phys 342 - lecture 6 notes - f12

8/11/2019 PHYS 342 - Lecture 6 Notes - F12

1/15

Lecture 6

Pole and Barn Paradox

L1 L2

v

L1> L2

P. 48-50, textbook


2/15

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


3/15

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,


4/15

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


5/15

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


6/15

Lecture 6

Casuality

Spacetime regions: Casuality zone:


7/15

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 ))


8/15

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 !+

+!=


9/15

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 !+

"!=


10/15

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

!

!

="

!

!

="

!

!="


11/15

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


12/15

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

""


13/15

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

!

!+

!"

!+

!=

##


14/15

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

"=

"=

"=


15/15

Lecture 6

Reading Assignments

Chapter 2, 2-1 and 2-2

phys 342 - lecture 6 notes - f12

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