1. Relativity
You can measure lengths with a ruler or meter stick. You can time events with a stopwatch. Einstein’s special theory of relativity questions deeply held assumptions about the nature of space and time.
Chapter Goal: To understand how Einstein’s theory of relativity changes our concepts of space and time.

2. Chapter 37. Relativity Topics: Relativity: What’s It All About?
Galilean Relativity
Einstein’s Principle of Relativity
Events and Measurements
The Relativity of Simultaneity
Time Dilation
Length Contraction
The Lorentz Transformations
Relativistic Momentum
Relativistic Energy

3. Reading Quizzes

4. In relativity, the Galilean transformations are replaced by the
Einstein tranformations.
Lorentz transformations.
Feynman transformations.
Maxwell transformations.
Laplace tranformations.
Answer: B

5. In relativity, the Galilean transformations are replaced by the
Einstein tranformations.
Lorentz transformations.
Feynman transformations.
Maxwell transformations.
Laplace tranformations.
IG36.1

6. A physical activity that takes place at a definite point in space and time is called
a gala.
a sport.
an event.
a happening.
a locale.
Answer: C

7. A physical activity that takes place at a definite point in space and time is called
a gala.
a sport.
an event.
a happening.
a locale.
IG36.2

8. Basic Content and Examples

9. Since relativity theory plays a major role in the study of atomic and nuclear physics, we shall discuss it in some detail.

10. Einstein’s Principle of Relativity

12. The Galilean Transformations
Consider two reference frames S and S'. The coordinate axes in S are x, y, z and those in S' are x', y', z'. Reference frame S' moves with velocity v relative to S along the x-axis. Equivalently, S moves with velocity −v relative to S'.
The Galilean transformations of position are:
The Galilean transformations of velocity are:

13. INVARIANCE OF CLASSICAL MECHANICS UNDER GALILEAN TRANSFORMATIONS
Conservation of Linear Momentum.
Invariance of Newton's Second Law.
Conservation of Energy.

14. Conservation of Linear Momentum
We suppose that an observer in system S2 and S1 watches a head-on collision between two particles of respective masses m and M(Fig.a and b)
For the observer in the inertial systemS2
Using the galilean velocity transformations

15. Invariance of Newton's Second Law.
Consider two bodies, of respective masses m and M, that interact with one another as a result of some force, such as the gravitational force. If no net external force is applied to the system (the two bodies), the system is isolated, and the total linear momentum of the system must remain constant.
Observer S1 states
which is Newton's third law. In a similar fashion observer S2would write
since therefore

16. From the invariance of Newton's second law and from the fact that the forces and acceleration are unchanged it immediately follows that, if S1 is an inertial system, then S2, which represents any coordinate system moving with respect to S1 at a constant velocity v, is also an inertial system. From the point of view of Newton's second law, all inertial systems, of which there are an infinite number, are equivalent and indistinguishable. Clearly,
any coordinate frame of reference that is accelerated with respect to some inertial system cannot itself be an inertial system, because no longer does
hold.

17. Conservation of Energy.
Observer S2
Kinetic energy before collision = kinetic energy after collision
observer S1
Hence using the invariance of the conservation of momentum law
We have found that the laws of classical mechanics (the conservation of momentum, Newton's law of motion, and the conservation of energy) are all invariant under a Galilean transformation.

18. THE FAILURE OF GALILEAN TRANSFORMATIONS
One might well ask whether the laws of electromagnetism are
invariant under a Galilean transformation?

19. THE FAILURE OF GALILEAN TRANSFORMATIONS
In general, the measured speed of a sound pulse depends on the relative speed obtaining between the observer and the medium through which the pulse travels.
This result is confirmed by experiments with sound 'waves.
However, the measured velocity of the sound pulse in the system S1,in which the air is at rest, does not depend on the velocity of a source of the sound. Of course, if a source of sound generates sinusoidal variations in the air pressure, rather than a pulse, the frequency and wavelength of the sound will, according to the Doppler effect, depend on the relative motion of the source and the medium, but the velocity of propagation of the disturbance will be independent of the source's relative motion.

20. A pulse of light travels to the right with respect to the medium through which it is propagated (ether ) at a speed
Because nineteenth-century physicists were firmly convinced that all physical phenomena were ultimately mechanical in origin, it was for them unthinkable that an electromagnetic disturbance could be propagated in empty space. Thus, the ether concept was invented.
From Galilean transformation
Therefore, the speed of light is certainly not invariant under galilean transformations
It is to measure the speed of light in a variety of inertial systems, noting whether the measured speed is different in the different systems and, most especially, whether there is evidence of a single, unique inertial system, "the ether," in which the speed of light is c.

21. Michelson and Morley: There was a change in the measured speed of light as the earth drifted through a conjectured ether in its axial rotation and its revolution about the sun.

23. It may be assumed that v/c << 1.
the maximum fractional change in the round-trip time interval for reorientation at 90° is
Performing this experiment many times, at various times of the year and in various locations, Michelson and Morley always found that was zero; that is, the result was always null.
Any inertial system S2 behaves as if it were the unique inertial system S1 ; or the measured speed of light in every inertial system is found to be the same, namely c, for all directions and for all observers.

24. Einstein’s Principle of Relativity
Maxwell’s equations are true in all inertial reference frames.
Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c = 3.00 × 108 m/s.
Therefore, light travels at speed c in all inertial reference frames.
Every experiment has found that light travels at 3.00 × 108 m/s in every inertial reference frame, regardless of how the reference frames are moving with respect to each other.

25. The first postulate, the principle of relativity, is basic also to classical, or Newtonian, mechanics; the second postulate, the constancy of the speed of light, is at seeming variance with it and also with Postulate I, if the classical concepts of space and time are adhered to.

27. Time Dilation

28. Time Dilation
The time interval between two events that occur at the same position is called the proper time Δτ. In an inertial reference frame moving with velocity v = βc relative to the proper time frame, the time interval between the two events is
The “stretching out” of the time interval is called time dilation.

29. EXAMPLE 37.5 From the sun to Saturn
QUESTIONS:

30. EXAMPLE 37.5 From the sun to Saturn

31. EXAMPLE 37.5 From the sun to Saturn

32. EXAMPLE 37.5 From the sun to Saturn

33. EXAMPLE 37.5 From the sun to Saturn

34. EXAMPLE 37.5 From the sun to Saturn

35. EXAMPLE 37.5 From the sun to Saturn

36. Length Contraction
The distance L between two objects, or two points on one object, measured in the reference frame S in which the objects are at rest is called the proper length ℓ. The distance L' in a reference frame S' is
NOTE: Length contraction does not tell us how an object would look. The visual appearance of an object is determined by light waves that arrive simultaneously at the eye. Length and length contraction are concerned only with the actual length of the object at one instant of time.

37. EXAMPLE 37.6 The distance from the sun to Saturn
QUESTION:

38. EXAMPLE 37.6 The distance from the sun to Saturn

39. EXAMPLE 37.6 The distance from the sun to Saturn

40. EXAMPLE 37.6 The distance from the sun to Saturn

41. The Lorentz Transformations
Consider two reference frames S and S'. An event occurs at coordinates x, y, z, t as measured in S, and the same event occurs at x', y', z', t' as measured in S' . Reference frame S' moves with velocity v relative to S, along the x-axis.
The Lorentz transformations for the coordinates of one event are:

42. The Lorentz Velocity Transformations
Consider two reference frames S and S'. An object moves at velocity u along the x-axis as measured in S, and at velocity u' as measured in S' . Reference frame S' moves with velocity v relative to S, also along the x-axis.
The Lorentz velocity transformations are:
NOTE: It is important to distinguish carefully between v, which is the relative velocity between two reference frames, and u and u' which are the velocities of an object as measured in the two different reference frames.

43. EXAMPLE 37.10 A really fast bullet
QUESTION:

44. EXAMPLE 37.10 A really fast bullet

45. EXAMPLE 37.10 A really fast bullet

46. EXAMPLE 37.10 A really fast bullet

47. Relativistic Momentum
The momentum of a particle moving at speed u is
where the subscript p indicates that this is γ for a particle, not for a reference frame.
If u << c, the momentum approaches the Newtonian value of p = mu. As u approaches c, however, p approaches infinity.
For this reason, a force cannot accelerate a particle to a speed higher than c, because the particle’s momentum becomes infinitely large as the speed approaches c.

48. EXAMPLE 37.11 Momentum of a subatomic particle
QUESTION:

49. EXAMPLE 37.11 Momentum of a subatomic particle

50. EXAMPLE 37.11 Momentum of a subatomic particle

51. EXAMPLE 37.11 Momentum of a subatomic particle

52. EXAMPLE 37.11 Momentum of a subatomic particle

53. Relativistic Energy The total energy E of a particle is
This total energy consists of a rest energy
and a relativistic expression for the kinetic energy
This expression for the kinetic energy is very nearly ½mu2 when u << c.

54. EXAMPLE 37.12 Kinetic energy and total energy

55. EXAMPLE 37.12 Kinetic energy and total energy

56. EXAMPLE 37.12 Kinetic energy and total energy

57. EXAMPLE 37.12 Kinetic energy and total energy

58. Conservation of Energy
The creation and annihilation of particles with mass, processes strictly forbidden in Newtonian mechanics, are vivid proof that neither mass nor the Newtonian definition of energy is conserved. Even so, the total energy—the kinetic energy and the energy equivalent of mass—remains a conserved quantity.
Mass and energy are not the same thing, but they are equivalent in the sense that mass can be transformed into energy and energy can be transformed into mass as long as the total energy is conserved.

60. Chapter 37. Summary Slides

61. General Principles

62. Important Concepts

63. Important Concepts

64. Important Concepts

65. Important Concepts

66. Important Concepts

67. Important Concepts

68. Applications

69. Applications

70. Applications

71. Chapter 37. Clicker Questions

72. Which of these is an inertial reference frames (or a very good approximation)?
A rocket being launched
A car rolling down a steep hill
A sky diver falling at terminal speed
A roller coaster going over the top of a hill
None of the above
Answer: C

73. Which of these is an inertial reference frames (or a very good approximation)?
A rocket being launched
A car rolling down a steep hill
A sky diver falling at terminal speed
A roller coaster going over the top of a hill
None of the above
STT36.1

74. Ocean waves are approaching the beach at 10 m/s
Ocean waves are approaching the beach at 10 m/s. A boat heading out to sea travels at 6 m/s. How fast are the waves moving in the boat’s reference frame?
4 m/s
6 m/s
16 m/s
10 m/s
Answer: C

75. Ocean waves are approaching the beach at 10 m/s
Ocean waves are approaching the beach at 10 m/s. A boat heading out to sea travels at 6 m/s. How fast are the waves moving in the boat’s reference frame?
4 m/s
6 m/s
16 m/s
10 m/s
STT36.2

76. A carpenter is working on a house two blocks away
A carpenter is working on a house two blocks away. You notice a slight delay between seeing the carpenter’s hammer hit the nail and hearing the blow. At what time does the event “hammer hits nail” occur?
Very slightly after you see the hammer hit.
Very slightly after you hear the hammer hit.
Very slightly before you see the hammer hit.
At the instant you hear the blow.
At the instant you see the hammer hit.
Answer: C

77. A carpenter is working on a house two blocks away
A carpenter is working on a house two blocks away. You notice a slight delay between seeing the carpenter’s hammer hit the nail and hearing the blow. At what time does the event “hammer hits nail” occur?
Very slightly after you see the hammer hit.
Very slightly after you hear the hammer hit.
Very slightly before you see the hammer hit.
At the instant you hear the blow.
At the instant you see the hammer hit.
STT36.3

78. A tree and a pole are 3000 m apart
A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is at rest under the tree. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2?
Answer: C
before event 2
after event 2
at the same time as event 2

79. A tree and a pole are 3000 m apart
A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is at rest under the tree. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2?
STT36.4
before event 2
after event 2
at the same time as event 2

80. A tree and a pole are 3000 m apart
A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is flying her rocket at v = 0.5c in the direction from the tree toward the pole. The lightning hits the tree just as she passes by it. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2?
Answer: B
before event 2
after event 2
at the same time as event 2

81. A tree and a pole are 3000 m apart
A tree and a pole are 3000 m apart. Each is suddenly hit by a bolt of lightning. Mark, who is standing at rest midway between the two, sees the two lightning bolts at the same instant of time. Nancy is flying her rocket at v = 0.5c in the direction from the tree toward the pole. The lightning hits the tree just as she passes by it. Define event 1 to be “lightning strikes tree” and event 2 to be “lightning strikes pole.” For Nancy, does event 1 occur before, after or at the same time as event 2?
STT36.5
before event 2
after event 2
at the same time as event 2

82. Molly flies her rocket past Nick at constant velocity v
Molly flies her rocket past Nick at constant velocity v. Molly and Nick both measure the time it takes the rocket, from nose to tail, to pass Nick. Which of the following is true?
Nick measures a shorter time interval than Molly.
Molly measures a shorter time interval than Nick.
Both Molly and Nick measure the same amount of time.
Answer: A

83. Molly flies her rocket past Nick at constant velocity v
Molly flies her rocket past Nick at constant velocity v. Molly and Nick both measure the time it takes the rocket, from nose to tail, to pass Nick. Which of the following is true?
Nick measures a shorter time interval than Molly.
Molly measures a shorter time interval than Nick.
Both Molly and Nick measure the same amount of time.
STT36.6

84. Beth and Charles are at rest relative to each other
Beth and Charles are at rest relative to each other. Anjay runs past at velocity v while holding a long pole parallel to his motion. Anjay, Beth, and Charles each measure the length of the pole at the instant Anjay passes Beth. Rank in order, from largest to smallest, the three lengths LA, LB, and LC.
LA = LB = LC
LB = LC > LA
LA > LB = LC
LA > LB > LC
LB > LC > LA
Answer: C

85. Beth and Charles are at rest relative to each other
Beth and Charles are at rest relative to each other. Anjay runs past at velocity v while holding a long pole parallel to his motion. Anjay, Beth, and Charles each measure the length of the pole at the instant Anjay passes Beth. Rank in order, from largest to smallest, the three lengths LA, LB, and LC.
LA = LB = LC
LB = LC > LA
LA > LB = LC
LA > LB > LC
LB > LC > LA
STT36.7

86. An electron moves through the lab at 99% the speed of light
An electron moves through the lab at 99% the speed of light. The lab reference frame is S and the electron’s reference frame is S´. In which reference frame is the electron’s rest mass larger?
Frame S, the lab frame
Frame S´, the electron’s frame
It is the same in both frames.
Answer: C

87. An electron moves through the lab at 99% the speed of light
An electron moves through the lab at 99% the speed of light. The lab reference frame is S and the electron’s reference frame is S´. In which reference frame is the electron’s rest mass larger?
Frame S, the lab frame
Frame S´, the electron’s frame
It is the same in both frames.
STT36.8