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Special Relativity
2
1
Topics

Recap

Length Contraction

Cosmic Ray Muons

Spacetime

Summary
2
Recap


Postulates of Relativity
 Laws of physics are the same in all inertial
(that is, non-accelerating) frames
 The speed of light in vacuum is independent
of the motion of source and observer
Relativity of Simultaneity
 Events simultaneous for one observer need
not be simultaneous for another moving
relative to the first
3
Recap - Time Dilation
If tV is the elapsed time measured by a
clock moving at speed v relative to a
clock that measures an elapsed time of tR
and is at rest relative to an observer then
tR   tV
where

1
2
v
1 2
c
4
Recap - Time Dilation
Very Important Point (VIP):
By Principle 1, the time dilation effect
must be perfectly symmetrical:
If Betty and Ann are moving at a constant
velocity relative to each other, Betty sees
Ann’s clock run slower and Ann sees Betty’s
clock run slower
5
Length Contraction
dP = v tP
B
You: dY = v tY
Prof
A
 dP 
 tY 
Consider the
d

t

d




Y
Y
P
measurement of the
tP 
tP 


length of a table by You and
your professor who moves at speed v from A to B
6
Length Contraction
Prof
B
dP = v tP
A
You: dY = v tY
The ratio of your
elapsed time tY to the elapsed
time tP measured by your
professor is tY / tP = , so
 tY 
dY    dP
 tP 
dY   dP
7
Time Dilation & Length Contraction
Summary: Moving clocks tick more slowly
Moving lengths contract
Again, it is important to understand that
these statements must hold true for all
inertial observers: every inertial observer
will see moving clocks run more slowly and
moving lengths contracted longitudinally
8
Cosmic Ray Muons
~ 9 km
Time to the ground
9km / 300,000 km/s
~ 30 microsecond
For every 15
microseconds of
our clock, the
Mean muon lifetime
muon’s “clock” advances
~ 2 microsecond
only 1 microsecond!
9
Muon Decay – Detailed Example
Muons decay according to the
statistical law of radioactivity
t /
N(t)  N0e
where N0 is the initial number
of muons, N(t) is the number
remaining after an elapsed time
t and  is the mean muon lifetime
of 2.2 ms
10
Muon Decay – Detailed Example
The time to reach the ground
is about 15 times the mean muon
lifetime. Therefore, according
to Newtonian physics only a
fraction
15
e
 310
7
will reach the ground. But
because of time dilation and
length contraction the fraction
reaching the ground is predicted
to be about 1
e  0.4
11
Space + Time = Spacetime
The distinction between past, present
and future is only an illusion, even if
a stubborn one.
Albert Einstein
Space: is a (3-D) hyper-surface through
spacetime on which all events are
simultaneous, for someone
Time: is the direction perpendicular to
this hyper-surface
12
Event: A place at a given
time
Earth’s Time Axis
Spacetime: The set of
all events
worldline
3008 AD
D
C now
B
(t,x,y,z)
2508 AD
A
x
O
y
2008 AD
13
Summary



Clocks and Rulers
 Moving clocks run more slowly
 Moving lengths contract longitudinally
Spacetime
 Space is a hyper-surface of simultaneous
events and time is the direction
perpendicular to it
Worldline
 A trajectory through spacetime
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