikes at x = 60 m Calculate and if v = 15ms -1

7. IB Physics – Relativity Calculating relative velocity A ball rolls on the floor of the train at 2ms -1 (with respect to the floor). The train moves with respect to the ground at (a) 12 ms -1 to the right and (b) 12 ms -1 to the left. Find the velocity of the ball relative to the ground. x An object rolling on the floor of the ‘moving’ frame appears to move faster as far as the ground observer is concerned y vt v v Tsokos, 2005, p553

8. IB Physics – Relativity Nature of Light Oscillating magnetic and electric fields at right angles to each other. Maxwell’s equations predict that the speed of light is independent of the velocity of the light source.

9. IB Physics – Relativity Conflict with Galilean relativity. Observer in the train measures the speed of light = C Observer on the ground measures the speed of light as C + V C V

10. IB Physics – Relativity The Michelson-Morley Experiment Read in Kirk p149 Explain how the null result from this experiment shows that the speed of light is unaffected by the motion of the Earth. Link to Michelson-Morley powerpoint Link to Michelson-Morley explanation

11. IB Physics – Relativity Modified equations

12. IB Physics – Relativity Postulates of Special Relativity The laws of physics are the same for all inertial observers. The speed of light in a vacuum is the same constant for all inertial observers Animations showing time dilation and length contraction.

13. IB Physics – Relativity Postulates of special relativity video

14. IB Physics – Relativity The laws of Physics are the same

15. IB Physics – Relativity Electromagnetism and relativity

16. IB Physics – Relativity Speed of light is constant video

17. IB Physics – Relativity Maxwells equations lead to constant c

18. IB Physics – Relativity Simultaneity Two events which happen together are simultaneous. If two events are simultaneous in one frame of reference they may not be in another. See train example in Kirk p143

19. IB Physics – Relativity Simultaneity video

20. IB Physics – Relativity B A Light clocks B A L B A B A S x = 0 x = v  t v  t

21. IB Physics – Relativity Time dilation video

22. IB Physics – Relativity Relativity Lesson 2 1.Time dilation 2.Lorentz Factor 3.Proper time 4.Lorentz contraction 5.Proper length 6.Twin paradox and symmetric situations 7.Muon decay; evidence for time dilation

23. IB Physics – Relativity Time dilation

24. IB Physics – Relativity Time dilation Think about what this means? ????

25. IB Physics – Relativity Proof of formula The proof of the time dilation formula is a standard requirement in the exam. Carefully work through the proof using Pythagoras’ Theorem. Make sure you understand each step. Hints The “prime” notation refers to measurements in the ‘moving’ frame The speed of light is the same for all observers.

26. IB Physics – Relativity The Lorentz factor v 0.1c0.2c0.3c0.4c0.5c0.6c0.7c0.8c0.9cc  For what values of v is  significant ? Work out  below

27. IB Physics – Relativity Example of time dilation If the train passengers measure a time interval of  t 1 = 6 s and the train moves at a speed v = 0.80c, calculate the length of the same time interval measured by a stationary observer outside the train standing on the ground

28. IB Physics – Relativity Atomic clocks prove time dilation!

29. IB Physics – Relativity Proper time A proper time interval is the time separating two events that take place at the same point in space observed time interval =  x proper time interval Note that the proper time interval is the shortest possible time separating two events

30. IB Physics – Relativity Examples of proper time 1.The time interval between the ticks of a clock carried on a fast rocket is half of what observers on Earth record. How fast is the rocket moving with respect to the Earth? What are the two events here? 2.A rocket moves past an observer in a laboratory with speed = 0.85c. The lab observer measures that a radioactive sample of mass 50 mg (which is at rest in the lab) has a half life of 2.0 min. What half-life do the rocket observers measure? Again, what are the two events? 3.In the year 2010, a group of astronauts embark on a journey toward Betelgeuse in a spacecraft moving at v = 0.75c with respect to the Earth. Three years after departure from the Earth (as measured by the astronaut’s clocks) one of the astronauts announces that she has given birth to a baby girl. The other astronauts immediately send a radio signal to Earth announcing the birth. When is the good news received on Earth (according to the Earth Clocks)? In each case first suggest in which frame the proper time is directly measured Tsokos, 2005, p562

31. IB Physics – Relativity Length Contraction Another consequence of the invariance of the speed of light is that the distance between two points in space contracts according to an observer moving relative to the two points. The contraction is in the same direction as the relative motion. Measured by observer who is stationary with respect to the object Measured by observer in a moving frame with respect to the object A paradox on length contraction

32. IB Physics – Relativity Proper Length The proper length of an object is the length recorded in a frame where the object is at rest Any observers moving relative to the object measure a shorter length (Lorentz contraction);

33. IB Physics – Relativity Examples 1.An unstable particle has a life time of 4.0 x 10 -8 s in its own rest frame. If it is moving at 98% of the speed of light calculate; a)Its life time in the lab frame b)The length traveled in both frames. 2.Electrons of speed v = 0.96c move down the 3 km long SLAC linear accelerator. a)How long does take according to lab observers? b)How long does it take according to an observer moving along with the electrons? c)What is the speed of the accelerator in the rest frame of the electrons? Tsokos, 2005, p566 Kirk, 2003, p145

34. IB Physics – Relativity The twin paradox Link to twin paradox Read about this in Kirk p 146. The paradox arises because both twins view a symmetrical situation. Explain why? Explain why it is not a paradox.

35. IB Physics – Relativity The muon experiment This offers direct experimental evidence of time dilation Key points Muons have an average lifetime of 2.2 x 10 -6 s in their own rest frame. They are created 10 km up in the atmosphere with velocities as large as 0.99c. Show that without special relativity muons are unlikely to be detected on Earth. Muon decay explanation

36. IB Physics – Relativity Relativity Lesson 3 1.Velocity addition 2.Rest mass and relativistic mass 3.Use of E = mc 2 as total energy 4.Acceleration of electrons by a p.d. 5.Use of MeVc -2 and MeVc -1

37. IB Physics – Relativity Galilean Velocity addition S ground stationary train stationary v Easy!

38. IB Physics – Relativity What if approaches c ? u would be bigger than c ….and my special theory of relativity does not allow this

39. IB Physics – Relativity Relativistic velocity formulae Just a small correction needed prove the inverse formulae from this; Notice the similarity to Galileo's formulae with a small correction

40. IB Physics – Relativity Now try this 1.An electron has a speed of 2.00 x 10 8 ms -1 relative to a rocket, which itself moves at a speed of 1.00 x 10 8 ms -1 with respect to the ground. What is the speed of the electron with respect to the ground. Answer; u = 2.45 x 10 8 ms -1 The trick is to clearly sort out your reference frames. ground rocket Tsokos, 2005, p567

41. IB Physics – Relativity And this; a bit harder 2. Two rockets move away from each other with speeds of of 0.9c to right and 0.8c to the left with respect to the ground. What is the relative speed of each rocket as measured from the other. Answer ± 0.988c Again the trick is to be very careful with your frames of reference 0.8c0.9c AB Tsokos, 2005, p567

42. IB Physics – Relativity rocket B ground rocket A u is then the velocity of A with respect to B. Write down then check the velocity of B with respect to A Learn this solution

43. IB Physics – Relativity Apply the same approach for the question in Kirk on p147 By now you should have realised that at low velocities the relativistic formulae approximate to the Galilean equations. Check this out. Show also that the relativistic formulae do not allow objects to have velocities greater than c............. but F = ma allows velocities > c so we might have to look at mass again!!!

44. IB Physics – Relativity Galileo and Newton allow v > c t v c constant acceleration decreasing acceleration as speed approaches c but I don’t allow this

45. IB Physics – Relativity Inertial mass Defined as m = F/a For a constant force Einstein predicted that as v approaches c the acceleration will decrease to zero. This ensures that speed can never exceed the speed of light This means the inertial mass must be approaching infinite.

46. IB Physics – Relativity Rest Mass The rest mass, m 0, of an object is the mass as measured in a frame where the object is at rest If mass is measured in a frame moving with respect to the object; m =  m 0 m/m 0

47. IB Physics – Relativity Try these 1.Find the speed of a particle whose relativistic mass is double its rest mass. Answer = 0.866c 2.A constant force, F, is applied to a particle at rest. Find the acceleration in terms of the rest mass, m 0. Find the acceleration again if the same force is applied to the same particle when it moves at a speed 0.8c. 3.Find the rest energy of an electron and its total energy when it moves at a speed equal to 0.8c. Give values in MeV. Tsokos, 2005, p571

48. IB Physics – Relativity Mass - Energy The energy needed to create a particle at rest is called the rest energy given by; E 0 = m 0 c 2 If this particle accelerates it gains kinetic energy and the mass also will increase so total energy; E = mc 2 or E =  m 0 c 2

49. IB Physics – Relativity Proper energy; mc 2

50. IB Physics – Relativity Kinetic energy (T) and momentum (p) The formulae; E k = ½ mv 2 and p = mv no longer work at high speeds. Total energy of a particle; E = E k + m 0 c 2 So total energy is the kinetic energy plus the rest mass energy

51. IB Physics – Relativity Now try this An electron is accelerated through a p.d. of 1.0 x 10 6 V. Calculate its velocity; (use the data book to find the electron rest mass and charge). Answer = 0.94c Hint; First calculate the energy gained then find the rest mass energy. This will give you a value for . ***??** Kirk, 2003, p148

52. IB Physics – Relativity Relativistic formulae At high speeds you must use the following formulae; p =  m 0 v for momentum E = E k + m 0 c 2 for kinetic energy E 2 = p 2 c 2 + m 0 2 c 4 for total energy E =  m 0 c 2 the derivations of these are not required.

53. IB Physics – Relativity Useful units The rest mass of an electron = 9.11 x 10 -31 Kg This is equivalent to energy E 0 = m 0 c 2 = 9.11 x 10 -31 x 9 x 10 16 = 8.199 x 10 -14 J This is more conveniently quoted in MeV. so E 0 = 8.199 x 10 -14 / 1.6 x 10 -19 = 0.512 MeV Working backwards we know E 0 /c 2 = m 0 So mass can be measured in MeVc -2  electron rest mass = 0.512 MeVc -2 Similarly KeVc -2 and GeVc -2 etc. may be used

54. IB Physics – Relativity Similarly MeVc -1 may be used for momentum and problems are much easier to solve.

55. IB Physics – Relativity So try these using the new units 1.Find the momentum of a pion (rest mass 135 MeVc -2 ) whose speed is 0.80c. Answer 180 MeVc -1 2.Find the speed of a muon (rest mass = 105 MeVc -2 ) whose momentum is 228 MeVc -1. Answer = 0.91c 3.Find the kinetic energy of an electron (rest mass 0.511 MeVc -2 ) whose momentum is 1.5MeVc -1. Answer = 1.07 MeV Work through the example in Kirk on p.150 Tsokos, 2005, p581

56. IB Physics – Relativity Relativity Lesson 4 General Relativity (without the maths) 1.The equivalence principle 2.Spacetime 3.Gravitational redshift 4.Black holes

57. IB Physics – Relativity The Nature of Mass Gravitational Mass m g = W/g Inertial Mass m i = F/a I do not think so! These are completely different

58. IB Physics – Relativity Einstein’s Happiest Thought Sitting in a chair in the Patent office at Berne (in 1907), a sudden thought occurred to me. " If a person falls freely he will not feel his own weight”. I was startled and this simple thought made a deep impression on me. It impelled me towards a theory of gravitation. It was the happiest thought in my life. I realised that........for an observer falling freely from the roof of a house there exists – at least in his immediate surroundings – no gravitational field. The observer therefore has the right to interpret his state as at rest or in uniform motion................... McEvoy & Zarate, 1995, p32

59. IB Physics – Relativity The equivalence principal a Planet 1.An object inside an accelerating rocket in outer space will “fall”. 2.An object in a stationary rocket inside a gravitational field will fall So both situations are equivalent Path of “falling” objects

60. IB Physics – Relativity and similarly Planet v 1. An object inside a rocket in outer space moving at constant velocity is weightless. 2. An object inside a rocket accelerating due to a gravitational field feels weightless Again both situations are equivalent Equivalence animations ?????

61. IB Physics – Relativity Einstein’s elevator

62. IB Physics – Relativity The path of light v Observer inside rocket Inertial observer outside rocket xyx y1y1

63. IB Physics – Relativity The bending of light a xyx y1y1 y2y2 y2y2 Observer inside rocket Inertial observer outside rocket

64. IB Physics – Relativity But according to the equivalence principle a xy y2y2 Is equivalent to Planet x y y2y2 So gravity bends light towards the planet

65. IB Physics – Relativity Space time Einstein viewed space time like a rubber sheet extending into the x, y, z and t dimensions Space time is “curved” by mass. An object moves along the path of least resistance. This means they take the shortest path between two points in curved space time. Flat space time. Objects move in a “straight line” McEvoy & Zarate, 1995, p32

66. IB Physics – Relativity General relativity Model of a planet in orbit An object is “just” captured by the depression. Model of a meteorite crashing into the Earth. McEvoy & Zarate, 1995, p32

67. IB Physics – Relativity Space-time movie

68. IB Physics – Relativity So what’s so great about general relativity? Matter tells space how to curve and then space tells matter how to move........ The beauty of this simple model is.... we don’t need forces. “objects move in a straight line in curved space-time”

69. IB Physics – Relativity More on space-time Events can be given an x, y, z and t coordinate in space-time to describe where and when they occurred. ct x space-time diagram 45 o a b c 1.Why is the time axis multiplied by c? 2.What is the gradient of a line on this axis? 3.What is the velocity of lines a, b and c? 4.What is wrong with d? d Think?

70. IB Physics – Relativity Gradient of a space-time diagram or now try the questions on the previous slide

71. IB Physics – Relativity Even light bends in space time General relativity predicts that the path of light is deviated by curved space-time. “There was to be a total eclipse of the Sun on 29 May 1919, smack in the middle of a bright field of stars in the cluster Hyades.”........Arthur Eddington led an expedition to the island of Principe off the coast of Africa to photograph the eclipse. Eddington found that the position of the stars appeared different from pictures taken at a different time. He concluded that the light had curved around the sun by the exact amount that Einstein had predicted. hyperphysics.com, 2006 McEvoy & Zarate, 1995, p32

72. IB Physics – Relativity Gravitational lensing Light from objects (e.g.quasars) which are very far away can be bent round massive galaxies to produce multiple images; the galaxy behaves like a lens. Researchers at Caltech have used the gravitational lensing afforded by the Abell 2218 cluster of galaxies to detect the most distant galaxy known (Feb, 15 th 2004) through imaging with the Hubble Telescope. Spot the multiple images Wikiepedia.org, 2006 Kirk, 2003, p155

73. IB Physics – Relativity Black Holes When a star uses up its nuclear fuel it collapses. If the remnant mass of the star is greater than 3 x the sun’s mass there is no mechanism to stop it collapsing to a singularity. The curvature of space-time near a singularity is so extreme that even light cannot escape. Escape velocity For a photon

74. IB Physics – Relativity Schwarzchild Radius; R Sch At a distance, R Sch, from a singularity the escape velocity is the speed of light. At a distance less than R Sch from a singularity; escape velocity > C  Light cannot escape R Sch is also called the event horizon. Everything trapped within the event horizon is not observable in our universe.

75. IB Physics – Relativity Try this 1. Calculate the Schwarzschild radius for a star of one solar mass; (M = 2 x 10 30 Kg) Tsokos, 2005, p591

76. IB Physics – Relativity Time in gravitational fields General relativity predicts that time runs slower in places where the gravitational field strength is stronger. For example; the Earth’s field weakens as you go further away from the Earth’s surface. This means that time runs more slowly at the ground floor than at the top floor of a building.

77. IB Physics – Relativity Gravitational Red Shift Gravitational time dilation, or clocks running slower in strong gravitational fields, leads us directly to the prediction that the wavelength of a beam of light leaving the Earth’s surface will increase with height. or but Period

78. IB Physics – Relativity Red-shift Explained As the light gains height so time runs more quickly because the gravitational field weakens. So as time speeds up, the period increases and hence the wavelength also increases Planet Large wavelength Short wavelength

79. IB Physics – Relativity Calculating frequency shifts. Consider a photon of frequency f 0 leaving the surface of the Earth. It gains potential energy given by mg  h and the frequency is reduced to f. Photons do not have mass but have an effective mass given by; Conserving energy we get; Do not get h the Planck's constant confused with  h the height gain.

80. IB Physics – Relativity Blue shift Similarly, a beam of light directed from outer space towards the Earth is blue-shifted i.e. the wavelength decreases as the gravitational intensity increases. The same formula is used to calculate red-shift or blue-shift. Examples to try 1. A photon of energy 14.4 KeV is emitted from the top of a 30 m tall tower toward the ground. What shift of frequency is expected at the base of the tower? Ans;  f = 1.16 x 10 4 Hz Tsokos, 2005, p589 2. A UFO travels at such a speed to remain above one point on the Earth at a height of 200 Km above the Earth’s surface. A radio signal of frequency 110MHz is sent to the UFO. (i) What is the frequency received by the UFO? (ii) If the signal was reflected back to Earth, what would be the observed frequency of the return signal? Explain your answer. Ans; (i) f = 1.1 x 108 Hz (shift is v. small). (ii) return signal is the same frequency as the emitted signal Kirk, 2003, p153

81. IB Physics – Relativity Experimental support In 1960 a famous experiment called the Pound-Rebka experiment was carried out at Harvard University to verify gravitational blue/red-shift. The frequencies of gamma ray photons were measured at the bottom and top of the Jefferson Physical Laboratory tower. Very small frequency shifts were detected as predicted by the theory. Atomic clocks are very sensitive and precise. Atomic clocks sent to high altitudes in rockets have been shown to run faster than a similar clocks left on Earth

82. IB Physics – Relativity An alternative view of black holes The gravitational intensity near black holes is very strong and approaches infinite at the event horizon. Time effectively stops and any object falling into a black hole will appear to stop at the event horizon according to a far observer. Light emitted from a black hole will therefore be infinitely red- shifted; hence no light is emitted. RsRs x ct Space-time diagram of the formation of a black hole This photon is trapped in the black hole. This photon will escape

83. IB Physics – Relativity Beyond relativity!

84. IB Physics – Relativity Bibliography Kirk, T; Physics for the IB diploma, OUP, UK, 2003 McEvoy, J.P. & Zarate, O; Stephen Hawking for beginners, Icon Books, U.K., 1995. Tsokos, K.A.; IB Physics, Cambridge University press, U.K., 2005. hyperphysics.com, 2006 http://en.wikipedia.org/wiki/Gravitational_lensinghttp://en.wikipedia.org/wiki/Gravitational_lensing, Mar 06 ©Neil Hodgson Sha Tin College

85. IB Physics – Relativity Videology 1.Postulates of special relativity 2.Speed of light is constant 3.Simultaneity 4.Time dilation 5.Atomic clocks prove time dilation