1. CHAPTER 2 Special 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 ( )
Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.
Albert Einstein
It's rotten To find I'd completely forgotten That by living so fast, All my future's my past, And I buried before I'm begotten.
Prof. Rick Trebino, Georgia Tech,

2. The Complete Lorentz Transformation
Length contraction
Simultaneity problems
Time dilation
If v << c, i.e., β ≈ 0 and g ≈ 1, yielding the familiar Galilean transformation.
Space and time are now linked.

3. 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.
Einstein pic:
Lightning and space ship:
Young Einstein

4. Simultaneity
Timing events occurring in different places can be tricky. Even without special relativity, different events will be perceived in different orders by different observers.
Prof. Filbert
Farrah
Fred -L 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; Farrah: 0; Prof. Filbert: +2L/c
But this obvious position-related simultaneity problem disappears if Fred and Prof. Filbert have synchronized watches and can tweet each other.

5. Synchronized Clocks in a Frame
It’s possible to synchronize clocks throughout space in each frame.
This will prevent position-dependent simultaneity problems, which we can forget about, as they have nothing to do with special relativity.
But there will be much more subtle simultaneity problems in special relativity due to velocity.

6. Simultaneity
So all stationary observers in the explosions’ frame measure these events as simultaneous: T = t2 – t1 = 0.
What about observers in a moving frame? We’ll need two of them!
Compute the interval T’ as seen by Mary and Melinda using the Lorentz time transformation. K’
Mary K’
Melinda
Event #1: (t1 = 0, x1 = -L)
Event #2: (t2 = 0, x2 = +L) -L L
Mary and Melinda experience the orange explosion before the yellow one. And note that T’ is independent of position (x’) because these times are properties of the entire K’ frame.

7. 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.

8. We must think about how we measure space and time.
In order to measure an object’s length in space, we should measure its leftmost and rightmost points at the same time.
If the object is moving, a single observer can’t, so he must ask someone else to stop by and be there to help out.
In order to measure an event’s duration in time, we should make the start and stop measurements at the same point in space.
If the object is moving, a single observer can’t, so he must ask someone else to stop by and be there to help out.
Ruler:
Clock:

9. 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) occurring at 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 time measurements.
What will they measure?

10. 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. K’
Mary
Melinda
Mary and Melinda are doing the best measurement that can be done in their frame. 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

11. Time Dilation
Mary and Melinda measure the times for the two explosions in system K’ as and By the Lorentz transformation: T0
T’ is the time interval as measured in the frame K’. It 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:

12. 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!

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

14. Reflection (continued)
We’ll solve for the time T’ in the moving frame K’ to verify time dilation.
First recall that: so
Substituting for L:
Multiplying by (2/c)2:
Isolating T’ terms:
Or:
So the event in the moving frame K’ occurs more slowly than in its rest frame K.

15. 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.

16. Proper Length
When both endpoints of an object are at rest in a given frame and 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.

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

18. 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.

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

20. 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:
And their average velocity is 0.98c!
No relativistic correction
With relativistic correction
Image and some text taken from Warren Rogers Modern Physics lectures
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.

21. Detecting Muons to See Time Dilation
It takes 6.8ms for a 2000m path at 0.98c, about 4.5 times the muon lifetime. So, without time dilation, of every 1000 muons seen at the top of a mountain, we expect only 1000 × = 45 muons at sea level.
Since 0.98c yields g = 5, instead of moving only 600m on average, they live 7.6ms and travel 3000m in the Earth’s frame.
In fact, we see 542, in agreement with relativity!
And how does it look to the muon? It knows its lifetime is 1.52ms.
But Lorentz contraction shortens the distance!

22. 2.8: The Twin Paradox Mary and Frank are twins.
Mary leaves on a trip many light-years from Earth at great speed and returns; Frank remains on Earth.
Frank knows that Mary’s time must run slow, so she will return younger than he. However, Mary claims that Frank is also moving relative to her, and so his clocks must run slow.
Who, in fact, is younger upon Mary’s return?
Image from

23. 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 remain in the same inertial system. Frank does, however, and Mary ages less than Frank.
Indeed, Frank’s calculation of the elapsed time is correct.
Mary’s is not.
Image from members.shaw.ca/vcofell3/myweb6/p.htm

24. Atomic Clock Measurement
vplane~ vrotation
Two airplanes traveled east and west, respectively, around the Earth as it rotated. The plane traveling west was effectively at rest, while the one going east had a speed of about twice that of the earth’s surface.
The clock in the eastward-flying airplane ran slower.
Predicted Observed
Time difference: ± 30ns 332 ± 12ns

25. There have been many rigorous tests of the Lorentz transformation and Special Relativity.
Particle Accuracy
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.

26. 2.6: Addition of Velocitiesu'
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?
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):

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

28. 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:

29. Relativistic Velocity Addition
Speed, u’
0.25c
Speed, u
0.50c
0.75c
v = 0.75c
1.0c
0.9c
0.8c
1.1c
Galilean velocity addition
Relativistic velocity addition
Note: this plot exaggerates the difference between the two laws; also, it’s not possible to relabel the u’ axis to fix it. But it’s not off by much (at most a few per cent).

30. Example: Lorentz Velocity Transformation
The Enterprise escapes from a hostile Romulan ship at 3c/4, but the Romulans follow at c/2, firing matter torpedos, whose speed relative to the Romulan ship is c/3.
Question: Does the Enterprise survive?
v = c/2
u’x = c/3
vE = 3c/4
Romulans
Torpedo
Enterprise
v = velocity of Romulans relative to galaxy
u’x = velocity of torpedo relative to Romulans
vE = velocity of Enterprise relative to galaxy

31. Galileo’s Addition of Velocities
We need to compute the torpedo's velocity relative to the galaxy (ux) and compare that with the Enterprise's velocity relative to the galaxy (vE).
Using the Galilean transformation, we simply add the torpedo’s velocity to that of the Romulan ship:

32. Einstein’s Addition of Velocities
Due to the high speeds involved, we really must relativistically add the Romulan ship’s and torpedo’s velocities:
The Enterprise survives to seek out new worlds and go where no one has gone before…

33. Addition of Velocities: Very Cool Example
Pions (p0) decay, emitting g rays, traveling at c in opposite directions. p0 p0
But what if the pion is traveling at c? Simply adding velocities yields speeds of ≈ 0 and ≈ 2c for the g rays in our frame! p0
Interestingly, the velocity-addition formulas work even for light.
Parallel velocities:
Anti-parallel velocities:

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

35. which was what Fizeau found.
“Aether Drag”
Armand Fizeau ( )
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
Picture from
which was what Fizeau found.

36. 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 and are often called Minkowski diagrams. Paths in Minkowski space-time are called world-lines.

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

38. The Light Cone
The past, present, and future are easily identified in space-time diagrams. And if we add another spatial dimension, these regions become cones.

39. Space-Time Interval and Metric
Recall that, since all observers see the same speed of light, all observers, regardless of their velocities, must see spherical wave fronts from a point source of light.
s2 = x2 + y2 + z2 – c2t2 = x’2 + y’2 + z’2 – c2t’2 = s’2
This interval can be written in terms of the space-time metric: x z y

40. Space-Time Invariants
The quantity Ds2 between two events is invariant (the same) in any inertial frame.
Ds is known as the space-time interval between two events.
There are three possibilities for Ds2:
Ds2 = 0: Dx2 = c2 Dt2, and the two events can be connected only by a light signal. The events are said to have a light-like separation.
Ds2 > 0: Dx2 > c2 Dt2, and no signal can travel fast enough to connect the two events. The events are not causally connected and are said to have a space-like separation.
Ds2 < 0: Dx2 < c2 Dt2, and the two events can be causally connected. The interval is said to be time-like.