8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Pole and Barn Paradox
L1 L2
v
L1> L2
P. 48-50, textbook
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Quiz
In class, we explained, from the perspective of an observer in
the laboratory, why muons were detected at a higher rate at the
sea level than expected. How would an observer in the frame
of the muons explain the same phenomenon?1.
The average lifetime of muons is longer due to time dilation;
2. The distance traveled is shorter due to length contraction;
3. The average speed of muons is greater than the speed of light;
4. There is something wrong with the experiment;
5.
It must be the aliens
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Lorentz Invariance
Assuming the origins of inertial frames Sand Scoincide only
at t = t= 0, when a light pulse is generated. According to
Einsteins second postulate, the wave front of the light pulse
should evolve spherical symmetrically around the respective
origin, as seen by an observer in either inertial frames.
22222
tczyx =++
In reference frame S,
22222 )()()()( tczyx !=!+!+!
In reference frame S,
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Mathematical Proof
[ ]
( )[ ]
( ) ( ) 2
2
22222
2
22
2
22
222
22222
2
2
222222
)(1)()()(1
2)(
)()()(2)(
)()()()(
tcc
vzyxc
v
cxv
cxvttc
zytvtvxx
cxvtczytvx
!"=!+!+!"
!+
!!+!=
!+!+!+!!+!
!+!=!+!+!+!
##
#
#
##
22222 )()()()( tczyx !=!+!+!
Using Lorentz transformation, we have
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Spacetime Interval
In general, we can define a spacetime intervalas follows:
])()()[()()( 22222 zyxtcs !+!+!"!=!
It can be shown, similarly, that the spacetime interval is Lorentz
invariant, or it has the same value in all inertial frames.
0)( 2 >!sIf , the interval is timelike. The spacetime intervalbetween any two causally connected events must be timelike,
since nothing can move faster than the speed of light.
0)( 2 =!sIf , the interval is lightlike.
0)( 2
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Lecture 6
Casuality
Spacetime regions: Casuality zone:
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Relativistic Velocity
)(
)(
2c
xvtt
zz
yy
tvxx
!+!=
!=
!=
!+!=
"
"
)(
)(
2c
vxtt
zz
yy
vtxx
!="
="
="
!="
#
#
Lorentz Transformation:
$& '=!
#$& '=
(x
uc
v
dt
dx
c
v
dt
td
22 11 ))
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Relativistic Velocity-Contd
2
2
22
2
2
1
1)(
c
vu
c
vuuvu
c
vu
ucvvu
dt
tdv
dt
td
td
xd
dt
tdv
dt
xd
dt
dxu
xxxxx
xx
x
!"!+"=#%
&( !
#$%&'( !)+"=
&'( "
+
"""
=#$%
&'( "
+
"==
**
**
21c
vu
vu
u
x
x
x !+
+!=
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Relativistic Velocity-Contd
2
2
2
2
2
2
22
2
2
1
1
1
1
1
1
c
vu
cv
dt
td
c
vu
cv
c
vu
c
vu
c
vu
vu
c
v
dt
td
x
x
xx
x
x
!+
"=
!
!+
"!"!+=
###
$
%
&&&
'
(
!+
+!"=!
)
)dt
td
td
yd
dt
dyu
y
!
!
!==
2
2
2
1
1
c
vu
c
v
uu
x
yy !+
"!=
2
2
2
1
1
c
vu
c
v
uu
x
zz !+
"!=
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Velocity Transformation
2
2
2
2
2
2
2
1
1
1
1
1
c
vu
c
v
uu
c
vu
c
v
uu
c
vu
vuu
x
zz
x
yy
x
x
x
!+
"
!=
!+
"
!=
!+
+!=
2
2
2
2
2
2
2
1
1
1
1
1
c
vu
c
v
uu
c
vu
c
v
uu
c
vu
vuu
x
zz
x
yy
x
x
x
!
!
="
!
!
="
!
!="
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Examples
1. Letcu
x=! we have
c
c
v
vc
c
vu
vu
u
x
x
x=
+
+=
!+
+!=
11 2
2. In a laboratory frame S, two particles move at the speed of
light in opposite directions. What is the velocity of a particle
as measured by an observer sitting on the other particle?
c c
x
S S
x
cvcux
=!= ,
c
cc
c
vu
vu
u
x
x
x!=
+
!!=
!
!="
111 2
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Relativistic Acceleration
2/1 cuv
vu
u
x
x
x
!+
+!=
td
uda
dt
dua
xx
xx
!
!=!= , and
( )( )( ) ( )22222
22
222
2
/1/1
/1
/1/1
cuv
ud
cuv
udcv
c
uvd
cuv
vu
cuv
uddu
x
x
x
x
x
x
x
x
xx
!+
!
=
!+
!"
=
!
!+
+!"!+
!=
#
tdcuvcxvdtddt x !!+=!+!= )/1()/( 22
""
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Relativistic Acceleration-Contd
( )323 /1 cuva
a
x
x
x
!+
!=
"
As for theyorzcomponent, the expression becomes cumbersome.
( )2
/1 cuv
uu
x
y
y!+
!=
"
( ) ( ) 2222 /1/1 cuvd
cuv
u
cuv
uddu x
x
y
x
yy
!
!+
!"
!+
!=
##
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Relativistic Acceleration-Contd
( ) ( )3
22
2
222
/1
/
/1 cuv
acuv
cuv
a
dt
dua
x
xy
x
yyy
!+
!!"
!+
!==
##
Now, imagine an inertial frame Sin which the object is at rest
instantaneously, i.e., its velocity is equal to 0, the acceleration
measured by an observer in Sis then
223 ,,
!!!
zz
y
y
x
x
aa
aa
aa
"=
"=
"=
8/11/2019 PHYS 342 - Lecture 6 Notes - F12
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Lecture 6
Reading Assignments
Chapter 2, 2-1 and 2-2
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