CHAPTER 2 Special Theory of Relativity

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1 CHAPTER 2 Special Theory of Relativity Fall 2018 Prof. Sergio B. Mendes 1

2 Topics Inertial Frames of Reference Conceptual and Experimental Inconsistencies The Michelson-Morley Experiment Einstein s Postulates The Lorentz Transformation Time Dilation and Length Contraction Addition of Velocities Experimental Verification Twin Paradox Space-Time Doppler Effect Relativistic Momentum Relativistic Energy Computations in Modern Physics Electromagnetism and Relativity Fall 2018 Prof. Sergio B. Mendes 2

3 How to describe an event? The precise description of an event must be characterized by its location in space rr and its time tt. zz The set of (calibrated) rulers and (synchronized) clocks form a frame of reference that can be used to characterize events. Fall 2018 Prof. Sergio B. Mendes 3

4 What is an Inertial Frame of Reference? One in which Newton s laws are valid!! aa mmgg + TT = FF nnnnnn = mm aa FF nnnnnn = 00 vv = 00 aa = 00 FF nnnnnn = 00 vv = cccccccccccccccc aa = 00 Such a frame of reference is established when a body subjected to a null net external force, FF nnnnnn = 00, is observed to move with constant velocity (either rectilinear motion at constant speed or at rest). Fall 2018 Prof. Sergio B. Mendes 4

5 A non-inertial observer reaches conclusions that don t agree with Newton s laws aa = 0 aa FF nnnnnn 00 FF nnnnnn = 00 aa = aa aa aa FF nnnnnn = 00 aa = aa Fall 2018 Prof. Sergio B. Mendes 5

6 Therefore, we must use an inertial frame of reference to describe the laws of Mechanics Fall 2018 Prof. Sergio B. Mendes 6

7 Q: Well, how many inertial frames of reference are out there? A: An infinite number. Once we have found one inertial frame of reference (K), then any frame of reference moving at constant velocity vv oo with respect to K is also an inertial frame of reference. Fall 2018 Prof. Sergio B. Mendes 7

8 Proof Consider that K is at rest. Consider that K is moving with constant velocity vv oo with respect to K. Consider that the two frames of reference coincide at tt = 0. rr = rr vv oo tt K rr vv oo tt K rr xx = xx vv oo,xx tt yy = yy vv oo,yy tt zz = zz vv oo,xx tt vv = vv vv oo aa = aa tt = tt Fall 2018 Prof. Sergio B. Mendes 8

9 aa = aa mmm aa = mmm aa FF = FF mm = mm If Newton s laws are valid in one frame of reference (K), then they are also valid in another frame of reference (K ) moving at a uniform velocity relative to the first system. So, both are inertial frames of reference. Fall 2018 Prof. Sergio B. Mendes 9

10 Principle of (Classical) Relativity The laws of Mechanics (Newton s laws) are the same in all inertial frames of reference All inertial frames of reference are equivalent. Inertial frames of reference are related by: rr = rr vv oo tt tt = tt Galilean Transformation Fall 2018 Prof. Sergio B. Mendes 10

11 2.1 Conceptual and Experimental Inconsistencies Fall 2018 Prof. Sergio B. Mendes 11

12 Conceptual Inconsistencies: Although Newton s laws of motion had the same form under the Galilean transformation, Maxwell s equations did not. SS EE rr. ddaa = QQ iiiiiiiiiiii εε 0 CC EE rr. ddss = ddφ BB dddd SS BB rr. ddaa = 0 CC BB rr. ddss = μμ oo II + μμ oo εε oo ddφ EE dddd FF = qq EE + qq vv BB Fall 2018 Prof. Sergio B. Mendes 12

13 Experimental Inconsistencies: 2 EE xx 2 = εε 2 EE oo μμ oo tt 2 2 BB xx 2 = εε 2 BB oo μμ oo tt 2 In Maxwell s theory, the speed of light in terms of the permeability and permittivity of free space was given by: cc = 1 εε oo μμ oo Fall 2018 Prof. Sergio B. Mendes 13

14 The carrier medium for light, ether: Following the tradition of the time (that every wave has a medium to carry its propagation) the luminiferous ether was considered as the carrier medium for light propagation, where cc = 1 εε oo μμ oo Ether had to have such a low density that the planets could move through it without loss of energy It also had to have an elasticity to support the high velocity of light waves Fall 2018 Prof. Sergio B. Mendes 14

15 An Absolute Inertial Frame of Reference Ether was proposed as an absolute reference system in which the speed of light was this constant and from which other measurements could be made. The Michelson-Morley experiment was an attempt to show the existence of ether. Fall 2018 Prof. Sergio B. Mendes 15

16 2.2 The Michelson-Morley Experiment Albert Michelson ( ) was the first U.S. citizen to receive the Nobel Prize for Physics (1907). He built an extremely precise device called an interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions. Fall 2018 Prof. Sergio B. Mendes 16

17 The Michelson Interferometer Fall 2018 Prof. Sergio B. Mendes 17

18 How does it work? 1. AC is parallel to the motion of the Earth inducing an ether wind 2. Light from source S is split by mirror A and travels to mirrors C and D in mutually perpendicular directions 3. After reflection the beams recombine at A slightly out of phase due to the ether wind as viewed by telescope E. Fall 2018 Prof. Sergio B. Mendes 18

19 A Typical Interference Pattern Fall 2018 Prof. Sergio B. Mendes 19

20 The Analysis assuming the Galilean Transformation!! Time tt 1 from A to C and back: tt 1 = ll 1 cc + vv + ll 1 cc vv Time tt 2 from A to D and back: tt 2 = = 2 cc ll 1 cc 2 vv 2 ll 2 cc 2 vv 2 + ll 2 cc 2 vv 2 So that the change in time is: tt = tt 2 tt 1 = 2 ll 2 cc = 2 ll 2 cc = 2 ll 1 cc Fall 2018 Prof. Sergio B. Mendes vv2 cc vv2 cc 2 1 vv2 cc 2 2 ll 1 cc 1 1 vv2 cc 2

21 The Analysis (continued) tt = tt 2 tt 1 = 2 ll 2 cc 1 1 vv2 cc 2 2 ll 1 cc 1 1 vv2 cc 2 Upon rotating the apparatus by 90º, the optical path lengths ll 1 and ll 2 are interchanged producing a different change in time: ttt = ttt 2 tt 1 = 2 ll 2 cc 1 1 vv2 cc 2 2 ll 1 cc 1 1 vv2 cc 2 Fall 2018 Prof. Sergio B. Mendes 21

22 Difference in times upon rotation: tt tt = 2 ll 1 + ll 2 cc 1 1 vv2 cc vv2 cc 2 ll 1 + ll 2 vv 2 cc 3 Fall 2018 Prof. Sergio B. Mendes 22

23 Increasing the pathlength: Fall 2018 Prof. Sergio B. Mendes 23

24 Crunching the numbers: vv = mm/ss cc = mm/ss ll 1 ll 2 = 11 mm tt tt = ll 1 + ll 2 vv 2 cc 3 = ss λλ = mm cc = mm/ss TT = λλ cc = mm mm/ss = ss ffffffffffffffff oooo aaaa iiiiiiiiiiiiiiiiiiiiiiii ffffffffffff = ss ss 0.4 iiiiiiiiiiiiiiiiiiii rrrrrrrrrrrrrrrrrrrr 0.01 Fall 2018 Prof. Sergio B. Mendes 24

25 The Experiments on the relative motion of the earth and ether have been completed and the result decidedly negative. The expected deviation of the interference fringes from the zero should have been 0.40 of a fringe the maximum displacement was 0.02 and the average much less than 0.01 and then not in the right place. As displacement is proportional to squares of the relative velocities it follows that if the ether does slip past the relative velocity is less than one sixth of the earth s velocity. Albert Abraham Michelson, 1887 Fall 2018 Prof. Sergio B. Mendes 25

26 2.3 Einstein s Two Postulates 1. The principle of relativity: The laws of physics are the same in all inertial frames of reference. There is no way to detect absolute motion and no preferred inertial system exists. 2. The constancy of the speed of light: Observers in all inertial frames of reference measure the same value for the speed of light when propagating in vacuum. Fall 2018 Prof. Sergio B. Mendes 26

27 Consequences: Source and Observer at rest: c Observer in motion with respect to the Source: c Source in motion with respect to the Observer: c -v v Fall 2018 Prof. Sergio B. Mendes 27

28 Consequences In Newtonian physics, we previously assumed that tt = tt. Therefore K and K would always agree if two events happen at the same time (simultaneous) or not. Einstein realized that events considered simultaneous in K may not be in K. Fall 2018 Prof. Sergio B. Mendes 28

29 Because speed of light is absolute then simultaneity is relative K K K K Fall 2018 Prof. Sergio B. Mendes 29

30 Two events that are simultaneous in one reference frame (K) are not necessarily simultaneous in another reference frame (K ) moving with respect to the first frame. This suggests that each coordinate system must have its own set of observers with their own set of synchronized clocks. Fall 2018 Prof. Sergio B. Mendes 30

31 2.4 Lorentz Transformations The special set of linear transformations that preserve the constancy of the speed of light between inertial observers. Fall 2018 Prof. Sergio B. Mendes 31

32 Two inertial frames of reference KK and KK 1. The axes along (xx, yy, zz) are parallel to the corresponding axes along (xxx, yyy, zzz). 2. The relative motion between the two inertial frames of reference is along the x-axis (and x -axis). 3. Consider that the origins OO and OO of the two systems coincide at tt = ttt = 0 As seen by KK As seen by KK OO OOO vv oo vv oo OO OOO Fall 2018 Prof. Sergio B. Mendes 32

33 As seen by KK KK: (xx, tt) vv oo vv oo vv oo KKK: (xxx, ttt) vv oo vv oo Fall 2018 Prof. Sergio B. Mendes 33

34 As seen by KK : KK: (xx, tt) vv oo vv oo vv oo KKK: (xxx, ttt) vv oo vv oo Fall 2018 Prof. Sergio B. Mendes 34

35 As seen by KK: A flashbulb goes off at the origins when tt = ttt = 0. vv oo Fall 2018 Prof. Sergio B. Mendes 35

36 As seen by KKK: A flashbulb goes off at the origins when tt = ttt = 00. vv oo Fall 2018 Prof. Sergio B. Mendes 36

37 According to Postulate 2, the speed of light will be c in both systems!! The wavefronts observed in both systems must be spherical with respect to their own coordinates and time. KK OO KKK OOO xx 2 + yy 2 + zz 2 = cc 2 tt 2 xxx 2 + yyy 2 + zzz 2 = cc 2 ttt 2 Fall 2018 Prof. Sergio B. Mendes 37

38 xx 2 + yy 2 + zz 2 cc 2 tt 2 = 0 0 = xxx 2 + yyy 2 + zzz 2 cc 2 ttt 2 yy = yyy zz = zzz as in Galilean transformation xx 2 cc 2 tt 2 = xxx 2 cc 2 ttt 2 Fall 2018 Prof. Sergio B. Mendes 38

39 xx = γγ xx vv oo tt linear modification in Galilean transformation due to symmetry xx = γγ xxx + vv oo ttt = γγ γγ xx vv oo tt + vv oo ttt solve for: tt = 1 γγ vv oo 1 γγ 2 xx + γγ tt Fall 2018 Prof. Sergio B. Mendes 39

40 xx 2 cc 2 tt 2 = xxx 2 cc 2 ttt 2 xx = γγ xx vv oo tt tt = 1 γγ vv oo 1 γγ 2 xx + γγ tt Fall 2018 Prof. Sergio B. Mendes 40

41 xx 2 : 1 = γ 2 cc2 γγ 2 2 vv 1 γγ2 2 1 oo 1 γγ 2 = cc2 γγ 2 vv oo 2 1 γγ 2 1 = vv oo cc 2 2 tt 2 : cc 2 = γγ vv oo cc 2 γγ 2 γγ 2 = 1 vv 2 oo cc 2 2 xx tt: 0 = γγ 2 vv oo cc 2 1 vv oo 1 γγ 2 γγ 2 vv oo = cc 2 1 vv oo 1 γγ 2 vv oo 2 cc 2 = 1 γγ 2 1 γγ = 1 1 ββ 2 ββ vv oo cc Fall 2018 Prof. Sergio B. Mendes 41

42 Lorentz Transformations: xx = γγ xx γγ vv oo tt tt = γγ vv oo cc 2 xx + γγ tt γγ = 1 1 ββ 2 ββ vv oo cc Fall 2018 Prof. Sergio B. Mendes 42

43 Inverse Lorentz Transformations: xx = γγ xx + γγ vv oo tt tt = γγ vv oo cc 2 xx + γγ ttt γγ = 1 1 ββ 2 ββ vv oo cc Fall 2018 Prof. Sergio B. Mendes 43

44 Relativistic Factor γγ ββ = Fall 2018 Prof. Sergio B. Mendes 44

45 2.5 Time Dilation and Length Contraction Fall 2018 Prof. Sergio B. Mendes 45

46 Time Dilation: vv oo Fall 2018 Prof. Sergio B. Mendes 46

47 Time Dilation: tt = γγ vv oo cc 2 xx + γγ tt ttt 1 xx 1 = xx 2 ttt 1 = γγ vv oo cc 2 xx 1 + γγ tt 1 ttt 2 = γγ vv oo cc 2 xx 2 + γγ tt 2 Fall 2018 Prof. Sergio B. Mendes 47

48 Time Dilation & Proper Time: ttt 1 ttt 2 ttt 2 tt 1 = γγ tt 2 tt 1 tt 2 tt 1 = proper time, time duration measured at same location Fall 2018 Prof. Sergio B. Mendes 48

49 Length Contraction: vv oo Fall 2018 Prof. Sergio B. Mendes 49

50 Length Contraction: xx = γγ xx + γγ vv oo tt xx 1 = γγ xxx 1 + γγ vv oo tt 1 xx 2 = γγ xxx 2 + γγ vv oo tt 2 tt 2 = ttt 1 xxx 2 xxx 1 = 1 γγ xx 2 xx 1 xx 2 xx 1 = proper length, length measured at rest Fall 2018 Prof. Sergio B. Mendes 50

51 Fall 2018 Prof. Sergio B. Mendes 51

52 Fall 2018 Prof. Sergio B. Mendes 52

53 2.6 Addition of Velocities Fall 2018 Prof. Sergio B. Mendes 53

54 Taking the differentials: tt = γγ vv oo cc 2 xx + γγ tt ddtt = γγ vv oo cc 2 ddxx + γγ dddd xx = γγ xx γγ vv oo tt ddxx = γγ dddd γγ vv oo ddtt yy = yy zz = zz ddyy = ddyy ddzz = dddd Fall 2018 Prof. Sergio B. Mendes 54

55 Along x-axis ddxx = γγ dddd γγ vv oo ddtt ddtt = γγ vv oo cc 2 ddxx + γγ dddd ddxx ddtt = γγ dddd γγ vv oo dddd γγ vv oo dddd + γγ dddd cc 2 uuu xx ddxx ddtt = uu xx vv oo 1 vv oo uu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 55

56 ddyy Along y-axis ddyy = ddyy ddtt = γγ vv oo cc 2 ddtt = γγ vv oo cc 2 ddxx + γγ dddd dddd dddd + γγ dddd uuu yy ddyy ddtt = uu yy γγ 1 vv oo uu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 56

57 Along z-axis ddzz = dddd ddzz ddtt = γγ vv oo cc 2 ddtt = γγ vv oo cc 2 ddxx + γγ dddd dddd dddd + γγ dddd uuu zz ddzz ddtt = uu zz γγ 1 vv oo uu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 57

58 In Summary, Addition of Velocities: uuu xx = uu xx vv oo 1 vv oo uu xx cc 2 uuu yy = uu yy γγ 1 vv oo uu xx cc 2 uuu zz = uu zz γγ 1 vv oo uu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 58

59 Inverted Relations: uu xx = uuu xx + vv oo 1 + vv oo uuu xx cc 2 uu yy = uuu yy γγ 1 + vv oo uuu xx cc 2 uu zz = uuu zz γγ 1 + vv oo uuu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 59

60 Example uu xx = cc vv oo = cc uuu xx = = uu xx vv oo 1 vv oo uu xx cc cc cc cc cc 1 cc 2 = cc Fall 2018 Prof. Sergio B. Mendes 60

61 2.7 Experimental Verification of Special Relativity Fall 2018 Prof. Sergio B. Mendes 61

62 Cosmic Rays and Muon Decay NN NN oo = ee llll 2 tt ττ ττ = 1.52 μμμμ Fall 2018 Prof. Sergio B. Mendes 62

63 (Incorrect) Classical Calculation h = 2000 mm vv = 0.98 cc tt = h vv = 6.80 μμμμ NN NN oo = ee llll 2 tt ττ ττ = 1.52 μμμμ llll μμμμ = ee 1.52 μμμμ = 4.5% Don t agree with experiment Fall 2018 Prof. Sergio B. Mendes 63

64 Relativistic Calculation vv = 0.98 cc γγ = tt = tt γγ 6.80 μμμμ = μμμμ NN NN oo = ee llll 2 tt ττ llll μμμμ = ee 1.52 μμμμ = 54% Agrees with experiment Fall 2018 Prof. Sergio B. Mendes 64

65 Atomic Clock Measurement Fall 2018 Prof. Sergio B. Mendes 65

66 Velocity Addition ππ 0 γγ + γγ vv oo = cc uu xx = uuu xx + vv oo 1 + vv oo uuu xx cc 2 uuu xx = cc vv xx = cc Fall 2018 Prof. Sergio B. Mendes 66

67 2.10 Relativistic Doppler effect for light waves in vacuum Although light velocity in vacuum is always constant c, the frequency will change for a relative motion between source and observer Source- Observer approaching Higher Frequency Source- Observer receding Lower Frequency Fall 2018 Prof. Sergio B. Mendes 67

68 Relativistic Doppler effect for light waves in vacuum cc vv vv TT cc TT lllllllllll oooo ttttt wwwwwwww tttttttttt = cc TT vv TT nn = cc TT vv TT λλ ff = cc λλ = cc nn cc TT vv TT Fall 2018 Prof. Sergio B. Mendes 68

69 nn = pppppppppppp tttttttt oooo ttttt ssssssssssss tttttttt pppppppppppp oooo ttttt ssssssssssss = TTT oo 1 fffoo TT = γγ TTT oo nn = TT fff oo γγ ff = cc nn cc TT vv TT = cc γγ cc vv fff oo = 1 + ββ 1 ββ fff oo Fall 2018 Prof. Sergio B. Mendes 69

70 Relativistic Doppler effect for light propagating in vacuum Source- Observer approaching Source- Observer receding ff = 1 + ββ 1 ββ fff oo ff = 1 ββ 1 + ββ fff oo blueshifted redshifted Fall 2018 Prof. Sergio B. Mendes 70

71 Rotation of Venus Fall 2018 Prof. Sergio B. Mendes 71

72 Laser Cooling Fall 2018 Prof. Sergio B. Mendes 72

73 Laser Radar Technology Fall 2018 Prof. Sergio B. Mendes 73

74 2.11 Relativistic Linear Momentum Fall 2018 Prof. Sergio B. Mendes 74

75 Classical Expressions from Galileo and Newton: pp mm uu Linear Momentum dd pp FF = dddd Newton s law Newton s law is invariant under Galilean transformations, but not under Lorentz transformations. Fall 2018 Prof. Sergio B. Mendes 75

76 Relativistic Linear Momentum Invariance of Newton s law under Lorentz transformations will be shown to lead to: pp = ΓΓ mm uu ΓΓ 1 1 uu cc 2 Fall 2018 Prof. Sergio B. Mendes 76

77 Conservation of Linear Momentum in an Elastic Collision: uu xx = 0 uu xx = uuu xx + vv 1 + vv uuu xx cc 2 uu xx = vv uu xx = 0 uu yy = uu oo uu yy = uuu yy γγ 1 + vv uuu xx cc 2 uu yy = uu oo γγ uu yy = 2 uu oo γγ mm mm pp = mm uu + uu 0 mm uu xx = 0 uu yy = ± uu oo uu xx = 0 uu yy = 2 uu oo Fall 2018 Prof. Sergio B. Mendes 77

78 Relativistic Linear Momentum pp = mm ΓΓ uu pp = mm ΓΓ uu + uu = 0 ΓΓ 1 1 uu cc 2 uu xx = vv uu yy = uu oo γγ ΓΓ uu xx = 0 ΓΓ uu yy = ΓΓ 2 uu oo γγ = 2 uu oo 1 uu oo 2 cc 2 ΓΓ = 1 1 vv 2 + uu oo 2 γγ 2 cc 2 uu xx = 0 ΓΓ uu xx = 0 uu yy = ± uu oo ΓΓ uu yy = 2 uu oo 1 uu oo 2 cc 2 ΓΓ = 1 1 uu oo 2 cc 2 Fall 2018 Prof. Sergio B. Mendes 78

79 Relativistic Linear Momentum pp = mm ΓΓ uu ΓΓ 1 1 uu cc 2 Fall 2018 Prof. Sergio B. Mendes 79

80 2.12 Relativistic Energy FF = dd pp dddd 2 2 dd pp WW 12 = KK 2 KK 1 = FF dd rr = 1 1 dddd dd rr Fall 2018 Prof. Sergio B. Mendes 80

81 pp = mm ΓΓ uu dd pp = mm dd ΓΓ uu uu = dd rr dddd dd rr = uu dddd 2 dd pp KK 2 KK 1 = 1 dddd dd rr 2 dd ΓΓ uu = mm 1 dddd uu dddd 2 = mm uu dd ΓΓ uu 1 Fall 2018 Prof. Sergio B. Mendes 81

82 Solving the integral: ΓΓ 1 1 uu cc 2 2 KK = mm uu dd ΓΓ uu 1 uu = mm 0 uu 1 uu cc dddd = mm cc uu cc 2 1 Fall 2018 Prof. Sergio B. Mendes 82

83 Kinetic Energy KK = mm cc uu cc 2 1 uu cc uu cc 2 1 uu cc KK mm uu2 2 Fall 2018 Prof. Sergio B. Mendes 83

84 Comparison of Classical and Relativistic Kinetic Energy Fall 2018 Prof. Sergio B. Mendes 84

85 Experimental Results Fall 2018 Prof. Sergio B. Mendes 85

86 Total Relativistic Energy KK = mm cc uu cc 2 1 KK + mm cc 2 mm cc 2 Total Energy: EE tttttttttt = = = ΓΓ mm cc 2 1 uu cc 2 Rest Energy: EE 0 = mm cc 2 Fall 2018 Prof. Sergio B. Mendes 86

87 Hydrogen Fusion and Solar Energy 1 HH + 1 HH 2 HH + ee + + νν ee MeV 2 HH + 1 HH 3 HHee + γγ MeV 3 HHHH + 3 HHHH 4 HHee HH + γγ MeV Fall 2018 Prof. Sergio B. Mendes 87

88 Total Energy and Linear Momentum EE tttttttttt = mm cc 2 1 uu cc 2 EE tttttttttt 2 = mm2 cc 4 1 uu cc 2 pp = mm uu 1 uu cc 2 pp 2 cc 2 = mm2 uu 2 cc 2 1 uu cc 2 EE tttttttttt 2 pp 2 cc 2 = mm 2 cc 4 Fall 2018 Prof. Sergio B. Mendes 88

89 Massless Particles: EE tttttttttt 2 pp 2 cc 2 = mm 2 cc 4 mm = 0 EE tttttttttt = pp cc uu = cc Fall 2018 Prof. Sergio B. Mendes 89

90 Units of Energy: WW = qq VV ee = CC VV = 1 VV WW = ee VV = JJ 1 eeee Fall 2018 Prof. Sergio B. Mendes 90

91 Units of Mass: EE 0 = mm cc 2 mm = EE 0 cc 2 1 kkkk = JJ/cc 2 1 uu 1 12 mmmmmmmm oooo nnnnnnnnnnnnnn 12 CC = kkkk = MMMMMM/cc 2 Particle Mass (MeV) / c 2 Mass (u) electron proton neutron Higgs boson 125, Fall 2018 Prof. Sergio B. Mendes 91

92 Units of Linear Momentum EE tttttttttt 2 pp 2 cc 2 = mm 2 cc 4 lliiiiiiiiii mmmmmmmmmmmmmmmm = eeeeeeeeeeee cc Fall 2018 Prof. Sergio B. Mendes 92

93 Binding Energy: EE 0, pp + 2 EE 0, nn EE 0, HHHH = EE BB,HHHH uu uu u cc 2 = uu cc 2 = 28.3 MMMMMM = EE BB,HHHH Fall 2018 Prof. Sergio B. Mendes 93

94 2.9 Spacetime Representation Fall 2018 Prof. Sergio B. Mendes 94

95 Conventional Representation: (tt, xx) Fall 2018 Prof. Sergio B. Mendes 95

96 Spacetime Representation (xx, cccc) When describing events in relativity, it is convenient to represent events on a spacetime diagram. In this diagram one spatial coordinate x, to specify position, is used and instead of time t, ct is used as the other coordinate so that both coordinates will have dimensions of length. Spacetime diagrams were first used by H. Minkowski in 1908 and are often called Minkowski diagrams. Paths in Minkowski spacetime are called worldlines. Fall 2018 Prof. Sergio B. Mendes 96

97 Worldline Fall 2018 Prof. Sergio B. Mendes 97

98 tan θθ = xx cc tt = uu cc = ββ xx cc tt uu tan θθ = uu cc = ββ 1 θθ θθ 45 Fall 2018 Prof. Sergio B. Mendes 98

99 Light Cone Fall 2018 Prof. Sergio B. Mendes 99

100 Spacetime and Inertial Frames of Reference xx = γγ xx γγ vv oo tt tt = γγ vv oo cc 2 xx + γγ tt cc tt θθ cc tt θθ xx xx tan θθ = vv oo cc Fall 2018 Prof. Sergio B. Mendes 100

101 Back to the Lorentz Transformations: ddtt = γγ vv oo cc 2 ddxx = γγ dddd γγ vv oo ddtt ddyy = ddyy ddzz = dddd ddxx + γγ dddd xx yy zz cccc four-vector ddxx 2 + ddyy 2 + ddzz 2 cc 2 ddtt 2 = ddxxx 2 + ddyy 2 + ddzz 2 cc 2 ddttt 2 Invariant: its value does not change among inertial frames of reference Fall 2018 Prof. Sergio B. Mendes 101

102 Example 2.13 KK = 2.00 GGGGGG KK = 2.00 GGGGGG EE tttttttttt =?? = KK + mm cc 2 = 2.00 GGGGGG GeV = 2.94 GGGGGG Fall 2018 Prof. Sergio B. Mendes 102

103 Example 2.13 KK = 2.00 GGGGGG KK = 2.00 GGGGGG pp =?? EE tttttttttt 2 pp 2 cc 2 = mm 2 cc 4 pp = 1 cc EE tttttttttt 2 mm cc 2 2 = 1 cc GGGGGG GGGGGG 2 = 2.78 GGGGGG cc Fall 2018 Prof. Sergio B. Mendes 103

104 Example 2.13 KK = 2.00 GGGGGG KK = 2.00 GGGGGG uu =?? ββ =?? ΓΓ =?? EE tttttttttt = KK + mm cc 2 = mm cc 2 2 = ΓΓ mm cc2 1 uu cc ΓΓ = EE tttttttttt mm cc 2 = GGGGGG GGGGGG = 3.13 ββ = ΓΓ2 1 ΓΓ 2 = Fall 2018 Prof. Sergio B. Mendes 104

105 Topics Inertial Frames of Reference Conceptual and Experimental Inconsistencies The Michelson-Morley Experiment Einstein s Postulates The Lorentz Transformation Time Dilation and Length Contraction Addition of Velocities Experimental Verification Twin Paradox Space-Time Doppler Effect Relativistic Momentum Relativistic Energy Computations in Modern Physics Electromagnetism and Relativity Fall 2018 Prof. Sergio B. Mendes 105

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