1. Relativity Part III
If twin Astrid aged 10 years during her high-speed trip and Eartha aged 50 years,
what is gamma ?
What is u ?
Ans: γ=5, u~0.980c
Twin Paradox
Quiz
Length Contraction
Relativistic motion perpendicular and parallel to the boost Lorentz transformations
Next time: read 37.6,37.7,37.8

2. Twin Paradox
Eartha and Astrid are twins. Eartha remains on Earth while Astrid travels at relativistic velocities throughout the galaxy. According to Eartha, Astrid’s heartbeat and life processes are proceeding more slowly than her own. When Astrid returns to Earth, she will be younger than Eartha.
But can’t Astrid make the same argument ?
Resolution: the situation is not symmetric. Astrid had to accelerate to attain relativistic velocities. Astrid is not in an inertial reference frame. 2

3. Q10.1
Mavis has a light clock with her on a train. The train is moving with respect to Stanley. Observing Mavis’ light clock, Stanley concluded that it is running
Fast
Slow
At the same rate
uncalibrated
B (delta t = gamma delta t_0; time dilation)

4. Mavis has a light clock with her on a train.
Q10.1
Mavis has a light clock with her on a train.
The train is moving with respect to Stanley.
Observing Mavis’ light clock, Stanley concluded that
it is running
Fast
Slow
At the same rate
Uncalibrated
B (delta t = gamma delta t_0; time dilation)

5. According to Stanley, Mavis’ clock is running slow.
Q10.2
According to Stanley, Mavis’ clock is running slow.
To verify this, Stanley places two of his clocks at two locations, one at the beginning of the light pulse, one at the end of the light pulse. The “proper time” refers to
The time interval of these two events as measured by Mavis’ clock.
The time interval of these two events as measured by Stanley’s two clocks at two locations.
A time interval measured by a clock in the inverted Lorentz frame.
A (proper time, time between events at the same spatial location in one frame)

6. According to Stanley, Mavis’ clock is running slow.
Q10.2
According to Stanley, Mavis’ clock is running slow.
To verify this, Stanley places two of his clocks at two locations, one at the beginning of the light pulse, one at the end of the light pulse. The “proper time” refers to
The time interval of these two events as measured by Mavis’ clock.
The time interval of these two events as measured by Stanley’s two clocks at two locations.
A time interval measured by a clock in the inverted Lorentz frame.
A (proper time, time between events at the same spatial location in one frame)

7. According to Stanley, Mavis’ clock is running slow.
Q10.3
According to Stanley, Mavis’ clock is running slow.
Mavis agrees with Stanley that her clock is slow.
Mavis disagrees with Stanley and claims that Stanley’s two clocks at two locations are not synchronized correctly.
Mavis agrees with Stanley that the length of her spaceship is Lorentz contracted.
B (Stanley measures at two different space-time points)

8. Stanley measures at two different space-time points
Q10.3
Stanley measures at two different space-time points
According to Stanley, Mavis’ clock is running slow.
Mavis agrees with Stanley that her clock is slow.
Mavis disagrees with Stanley and claims that Stanley’s two clocks at two locations are not synchronized correctly.
Mavis agrees with Stanley that the length of her spaceship is Lorentz contracted.
B (Stanley measures at two different space-time points)

9. Another time dilation example (not a clicker)
Mavis boards a spaceship and zips past Stanley on earth at a relative speed of 0.600c. At the instant she passes him both start timers.
a) A short time later Stanley measures that Mavis has traveled 9.00 x 107m and is passing a space station. What does Stanley’s timer read as she passes the station ? What does Mavis’ timer read ?
Reference frame S of Stanley; reference frame S’ of Mavis, speed of S’ relative to S is u=0.600c
In S, Mavis passes at c = 1.80 x 108m/s and covers the distance in x 107m/1.80 x 108m/s= 0.500s
Mavis is measuring proper time 9

10. Recall meaning of “Proper time”
Proper time is the time interval between two events that occur at the same point in space.
A frame of reference can be pictured as a coordinate system with a grid of synchronized clocks, as in the Figure at the right. 10

11. Relativity of length 11

12. Relativity of length Measurement of length of ruler in Mavis’ frame
Note that this is a proper time interval, start and stop are measured at the same point in space.
Note “zero” subscripts on t and l 12

13. Relativity of length Stanley: distance from source to mirror
Stanley: distance from mirror back to source
Why did the sign change ? 13

14. Relativity of length 14

15. Relativity of length
Insert delta t back in terms of l_0 and c. 15

16. Length contraction (Lorentz contraction)
Cancel factors of c. collect factor of (1-u^2/c^2)
Length contraction 16

17. Length contraction (Lorentz contraction)
Check: if u is such that Δt0 = ½ Δt1, what is l ? (in terms of proper length)
¼ l0
½ l0 l0
2 l0
Cancel factors of c. collect factor of (1-u^2/c^2) 17

18. Length contraction (Lorentz contraction)
Check: if u is such that Δt0 = ½ Δt1, what is l ? (in terms of proper length)
Here γ = 2
¼ l0
½ l0 l0
2 l0
E. 4 l0
Cancel factors of c. collect factor of (1-u^2/c^2) 18

19. English translation of the original (not so different from the textbook).
English translation of Einstein’s 1905 paper on special relativity 19

20. History: Special Relativity‘s impact on 20th century art
  “In the intellectual atmosphere of 1905 it is not surprising that Einstein and Picasso began exploring new notions of space and time almost coincidentally. The main lesson of Einstein's 1905 relativity theory is that in thinking about these subjects, we cannot trust our senses. Picasso and Einstein believed that art and science are means for exploring worlds beyond perceptions, beyond appearances. Direct viewing deceives, as Einstein knew by 1905 in physics, and Picasso by 1907 in art. Just as relativity theory overthrew the absolute status of space and time, the cubism of Georges Braque and Picasso dethroned perspective in art.”
Picasso, Les Demoiselles d’Avignon, 1907 20

21. Review: Time dilation vs velocity (the “gamma factor”)
Introduce the γ factor. 21

22. Review: Length contraction (Lorentz contraction)
Time dilation
Cancel factors of c. collect factor of (1-u^2/c^2)
Limit u0 ?? 22

23. Lengths perpendicular to the direction of motion
If Mavis has a meter stick as shown, what does Stanley think the height is?
Shorter than 1m 1m
Longer than 1m
g * 1m
Complicated, depends on u 23

24. Lengths perpendicular to the direction of motion
Very important point: There is no length contraction for lengths perpendicular to the direction of relative motion.
Note the lengths of the two meter sticks oriented perpendicular to the length of motion (they are equal). 24

25. Lengths perpendicular to the direction of motion
Very important point: There is no length contraction for objects perpendicular to the direction of relative motion.
S’ is moving along the x axis with respect to S L0 S’
L0sin(θ0) S
L0sin(θ0) θ0
Note the y projection is unchanged
L0cos(θ0)
(L0/γ)cos(θ0)
Question Is the angle of the meter stick still θ0 in S ?
More on this in Lorentz transformation section
No in S, it is now tan-1(γsin(θ0)/ cos(θ0) 25

26. Example of length contraction (or Lorentz contraction)
A spaceship flies past earth at 0.990c.
A crew member on-board measures its length and obtains 400m.
32.7m
56.3m
74.4m
400m
B 56.4
Question: What do observers on earth measure ? 26

27. Example of length contraction (or Lorentz contraction)
A spaceship flies past earth at 0.990c.
A crew member on-board measures its length and obtains 400m.
Question: What do observers on earth measure ? 27

28. The Lorentz transformations
Lorentz transformations relate the coordinates and velocities in two inertial reference frames. They are more general than the Galilean transformations and are consistent with the principle of relativity.
Galilean transformations. Do not work at relativistic velocities. 28

29. The Lorentz transformations (“boost along x”)
Space and time are intertwined: 4 dimensional “space-time”
Intellectual revolution
Note the coordinates perpendicular to the “boost“ are unmodified 29

30. The Lorentz transformations (“boost along x”)
Space and time are intertwined: 4 dimensional “space-time”
Intellectual revolution
Time dilation
Caution!
Don’t confuse these 2! 30

31. For next time Relativity continues (37.6,37.7,37.8) Read in advance
Concepts require wrestling with material