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Special Relativity - New Mexico Institute of Mining and ...
Transcript of Special Relativity - New Mexico Institute of Mining and ...
May 2005 New Mexico Tech Physics -- R.Sonnenfeld
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Special Relativity• Postulates
– The laws of physics are the same in every inertial frame of reference
– The speed of light in a vacuum is the same in every frame of reference and independent of the motion of the light-source
• Reference Frame– Inertial Reference Frame
• History and Experimental Support• “Beta” and “Gamma”• Space-time events• Time Dilation / Proper Time• Length Contraction• Simultaneity• Relativistic Velocity Addition• Galilean and Lorentz transformations• Spacetime interval / Relativistic invariants• Relativistic Force and Acceleration• Relativistic Momentum & Kinetic Energy• Twin Paradox• Energy and mass conversion
May 2005 New Mexico Tech Physics -- R.Sonnenfeld
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Special Relativity• Reference Frame
– A reference frame is an environment in which we make measurements. It’s important features are its origin, angle, and relative velocity.
– Length and time within a reference frame are simple. The difficulty comes in transforming between reference frames.
– Inertial Reference Frame – A reference frame that is moving at constant velocity. The ONLY type of reference frame that special relativity covers.
May 2005 New Mexico Tech Physics -- R.Sonnenfeld
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Coordinate Transformations
u
Galilean
Lorentz
2
'
''
'
( )
( )
x utyz
xy
ut
z
t xc
γ
γ
= −==
= −
'
'''
x x utyz
yzt t
= −===
May 2005 New Mexico Tech Physics -- R.Sonnenfeld
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Proper Time
• Proper Time (“Tau”)– If you time two events that are at rest relative to
you, you have measured the “proper time” between the events.
– If you are measuring the time between events occurring on a rocket flying by, you are NOT measuring the proper time between those events.
– The “proper time interval” is always the shortest time interval.
( )'t γ τ∆ = ∆proper time−
May 2005 New Mexico Tech Physics -- R.Sonnenfeld
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Space-time Events(time interval)
1 1
2 2
0, 0
0, 1 sec
x t
x t micro
= == =
'
'1
2
'2
2
2
2
2
( )
( )
(0 (0)) 0
(0 (0)) 0
(0
'
'
'
(1 sec))
(1 sec (0))
x utu
t xcuucu micro
x
t
x
umicr
x
otc
t
γ
γ
γ
γ
γ
γ
= −
= −
= − =
= − =
= −
= −
May 2005 New Mexico Tech Physics -- R.Sonnenfeld
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Classical Relativistic
2
0
0
0
20
0
30
2
2 2 2
'
2 20
( )
( )/
( 1)
''1
'
'
1 '
( )
u
u
x ut
t x u cL L
t t
p m v
K m c
F m a
F m a
v uv
u
x
vc
E p c m
t
c
γ
γγγ τγ
γ
γγ
γ
β ββ
β β
⊥ ⊥
= −
= −=
∆ = ∆ = ∆
=
= −=
=
+=
+
+=
+
− =
P P
'
'
0
2
( )( )
(1 2)
'x utt
L L
tp mv
K mvF maF ma
v u
xt
v
τ
= −==
∆ = ∆=
===
= +
May 2005 New Mexico Tech Physics -- R.Sonnenfeld
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Beta and Gammaandβ γ
8
8
6
2 10 / 0.66
1.5 10 / 1/ 2
3 10 / 0.01
vc
v m s
v m s
v m s
β
β
β
β
≡
= × =
= × =
= × =
22
1 11
1
0.6 1.250.8 1.660.99 7.10.9999 70.7
vc
γβ
β γβ γβ γβ γ
≡ =− −
= == == == =
Gamma vs. Beta (Relativistic "Weirdness factor")
1
3
5
7
9
11
13
15
0 0.2 0.4 0.6 0.8 1
Beta (v/c)
Gam
ma
(1/s
qrt
(1-
bet
a^2)
)