1. Some places where Special Relativity is needed
Particle accelerator: electrons and positrons moving at 99.99% c
Supernova in a distant galaxy. Remember the universe is expanding !
GPS satellite in orbit (v~3.9 km/sec, 7μs/day SR correction)

2. Review from Friday: Einstein’s postulates
Einstein’s first postulate: The laws of physics are the same in all inertial reference frames
Einstein’s second postulate is that the speed of light in vacuum is the same in all inertial frames of reference and is independent of the motion of the source.
We must modify determination of space and time intervals when a frame of reference is moving relative to us at high velocity. (Today’s class: tough deep material, please slow me down if I start going too fast. Reread the textbook when you go home).
Another consequence: it is impossible for an inertial observer to travel at c, the speed of light in the vacuum.
We will do two derivations (time dilation and length contraction). These are quite deep (not just algebra).

3. Relativity of time intervals and Time Dilation
The two observers (Mavis and Stanley) measure different time intervals due to their relative motion. Let’s work out this important example in full detail.

4. Relativity of time intervals and Time Dilation
The two observers (Mavis and Stanley) measure different time intervals due to their relative motion. Let’s work out this example.
Mavis in reference frame S’ fires a light source at a mirror and measures the time interval. What does she find ?
This is a “proper time interval” – a time interval between events at the same space point.

5. Relativity of time intervals and Time Dilation
Stanley in reference S measures this time interval. What does he find ?
What length does Stanley measure ?

6. Relativity of time intervals and Time Dilation
The time intervals measured by Mavis (frame S’) and Stanley (frame S) are different !
Let’s compare the two results

7. Relativity of time intervals and Time Dilation
The time intervals measured by Mavis (frame S’) and Stanley (frame S) are different ! Let’s work out how they are related.
Now square this and collect Δt terms
Insert d into the delta t expression.

8. Relativity of time intervals and Time Dilation
Question: does Stanley measure a longer or shorter time interval ?
Stanley measures a longer round trip time than Mavis !
This is called relativistic time dilation

9. Time dilation vs velocity (let’s plot the “gamma factor”)
Introduce the γ factor.

10. Do muons produced in cosmic rays make it to ground level ?
Flux of muons at ground level ~1/cm2/min
No muons in my lab course PHYS481L
Apparent Paradox ?
The lifetime of the muon is 2.2 microseconds. It is moving close to the speed of light (3 x 108m/s). Therefore it will travel about 6.6 x 102 m before decaying. But the earth’s atmosphere is over 20 km high. So no muons will be found at ground level.

11. Time dilation example
A muon decays into other particles with a mean lifetime of 2.20 μs = 2.20 x 10-6 sec as measured in a reference frame in which it is at rest.
If a muon is moving at 0.990c relative to the earth, what will an observer on earth measure its mean lifetime to be ?
Reference frame S’ of the muon, proper lifetime Δt0= 2.20 μs
Reference frame S of the earth; relative speed of S’ and S is u=0.990c
Question: What is the gamma factor here ?
7.1 !

12. Another time dilation example
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

13. 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.

14. 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.

15. Relativity of length

16. 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

17. Relativity of length Stanley: distance from source to mirror
Stanley: distance from mirror back to source

18. Relativity of length

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

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