1. RELATIVITY III SPECIAL THEORY OF RELATIVITY
PHYS 420-SPRING 2006
Dennis Papadopoulos
LECTURE # 3
RELATIVITY III
SPECIAL THEORY OF RELATIVITY

3. Special Relativity I Einstein’s postulates Simultaneity Time dilation
Length contraction
Lorentz transformation
Examples
Doppler effect

4. I: EINSTEIN’S POSTULATES OF RELATIVITY
Postulate 1 – The laws of nature are the same in all inertial frames of reference
Postulate 2 – The speed of light in a vacuum is the same in all inertial frames of reference.
Let’s start to think about the consequences of these postulates.
We will perform “thought experiments” (Gedanken experiment)…
For now, we will ignore effect of gravity – we suppose we are performing these experiments in the middle of deep space

5. INVARIANCE OF SPEED OF LIGHT

6. REFERENCE FRAME -> GRID (3D) + SET OF CLOCKS CLOCKS SYNCHRONIZED
Figure 1.8 In relativity, we use a reference frame consisting of a coordinate grid and a set of synchronized clocks.
Fig. 1-8, p. 13

7. SIMULTANEITY NEWTON -> UNIVERSAL TIMESCALE FOR ALL OBSERVERS
“Absolute, true time, of itself and of its own nature, flows equably, without relation to anything external”
EINSTEIN
“A time interval measurement depends on the reference frame the measurement is made”

8. II. SIMULTANEITY

9. Figure 1. 9 Two lightning bolts strike the ends of a moving boxcar
Figure 1.9 Two lightning bolts strike the ends of a moving boxcar. (a) The events appear to be simultaneous to the stationary observer at O, who is midway between A and B. (b) The events do not appear to be simultaneous to the observer at O’, who claims that the front of the train is struck before the rear.
Fig. 1-9, p. 14

10. (a) The events appear to be simultaneous to the stationary observer at O, who is midway between A and B.
Fig. 1-9a, p. 14

11. (b) The events do not appear to be simultaneous to the observer at O’, who claims that the front of the train is struck before the rear.
Fig. 1-9a, p. 14

12. III: TIME DILATION
A light clock consists of two parallel mirrors and a photon
bouncing back and forth over the distance D. An observer
at rest with the clock will measure a click at times
Dto= 2H/c H

13. Now suppose that we put the clock on a platform sliding at constant
speed v. Use Einstein’s postulate d H
vDt
d2=H2+(vDt/2)2
Proper time- observer sees events at the same spatial point – Rides with the clock

14. Clock appears to run more slowly. An astronaut in the spacecraft
the inside of the spacecraft is also an inertial frame of reference – Einstein’s postulates apply…
So, the astronaut will measure a “tick” that lasts
So, different observers see the clock going at different speeds!
Time is not absolute!
Moving clocks run slow
Dto=H/c

15. Lorentz factor

16. Figure 1. 11 (a) Muons traveling with a speed of 0
Figure 1.11 (a) Muons traveling with a speed of 0.99c travel only about 650 m as measured in the muons’ reference frame, where their lifetime is about 2.2 s. (b) The muons travel about 4700 m as measured by an observer on Earth. Because of time dilation, the muons’ lifetime is longer as measured by the Earth observer.
Fig. 1-11, p. 17

17. (a) Muons traveling with a speed of 0
(a) Muons traveling with a speed of 0.99c travel only about 650 m as measured in the muons’ reference frame, where their lifetime is about 2.2 s.
Fig. 1-11a, p. 17

18. (b) The muons travel about 4700 m as measured by an observer on Earth
(b) The muons travel about 4700 m as measured by an observer on Earth. Because of time dilation, the muons’ lifetime is longer as measured by the Earth observer.
Fig. 1-11b, p. 17

19. Figure 1. 12 Decay curves for muons traveling at a speed of 0
Figure 1.12 Decay curves for muons traveling at a speed of c and for muons at rest.
Fig. 1-12, p. 17

20. Figure 1.10 (a) A mirror is fixed to a moving vehicle, and a light pulse leaves O’ at rest in the vehicle. (b) Relative to a stationary observer on Earth, the mirror and O’ move with a speed v. Note that the distance the pulse travels measured by the stationary observer on Earth is greater than 2d. (c) The right triangle for calculating the relationship between t and t’.
Fig. 1-10, p. 15

21. (a) A mirror is fixed to a moving vehicle, and a light pulse leaves O’ at rest in the vehicle.
Fig. 1-10a, p. 15

22. (b) Relative to a stationary observer on Earth, the mirror and O’ move with a speed v. Note that the distance the pulse travels measured by the stationary observer on Earth is greater than 2d.
Fig. 1-10b, p. 15

23. IV: LENGTH CONTRACTION
The only way observers in motion relative to each other can measure a single light ray to travel the same distance in the same amount of time relative to their own reference frames is if their ``meters'' are different and their ``seconds'' are different! Seconds and meters are relative quantities.

24. Figure 1. 13 A stick moves to the right with a speed v
Figure 1.13 A stick moves to the right with a speed v. (a) The stick as viewed in a frame attached to it. (b) The stick as seen by an observer who sees it move past her at v. Any inertial observer finds that the length of a meter stick moving past her with speed v is less than the length of a stationary stick by a factor of (1 - v2/c2)1/2.
Fig. 1-13, p. 19

25. (a) The stick as viewed in a frame attached to it.
Fig. 1-13a, p. 19

26. (b) The stick as seen by an observer who sees it move past her at v
(b) The stick as seen by an observer who sees it move past her at v. Any inertial observer finds that the length of a meter stick moving past her with speed v is less than the length of a stationary stick by a factor of (1 - v2/c2)1/2.
Fig. 1-13b, p. 19

27. Figure 1.14 Computer-simulated photographs of a box (a) at rest relative to the camera and (b) moving at a speed v = 0.8c relative to the camera.
Fig. 1-14, p. 20

28. Figure 1.15 (Example 1.5) (a) When the spaceship is at rest, its shape is as shown. (b) The spaceship appears to look like this when it moves to the right with a speed v. Note that only its x dimension is contracted in this case. The 25-m vertical height is unchanged because it is perpendicular to the direction of relative motion between the observer and the spaceship. Figure 1.15b represents the shape of the spaceship as seen by the observer who sees the ship in motion.
Fig. 1-15, p. 21

29. Length contraction… also called
So, moving observers see that objects contract in the direction of motion.
Length contraction… also called
Lorentz contraction
FitzGerald contraction

30. Everything is slowed/contracted by a
factor of:
in a frame moving with respect to the observer.
Time always runs slower when measured by an observer moving with respect to the clock.
The length of an object is always shorter when viewed by an observer who is moving with respect to the object.

31. Galilean transformation from S to S’: xv x’
Galilean transformation from S to S’: x
Guess that the relativistic version has a similar form but differs by some dimensionless factor G:
(we know that this must reduce to the Galilean transformation as v/c ->0)
The transformation from S’ to S must have the same form:
From first postulate of relativity-laws of physics must have the same form in S and S’
substitute
into
solve for t’, you get:

32. u=dx/dt=c u’=dx’/dt’=c
Now we need an expression for the velocity dx’/dt’ in the moving frame:
Take derivatives of:
where u=dx/dt
From second postulate of relativity- the speed of light must be the same for an observer in S and S’
u=dx/dt=c u’=dx’/dt’=c
Plug this into:
and get:
Solve for G:

33. The Lorentz transformations!
The transformation: v x’ x
To transform from S’ back to S: