2.1 The Need for Aether 2.2 The Michelson-Morley Experiment 2.3 Einstein’s Postulates

1. CHAPTER 2Special Theory of Relativity 2 • 2.1 The Need for Aether • 2.2 The Michelson-Morley Experiment • 2.3 Einstein’s Postulates • 2.4 The Lorentz Transformation • 2.5 Time Dilation and Length Contraction • 2.6 Addition of Velocities • 2.7 Experimental Verification • 2.8 Twin Paradox • 2.9 Space-time • 2.10 Doppler Effect • 2.11 Relativistic Momentum • 2.12 Relativistic Energy • 2.13 Computations in Modern Physics • 2.14 Electromagnetism and Relativity Albert Einstein (1879-1955) • Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. • Albert Einstein

2. Gedanken (Thought) experiments It was impossible to achieve the kinds of speeds necessary to test his ideas (especially while working in the patent office…), so Einstein used Gedanken experiments or Thought experiments. Young Einstein

3. Re-evaluation of Time! • In Newtonian physics, we previously assumed that t’ = t. • With synchronized clocks, events in K and K’ can be considered simultaneous. • Einstein realized that each system must have its own observers with their own synchronized clocks and meter sticks. • Events considered simultaneous in K may not be in K’. • Also, time may pass more slowly in some systems than in others.

4. Length contraction Simultaneity problems Time dilation The complete Lorentz Transformation If v << c, i.e., β≈ 0 and g ≈ 1, yielding the familiar Galilean transformation. Space and time are now linked, and the frame velocity cannot exceed c.

5. Frank Fil Fred Simultaneity Timing events occurring in different places can be tricky. Depending on how they’re measured, different events will be perceived in different orders by different observers. -L 0 L Due to the finite speed of light, the order in which these two events will be seen will depend on the observer’s position. The time intervals will be: Fred: -2L/c; Frank: 0; Fil: +2L/c But this obvious position-related simultaneity problem disappears if Fred and Fil have synchronized watches.

6. Simultaneity • If light is always the same speed, then observers do not agree on when two events are simultaneous • Can only tell if lightning hit A and B (A’ and B’) simultaneously by getting (light) signals from each!

7. Synchronized clocks in a frame It’s possible to synchronize clocks throughout space in each frame. This will prevent the position-dependent simultaneity problem in the previous slide. But there will still be simultaneity problems due to velocity.

8. K’ Mary Simultaneity So all stationary observers in the explosions’ frame measure these events as simultaneous. What about moving ones? Compute the interval as seen by Mary using the Lorentz time transformation. -L 0 L Mary experiences the explosion in front of her before the one behind her. And note that Dt’ is independent of Mary’s position!

9. 2.5: Time Dilation and Length Contraction More very interesting consequences of the Lorentz Transformation: • Time Dilation: • Clocks in K’run slowly with respect to stationary clocks in K. • Length Contraction: • Lengths in K’contract with respect to the same lengths in stationary K.

10. We must think about how we measure space and time. In order to measure an object’s length in space, we must measure its leftmost and rightmost points at the same time if it’s not at rest. If it’s not at rest, we must ask someone else to stop by and be there to help out. In order to measure an event’s duration in time, the start and stop measurements can occur at different positions, as long as the clocks are synchronized. If the positions are different, we must ask someone else to stop by and be there to help out.

11. Proper Time • To measure a duration, it’s best to use what’s called Proper Time. • The Proper Time, T0, is the time between two events (here two explosions) occurringat the same position (i.e., at rest) in a system as measured by a clock at that position. Same location Proper time measurements are in some sense the most fundamental measurements of a duration. But observers in moving systems, where the explosions’ positions differ, will also make such measurements. What will they measure?

12. Melinda Mary K’ Time Dilation and Proper Time Frank’s clock is stationary in K where two explosions occur. Mary, in moving K’, is there for the first, but not the second. Fortunately, Melinda, also in K’, is there for the second. Mary and Melinda are doing the best measurement that can be done. Each is at the right place at the right time. If Mary and Melinda are careful to time and compare their measurements, what duration will they observe? K Frank

13. Time Dilation • Mary and Melinda measure the times for the two explosions in system K’ as and . By the Lorentz transformation: This is the time interval as measured in the frame K’. This is not proper time due to the motion of K’: . Frank, on the other hand, records x2 – x1 = 0 in K with a (proper) time: T0 = t2 – t1, so we have:

14. Time Dilation • 1)  T’ > T0: the time measured between two events at different positions is greater than the time between the same events at one position: this is time dilation. • 2) The events do not occur at the same space and time coordinates in the two systems. • 3) System K requires 1 clock and K’requires 2 clocks for the measurement. • 4) Because the Lorentz transformation is symmetrical, time dilation is reciprocal: observers in K see time travel faster than for those in K’. And vice versa!

15. Mirror Mirror cT/2 L vT/2 v Time Dilation Example: Reflection Let T be the round-trip time in K Mary Frank Fred K’ K

16. Time Dilation • Consider two observers again • In the train and on the ground • How long does it take the light to go from the flashlight to the mirror and back? • For O’:

17. Time Dilation • Consider two observers again • In the train and on the ground • How long does it take the light to go from the flashlight to the mirror and back? • For O’: • For O:

18. Time Dilation • Consider two observers again • In the train and on the ground • How long does it take the light to go from the flashlight to the mirror and back?

19. But or Reflection (continued) The time in the rest frame, K, is: or or So the event in its rest frame (K’) occurs faster than in the frame that’s moving compared to it (K). or or

23. Time stops for a light wave Because: And, when v approaches c: For anything traveling at the speed of light: In other words, any finite interval at rest appears infinitely long at the speed of light.

24. Proper Length When both endpoints of an object (at rest in a given frame) are measured in that frame, the resulting length is called the Proper Length. We’ll find that the proper length is the largest length observed. Observers in motion will see a contracted object.

25. L0 Frank Sr. where Mary’s and Melinda’s measured length is: Length Contraction • Frank Sr., at rest in system K, measures the length of his somewhat bulging waist: • L0 = xr -xℓ • Now, Mary and Melinda measure it, too, making simultaneous measurements ( ) of the left, , and the right endpoints, • Frank Sr.’s measurement in terms of Mary’s and Melinda’s: ← Proper length Moving objects appear thinner!

26. Length contraction is also reciprocal. So Mary and Melinda see Frank Sr. as thinner than he is in his own frame. But, since the Lorentz transformation is symmetrical, the effect is reciprocal: Frank Sr. sees Mary and Melinda as thinner by a factor of g also. Length contraction is also known as Lorentz contraction. Also, Lorentz contraction does not occur for the transverse directions, y and z.

27. v = 80% c v = 99% c v = 99.9% c Lorentz Contraction v = 10% c A fast-moving plane at different speeds.

28. v 2.6: Addition of Velocities • Taking differentials of the Lorentz transformation [here between the rest frame (K) and the space ship frame (K’)], we can compute the shuttle velocity in the rest frame (ux = dx/dt): Suppose a shuttle takes off quickly from a space ship already traveling very fast (both in the x direction). Imagine that the space ship’s speed is v, and the shuttle’s speed relative to the space ship is u’. What will the shuttle’s velocity (u) be in the rest frame?

29. The Lorentz Velocity Transformations • Defining velocities as: ux = dx/dt, uy= dy/dt, u’x= dx’/dt’, etc., we find: with similar relations for uyand uz: Note the g’s in uy and uz.

30. The Inverse Lorentz Velocity Transformations • If we know the shuttle’s velocity in the rest frame, we can calculate it with respect to the space ship. This is the Lorentz velocity transformation for u’x, u’y, and u’z. This is done by switching primed and unprimed and changing v to –v:

31. Speed, u 0.3c 0.6c 0.9c Speed, u’ v = 0.7c Relativistic velocity addition plot

32. Example: Lorentz velocity transformation As the outlaws escape in their really fast getaway ship at 3/4c, the police follow in their pursuit car at a mere 1/2c, firing a bullet, whose speed relative to the gun is 1/3c. Question: does the bullet reach its target a) according to Galileo, b) according to Einstein? vbp = 1/3c vog = 3/4c vpg = 1/2c outlaws police bullet vpg = velocity of police relative to ground vbp = velocity of bullet relative to police vog = velocity of outlaws relative to ground

33. Galileo’s addition of velocities In order to find out whether justice is met, we need to compute the bullet's velocity relative to the ground and compare that with the outlaw's velocity relative to the ground. In the Galilean transformation, we simply add the bullet’s velocity to that of the police car:

34. Einstein’s addition of velocities Due to the high speeds involved, we really must relativistically add the police ship’s and bullet’s velocities:

35. Example: Addition of velocities We can use the addition formulas even when one of the velocities involved is that of light. At CERN, neutral pions (p0), traveling at 99.975% c, decay, emitting g rays in opposite directions. Since g rays are light, they travel at the speed of light in the pion rest frame. What will the velocities of the g rays be in our rest frame? (Simply adding speeds yields 0 and 2c!) Parallel velocities: Anti-parallel velocities:

36. “Aether Drag” In 1851, Fizeau measured the degree to which light slowed down when propagating in flowing liquids. Fizeau found experimentally: This so-called “aether drag” was considered evidence for the aether concept.

37. Armand Fizeau (1819 - 1896) “Aether Drag” Let K’ be the frame of the water, flowing with velocity, v. We’ll treat the speed of light in the medium ( u, u’ ) as a normal velocity in the velocity-addition equations. In the frame of the flowing water, u’ = c / n which was what Fizeau found.

38. With relativistic correction 2.7: Experimental Verification of Time Dilation Cosmic Ray Muons: Muons are produced in the upper atmosphere in collisions between ultra-high energy particles and air-molecule nuclei. But they decay (lifetime = 1.52 ms) on their way to the earth’s surface: No relativistic correction Top of the atmosphere Now time dilation says that muons will live longer in the earth’s frame, that is, t will increase if v is large. And their average velocity is 0.98c!

39. Detecting muons to see time dilation • It takes 6.8 ms for the 2000-m path at 0.98c, about 4.5 times the muon lifetime. So, without time dilation, we expect only 1000 x 2-4.5 = 45 muons at sea level. Since 0.98c yields g = 5, instead of moving 600 m on average, they travel 3000 m in the Earth’s frame. In fact, we see 542, in agreement with relativity! And how does it look to the muon? Lorentz contraction shortens the distance!

42. 2.8: The Twin Paradox • The Set-up • Mary and Frank are twins. Mary, an astronaut, leaves on a trip many lightyears (ly) from the Earth at great speed and returns; Frank decides to remain safely on Earth. • The Problem • Frank knows that Mary’s clocks measuring her age must run slow, so she will return younger than he. However, Mary (who also knows about time dilation) claims that Frank is also moving relative to her, and so his clocks must run slow. • The Paradox • Who, in fact, is younger upon Mary’s return?

43. t x The Twin-Paradox Resolution • Frank’s clock is in an inertial system during the entire trip. But Mary’s clock is not. As long as Mary is traveling at constant speed away from Frank, both of them can argue that the other twin is aging less rapidly. • But when Mary slows down to turn around, she leaves her original inertial system and eventually returns in a completely different inertial system. • Mary’s claim is no longer valid, because she doesn’t remainin the same inertial system. Frank does, however, and Mary ages less than Frank.

46. There have been many rigorous tests of the Lorentz transformation and Special Relativity. ParticleAccuracy Electrons 10-32 Neutrons 10-31 Protons 10-27 Quantum Electrodynamics also depends on Lorentz symmetry, and it has been tested to 1 part in 1012.

47. 2.9: Space-time • When describing events in relativity, it’s convenient to represent events with a space-time diagram. • In this diagram, one spatial coordinate x, specifies position, and instead of time t, ct is used as the other coordinate so that both coordinates will have dimensions of length. • Space-time diagrams were first used by H. Minkowski in 1908 and are often called Minkowski diagrams. Paths in Minkowski space-time are called world-lines.

48. Particular Worldlines x Stationary observers live on vertical lines. A light wave has a 45º slope. Slope of worldline is c/v.

49. Worldlines and Time Observers at x1 and x2. see what’s happening at x = x3 at t = 0 simultaneously. Alternatively, an event occurring at x3 can be used to synchronize clocks at x1 and x2.

50. Moving Clocks Observers in a frame moving at velocity, v, will see the event happening at x = x3 at t = 0 at different times.