1. “Epiphenomena” of relativity
• moving clocks run slow
• moving rulers shrink (in the direction of motion)
• moving masses get more massive
But motion is relative! If you are moving relative to me, your meter stick is shorter than one meter, and is shorter than my meter stick.
But since I am moving relative to you, my meter stick is shorter than one meter, and is shorter than your meter stick.
You cannot have A<B and B<A.

2. The right geometry can tell us about physics
A hunter leaves his camp, walks five miles west, and then five miles north. He shoots a bear, and then drags it five miles south back to his camp.
What color was the bear?
Brown
White
Black
Pink
Puce

3. The laws of physics are the same in any inertial (non-accelerating) frame of reference Galileo & Einstein would both agree (at terrestrial speeds.) F=ma is true, regardless of whether you are on a train, or stationary. x’ = x - vF t, take two derivatives. The speed of light is independent of the speed of the observer or source. Galileo would say that this is impossible. If you are on a train and throw a ball forward with speed v, then on the ground the ball moves with speed v + vtrain. It’s crazy to suggest that it could move at speed v when measured by me, relative to the train, and also by you, when measured relative to the ground. It may be crazy, but it is true, if the ball is moving at speed c. This is an experimental result, repeated tens of thousands of times.

4. Note: If by “laws of physics” you mean to include Maxwell’s equations, then postulate 2 is automatically included, since M.E.s predict the speed of light. (Of course, Galileo (d. 1642) was woefully unfamiliar with Maxwell. (b. 1831)

5. A reference frame is a coordinate system, plus synchronized clocks located at all points in space. An inertial frame is one that is not accelerating.

6. A reference frame is a coordinate system, plus synchronized clocks located at all points in space. An inertial frame is one that is not accelerating. Acceleration gives “fictitious forces” in the frame.
Which of the following is not a fictitious force?
Choose any correct answer.
A] centrifugal force
B] coriolis force
C] gravity
D] electric force

7. In an inertial frame, no force still means no acceleration.

8. Spacetime diagrams & Worldlines
What is the worldline of a particle that is accelerating (+ direction)?
Which worldline is
(that of a stationary particle in) an inertial reference frame?

9. A train is moving at constant speed
A train is moving at constant speed. It is shown as the primed frame; the earth is unprimed. Times are in seconds, distances in meters. How fast is the train moving, in m/s? A] 0.2 B] 0.5 C] 1 D] 2 E] 5

10. A little boy throws a ball on the train
A little boy throws a ball on the train. The world line of the ball is shown. How fast does the ball move with respect to the earth (in m/s)? A] 0.2 B] 0.5 C] 1 D] 2 E] 5
ball

11. A little boy throws a ball on the train
A little boy throws a ball on the train. The world line of the ball is shown. How fast does the ball move with respect to the train (in m/s)? A] 0.2 B] 0.6 C] 1.5 D] 2 E] 5
ball

12. t=t’
These are vector equations, though this example is 1D
ball

13. An “event” is something that happens at a point in space and at an instant in time.
For example, the child throwing the ball can be thought of as an event.
The ball hitting the conductor is another event. e2
ball e1

14. Invariant - a quantity that is the same for all inertial observers.
In Galilean relativity, what is invariant? Choose any correct answer.
A] the distance between two events
B] the time between two events
C] the kinetic energy transferred in a collision
D] the mass of a particle

15. These clickers aren’t properly registered. Let me know who you are.

16. Galilean velocity addition & homework
13. Suppose light moves through ether with speed c and the earth moves through the ether with speed v.
If a beam of light moves at an angle ’ wrt to v in the earth frame, how fast does it move?

17. Michelson Morley Experiment The speed of light is independent of the motion of the earth. It has also been performed with starlight, so “ballistic” theories are ruled out. Light does not move faster if the source is moving! How long does it take to swim across a river & back, vs. upstream and downstream?

18. Let’s figure out what takes longer: a swim upstream and back downstream, or a swim across and back.
Up and down
Across and back
They take the same time

19. Michelson Morley Experiment Interferometer
Michelson Morley Experiment Interferometer. Remember the frequency of light is fixed. If we find out how long light takes to swim up and down, vs. across and back, we can find the phase difference between the beams at A.

20. Now let’s get back to spacetime diagrams.
We now know the speed of light is the same in any inertial reference frame.
Let’s scale our x and t axes so that a light beam moves with a 45° world line. We will write ct on the t axis.
We can label the x axis in light-seconds (a distance) and the t axis in seconds, with the understanding that we will multiply t by c.

21. In a spacetime diagram, the x’-axis is always: a) perpendicular to the t’ axis at t’=0 b) the collection of all events that happen at t’=0 c) both d) neither

22. Spacetime diagrams : revisited. What is the x-axis?
It is the set of all events that happen at t=0.
These are simultaneous.
Any line parallel to the x-axis represents the set of all events happening at time t.

25. Consider a long spaceship moving in the +x direction
Consider a long spaceship moving in the +x direction. When the middle of the spaceship is at the origin, a light flashes. When does the flash reach the back and the front of the ship? c c
WRONG X’

26. The speed of light is independent of the speed of the observer
The speed of light is independent of the speed of the observer. So, according to the spaceman, the light reaches the front and the back of his ship at the same time. c c
Right
A and B are simultaneous to the spaceman.
Thus, the line AB is parallel to the x’ axis!

27. Events that are simultaneous to one observer are NOT to a different observer.c c
Right
A and B are simultaneous to the spaceman.
Thus, the line AB is parallel to the x’ axis!

28. Note well: you do NOT “drop a perpendicular” to find a coordinate value.

29. The primed reference frame could best represent: a) a spaceship moving toward +x b) a spaceship moving toward -x c) a stationary spaceship

30. Two firecrackers explode at exactly the same time, as seen in the x-ct (“lab” or “earth”) frame. What is true in the spaceship frame? A) A explodes first B) B explodes first C) they are simultaneous

31. Monday
A train moving at c/2 turns on its headlight. How fast does the beam of light move forward, according to the engineer?
c/2 c
3c/2 2c
Other value

32. Monday
A train moving at c/2 turns on its headlight. How fast does the same beam of light move forward, according to the a person on the ground?
c/2 c
3c/2 2c
Other value

33. The constancy of the speed of light means we must abandon the idea of simultaneity of distant events.
What is simultaneous to one observer is not to a different observer.
In our spacetime diagrams, this means the x’ axis tilts.
The tilt of the x’ axis is the same as the ct’ axis (which is the worldline of the primed frame.)

34. Two firecrackers explode at exactly the same time, as seen in the x-ct (“lab” or “earth”) frame. What is true in the spaceship frame? A) A explodes first B) B explodes first C) they are simultaneous

35. Two firecrackers explode at exactly the same time, as seen in the x-ct (“lab” or “earth”) frame. What would be observed by a spaceship moving toward -x? A) A explodes first B) B explodes first C) they are simultaneous

36. If we arrange that the light from firecracker A causes firecracker B to to explode (without any delay, other than the light travel time), what spacetime diagram shows that? Choose A, B, C, or D -- it would look different in different frames

37. Slope of the ct’ axis = c/vss. What is the slope of the x’ axis
Slope of the ct’ axis = c/vss. What is the slope of the x’ axis? A) c/vss B) vss/c C) xss/t D) vss/xss

38. The general form of coordinate transformations

39. The general form of coordinate transformations
Note that every term has units of length

40. The Lorentz transformation. How do we “calibrate” the axes
The Lorentz transformation. How do we “calibrate” the axes? I like the light clock. The speed of light is c. Let’s work this on the board.
Which is larger, t or t’? t t’
They are the same

41. Which clock runs slow (in the picture of the light clock we are using)?
spaceship
earth
They are the same

42. Since the spaceship clock runs slow compared to the earth clock, when the spaceman looks at earth clocks he should see:
They are running fast, compared to his own
They are running slow, compared to his own
They are the same as his own

43. Why is there no contradiction here?
The fundamental entity in relativity is the “event”… something that happens at a specific place and time.
Suppose I ask for the time between event A and event B.
Observer P says this time is tP. Observer Q says this time is tQ. Which is smaller tp or tQ ?
I don’t know. But if tp < tQ, it is certainly NOT possible that tQ < tP !!!!!

44. What has this got to do with clocks?
The events that mark a tick on a light clock are
The departure of the pulse from the floor and
The return of the pulse to the floor.
These happen at the same place in the spaceship, but at different places in the earth frame.
What does the spaceman SEE on the earth clocks (that are running slow)?
He sees that the earth clocks are NOT PROPERLY SYNCHRONIZED! He says to his earth friend, “of course you think MY clocks are slow… the earth clock that was next to my ship when the light pulse returned to the floor was AHEAD of the clock next to my ship when the pulse first left the floor!
Can we see this on the spacetime diagram?

45. What will different observers disagree about in relativity?
Wednesday - Feb 26
What will different observers disagree about in relativity?
Alice and Bob may disagree about when events actually occurred. They figure out when an event (think firecracker explosion) actually occurred by subtracting the light travel time from when they saw the event.
e.g.they will disagree about where wavefronts are. Each will observe wavefronts that are equidistant from where the emitter was. In the emitter frame, these are centered on the emitter always. In the lab frame, they are not.
What can they NOT disagree about?
Alice cannot disagree with Bob about what Bob’s clock says when Bob sees an event. Bob cannot disagree about what Alice’s clock says when Alice sees an event.
By the way, there can be no sideways contractions of objects. Why? Pipes!

46. “Moving clocks run slow”… the light clock.
In the spaceship, the clock moves 1 second when the light moves from floor to ceiling and back.
In the earth frame, this same light traveled a much greater distance, so the time is longer, by the factor 
But since the earth is the moving frame, according to the spaceship, earth clocks must run slow, as measured by the spaceship.
In the light clock picture on the board, what does the earth-frame clock show when the light returns to the floor?
Since Earth clocks run slow, it will show 1/
It will show 1
It will show 

47. c) It will show 
It will show a larger elapsed time.
BUT!!! This is a different clock from the one used to time the start of the light pulse! Since the spaceship has moved, the pulse returns to the detector in a different place from where it was emitted.
So according to the spaceman, the Earth’s clocks run slowly, BUT THEY ARE IMPROPERLY SYNCHRONIZED. Earth clocks that are farther ahead in space (in the direction of the the spaceship’s motion) are set too much ahead.

48. Note that the spaceman IS NOT WRONG.
The earth clocks are improperly synchronized.
But the earthman IS NOT WRONG.
The earth clocks are properly synchronized in the Earth reference frame.
----
Now, let’s finish calibrating our axes and deriving the Lorentz transformation.

49. Lorentz contraction
If we measure a time interval between two events at the same place in the spaceship (light the light clock), we get a bigger interval than the spaceship clock shows.
If we measure a space interval (length of a meter stick), we get a smaller interval than the spaceship gets.

50. What is the length of a meter stick (if it’s moving)?
Answer: it is the distance between the ends, measured at the same time!

51. The spacetime interval is invariant.
For half a light clock cycle, this interval is L, which is manifestly invariant

52. Spacelike separated events are simultaneous in some frame.
They occur in the opposite order in some frames.

53. Inverse transform has V -> -V
If a particle has velocity vx in the unprimed frame, what is its velocity in the primed frame? Just take differentials of Lorentz.
Inverse transform has V -> -V

54. Lengths in y,z don’t change, but velocities do. Why?