1. SPECIAL RELATIVITY
-Postulates of Special Relativity -Relativity of time –> time dilation -Relativity of length –> length contraction
© 2005 J. F. Becker

2. An event takes place at a certain time and place as measured in a frame of reference, or coordinate system (x, y, z, t)
INERTIAL FRAME OF REFERENCE: Newton’s law of inertia is valid in the frame of reference. The acceleration of a body is zero when measured in the coordinates.
C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.
C 2005 J. F. Becker

3. Two inertial frames of reference moving with constant relative velocity vvt
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4. POSTULATES (ASSUMPTIONS) of Special Relativity
The relativity postulate: The laws of physics are the same in every inertial reference frame.
The speed of light postulate: The speed of light in a vacuum, measured in any inertial reference frame, always has the same value of c, no matter how fast the source of light and the observer are moving relative to each other.
C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.
C 2005 J. F. Becker

5. The speed of light postulate
Both the person on the truck and the observer on the earth measure the speed of light to be c, regardless of the speed of the truck.
The speed of light postulate
C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.
C 2005 J. F. Becker

6. MOVING CLOCKS TICK SLOW! (Time dilation)
We will see some mind-boggling effects of the theory of special relativity, such as:
MOVING CLOCKS TICK SLOW! (Time dilation)
MOVING RULERS APPEAR CONTRACTED! (Length contraction)
But first, let’s consider the concept of “simultaneity”
(two events happening at the same time)
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7. SIMULTANEITY IS RELATIVE
SIMULTANEITY IS RELATIVE! Whether two events are simultaneous depends on the frame of reference. Two lightning bolts strike the railroad car and ground at each end. u u u u
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8. Dt > Dto and Dt indicates a slow clock!
TIME DILATION The proper time interval Dto between two events is the time interval measured by an observer who is at rest relative to the events and views them as occurring at the same place.
An observer who is in motion with respect to the events and who views the events as occurring at different places measures a dilated (expanded) time interval Dt. The dilated time interval is greater than the proper time interval (Dt > Dto) according to the time-dilation equation: Dt = Dto / (1 – u2/c2) ½
Dt > Dto and Dt indicates a slow clock!
V = relative speed between the two observers. Clocks that appear to be in motion tick slower.

9. Time dilation: Dt = Dto / (1 – u2/c2) ½
A spacecraft speeds past Earth at a constant speed of u = 0.92c and the astronauts measure the time between ticks of the spacecraft clock to be 1.0 sec. (Dto = 1.0 sec.).
What time interval do observers on Earth measure? Dt = ?
Dt = Dto / [1 – (0.92)2]1/2 = Dto / =2.6 Dto
Dt = 2.6 sec. MOVING CLOCKS TICK SLOW!
V = relative speed between the two observers. Clocks that appear to be in motion tick slow.
C 2005 J. F. Becker

10. Velocity = distance / time
A light clock: One “tick” is the time interval it takes for the light pulse to travel the round trip distance / speed of light or (2 d = c x t)
time interval = 2 d / c
For stationary clock: Dto = (2 d) / c
For moving clock: Dt = (distance) / c d
Velocity = distance / time
C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.
C 2005 J. F. Becker

11. The path observed by Stanley is longer than d.
Time dilation
u Dt/2 L L d u
u Dt
Dto Dt
L = [ d2 + (u Dt/2) 2 ] ½
The path observed by Stanley is longer than d.
C 2005 J. F. Becker

12. So c Dt = 2 L = 2 [ (c Dto / 2)2 + (u Dt / 2)2 ] ½
Mavis measures (proper) Dto (= 2 d /c). (b) Stanley measures Dt. The path observed by Stanley is longer:
2 L = 2 [ (d2 + (u Dt / 2)2 ] ½
So c Dt = 2 L = 2 [ (c Dto / 2)2 + (u Dt / 2)2 ] ½
(Dt)2 = (2 L/c)2 = (2/c)2 [ (c Dto/2)2 + (u Dt/2)2]
(Dt)2 = [ (Dto)2 + (u Dt/c)2]
(Dt)2 - (u Dt/c)2 = [ (Dto) 2 ]
(Dt)2 { 1 - (u /c)2 } = [ (Dto) 2 ]
Time dilation: Dt = Dto / {1 – u2/c2} ½
A consequence of the speed of light postulate.

13. How far does a muon travel before it disintegrates?
Experimental verification of time dilation – Muons are particles observed on earth after being created in the upper atmosphere when cosmic rays from the Sun collide with atoms in our atmosphere. The muon quickly decays into an electron and a neutrino particle. These muons travel toward earth with speed u = c (Lifetime of a muon at rest = 2.2 (10)-6 s.)
How long does one of these muons live according to an observer on earth?
How far does a muon travel before it disintegrates?
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14. Verification of time dilation – How long does one of these muons live according to an observer on earth? The two events are the generation and disintegration of the muon. When the muon is at rest the events take place at the same place so the lifetime is the proper time interval Dto = 2.2 (10)-6 s. The muon moves at u = c so Earthlings measure a dilated time interval Dt.
Dt = Dto / (1 – u2/c2) ½
Dt = 35 (10)-6 s.
C 2005 J. F. Becker

15. Verification of time dilation –
How far does a muon travel before it disintegrates?
If we mistakenly neglected relativistic effects the calculated distance would be d = ut =0.998 [300(10)6 m/s] 2.2(10)-6s =659 m (distance too short; muons never reach earth!)
An Earthling measures a distance of d = u Dt =0.998 [300(10)6 m/s] 35(10)-6s
d = u Dt = 10,500 m
(the distance to top of atmosphere)
C 2005 J. F. Becker

16. Length contraction – As measured by an Earthling the Earth-to-star distance is Lo, the proper length as measured by an observer at rest with respect to the ends of the ruler, and the time to make the trip is Dt. An Astronaut measures the distance to be L & the time Dto.
Voyage to a star! u
Dto Dt u u Lo L
C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.
C 2005 J. F. Becker

17. Length contraction: u = 0.95 c = distance/time
The relative velocity is: u = L / Dto = Lo / Dt L = (Lo / Dt) Dto
And substituting Dt = Dto / (1 – u2/c2) ½
we get L = (Lo / Dt) Dt (1 – u2/c2) ½
L = Lo (1 – u2/c2) ½
Lo = the proper length between two points as measured by an observer at rest with respect to the two points, i.e., a stationary ruler.
MOVING RULERS APPEAR CONTRACTED!
C 2005 J. F. Becker

18. Length contraction from the muon’s frame:
Recall the muons speeding toward earth at u = c. In the muon’s reference frame it lives Dto = 2.2 (10)-6 s and it measures the distance to earth as L. But, as measured by Earthlings the distance is Lo = proper length = 10,500 m. Since moving rulers are contracted the muon measures L = Lo (1 – u2/c2) ½ L = 10,500 m. (1 – u2/c2) ½ = 632 m., a short enough distance to earth to cover in 2.2 ms.
Lo = the proper length between two points as measured by an observer at rest with respect to a ruler (a stationary ruler).
C 2005 J. F. Becker

19. MOVING CLOCKS TICK SLOW!
Time dilation:
MOVING CLOCKS TICK SLOW!
Length contraction: MOVING RULERS APPEAR CONTRACTED!
C 2005 J. F. Becker

20. Graph shows how the factor (1 – u2/c2) ½
increases as the relative speed approaches c.
c ~ 300 (10)6 km/s
c ~ 670 (10)6 mi/hr u
C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.

21. EQUIVALENCE OF MASS AND ENERGY
Total energy of an object: E = mc2 / (1 – u2/c2) ½
Rest energy of an object: Eo = mc2 / (1 - 0) ½ Eo = mc2
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22. REVIEW
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23. Length contraction – light pulse reflected off mirror at end of ruler

26. (a) Astronaut measures (proper) Dto. (b) Earthling measures Dt
(a) Astronaut measures (proper) Dto. (b) Earthling measures Dt. The path observed by Earthling is longer: s = [ D2 + L2 ] ½ = [ (D2 + (u Dt/2) 2 ] ½ .
L = u Dt/2
C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.