class23 lorentz 1

7/25/2019 Class23 Lorentz 1

1/23

PH424 Electromagnetic Theory I

Four Vector


2/23

!

ENDSEM Exam: 21/04/2015 : 9.30 12.30

M.Sc., M.Sc.+M.Tech, M.Sc.+Ph.D. IC3

B.Tech., DD (B.Tech.+M.Tech.) IC4


3/23

#

!Four Vectors

" Lorentz Transformation

!x = ! x"!ct

( )

" Let us define the four vectors in 4-dMinkowski space.

x

=!!

x

!

x = (ct,!

r ) = (ct,x,y,z), = 0,1, 2,3

!y = y

!z = z

!t = ! t""c

x#$% &

'(

x

y !y

!x

v

" These four vectors would transform under Lorentz transformation

as follows.

Contravariant


4/23

$

!Four Vectors

" Position four vectors: x

= (ct,!

r )= (ct,x,y,z), = 0,1, 2,3

! x="

!

x

!

,=

! !!" 0 0

!!" ! 0 0

0 0 1 0

0 0 0 1

"

#

$$$$$

%

&

'''''

" Lorentz Transformation:

!x = ! x"!ct( )!y = y

c !t = ! ct"!x( ) = (!)ct+ (!!")x + (0)y+ (0)z

= (!!")ct+ (!)x+ (0)y+ (0)z

= (0)ct+ (0)x + (1)y+ (0)z

!z = z = (0)ct+ (0)x + (0)y+ (1)z

c!t

!x

!y

!z

"

#

$$$$

%

&

''''

ct

x

y

z

!

"

####

$

%

&&&&


5/23

%

!Four Vectors

" Position four vectors: x = (ct,!

!

r ) = (ct,!x,!y,!z), = 0,1, 2,3

=

! !" 0 0

!" ! 0 0

0 0 1 0

0 0 0 1

!

"

#####

$

%

&&&&&

" Lorentz Transformation:

!x = ! x"!ct( )!y = y

c !t = ! ct"!x( ) ! c "t = !(ct)+!"(#x)+0(#y)+ 0(#z)

!" #x = !"(ct)+!("x)+ 0("y)+0("z)

!" #y = 0(ct)+ 0("x)+1("y)+ 0("z)!z = z ! " #z = 0(ct)+0("x)+ 0("y)+1("z)

c!t

" !x

" !y

" !z

#

$

%%%%

&

'

((((

ct

!x

!y

!z

"

#

$$$$

%

&

''''

Covariant

! x =

!"

!

x!


6/23

&

!Four Vectors

!!=

" ""# 0 0

""# " 0 0

0 0 1 0

0 0 0 1

#

$

%

%%%%

&

'

(

((((

!!! =

" "# 0 0

"# " 0 0

0 0 1 0

0 0 0 1

"

#

$

$$$$

%

&

'

''''

"

These two matrices are inversely related.

!!

!!"

!( )T

=#"

det !!

( ) =1= det !!!

( )" These Lorentz transformation is very much similar to the rotation

of a vector in 3-dimensions.


7/23

'

!Four Vectors

"

In rotation, the magnitude of a vector remains invariant.

xx

= c

2t2! r

2= x

2

"

Here, the question iswhat remains invariant underLorentz transformation?

"

We have seen that

is invariant under Lorentz transformation.

"

We can also define x2=!

" x

x

"=!

"x

x

"

!"

=

1 0 0 0

0 !1 0 0

0 0 !1 0

0 0 0 !1

"

#

$$$

$

%

&

'''

'

=!"


8/23

(

!Four Vectors

" It follows that these matrices are inverse of each other.

!"!"# =$#

!"

=

1 0 0 0

0 !

1 0 0

0 0 !1 0

0 0 0 !1

"

#

$

$$$

%

&

'

'''

=!"

"

The metric tensor allows us to raise or lower the Lorentz indices.

x=!

"x

"x

=!

"x

"


9/23

)

!Magnetism as a Relativistic Phenomenon

++++++++++

!!!!!!!!!!

++++++++++

!!!!!!!!!!

S !S

v

v v!

v+

u

u

d d

#

Frame S:

x

Line of +ve charge moving with velocity v

Line of ve charge moving with velocity v

Test Charge +qis moving with velocity u

( u < v )

Line of +ve charge moving with velocity v+

Line of ve charge moving with velocity v-

Test Charge +qis at rest

#

Frame S:


10/23

*+

!Magnetism as a Relativistic Phenomenon

++++++++++

!!!!!!!!!!

++++++++++

!!!!!!!!!!

S !S

v

v v!

v+

u

u

d d

#

Frame S:Analysis

x

#

Total current, I = 2!v

#

Because of current, there will be magnetic field

#Since the charge +qis moving in the presence of magnetic field,

it will experience Magnetic force.


11/23

**

!

Magnetism as a Relativistic Phenomenon

++++++++++

!!!!!!!!!!

++++++++++

!!!!!!!!!!

S !S

v

v v!

v+

u

u

d d

#

Frame S:Analysis

x

#

Since the charge +qis at rest, there will not be magnetic force.

#

Because of relativity, the velocities of +ve and ve chargesare different.

#Due to this there is net charge in the conductor, which will generateE.

#Since the charge +qis at rest in the presence of electric field,

it will experience Electrostatic force.


12/23*!

!


++++++++++

!!!!!!!!!!

++++++++++

!!!!!!!!!!

S !S

v

v v!

v+

u

u

d d

#

Frame S:Magnetic force, No Electrostatic force

x

#

Frame S:Electrostaticforce, No Magnetic force

#

Q: Can one derive Magnetic force from the Electrostatic force?


13/23*#

!


++++++++++

!!!!!!!!!!

!S

v+=

v!

u

1!vu

c2

v!

v+

u

d

#Velocity of +ve and ve charge

x

#

In Frame S, +ve particles will slow down and ve particles will

move fast.

v!

=

v+u

1+vu

c2

v+< v

!

#

There will be a net charge per unit time.

#

We need to calculate how the line charge density transforms.


14/23*$

!


++++++++++

v

x

#Say, +ve charges are moving with

velocity +v.

#One can say that in the charge

frame the charge is at rest and

the conductor is moving in the

opposite direction.

#Because of this the length of the conductor will be contracted.

#The line charge density will increase by the Lorentz contraction factor.

! = " !0

!=1

1! v c

( )

2

And !0is the line charge density when charged particle

and the conductor both are at rest.

!


15/23*%

!


++++++++++

!!!!!!!!!!

!S

!+= "

+!

0

v!

v+

u

d

#Frame S:

x

#

+ve line charged density

#

Electric field due to this line charge at d:

#

-ve line charged density

!!

= !"!

!0

!+=

1

1! v+ c( )

2

!!

=

1

1! v!

c( )2

!tot =!

++!

!

=!0 "

+!"

!( ) =!0 !2"uv

c21!u2 c2

"

#

$$

%

&

'' =

!2!uv

c21!u

2c2

E(d)=2!

tot

d

! i l i i i h


16/23*&

!


#Frame S:

#

Electric field due to this line charge at d: E(d)=2!

tot

d

#Force (electrostatic) on the test charge q: F!S =E q =!

4!uvq

dc2

1!u2c

2

#

We can transform this force to S:

FS =

1

!u

F!S =!

4!uvq

dc2

=!qu2(2!v)

dc2

"

#$%

&' =!qu

2I

dc2

"

#$%

&' =!q

u

c

2I

dc

"

#$%

&'

FS = !q!

u

c"

!

B#$% &

'( This is the force due to the magnetic field.

! F V


17/23*'

!

Four Vectors

" It follows that these matrices are inverse of each other.

!"!"# =$#

!" =

1 0 0 0

0

!1 0 0

0 0 !1 0

0 0 0 !1

"

#

$

$$$

%

&

'

'''

=!"

"

The metric tensor allows us to raise or lower the Lorentz indices.

x=!

"x

"x

=!

"x

"

! F V


18/23*(

!

Four Vectors

" In Euclidean space (3-d), the length of a vector is always positive.

1. Time like:

"

In Minkowski space (4-d), the length of a vectoris not necessarily positive.

x2 = xx

= c2t2

!

r2 > 0

2. Space like:

x2= x

x

= c

2t2! r

2< 0

3. Light like:

x2= x

x

= c

2t2! r

2= 0

t

x

Time-like

Space-like

! F V t


19/23*)

!

Four Vectors

1. Contravariant

" In Minkowski space (4-d), there are two kinds of vectors.

" IfAandBare two arbitrary vectors in M.S., the scalar (inner)

product of two vectors can be defined by

A.B =!"A

B

"= A

0B

0!

!

A.!

B

"

Both of these are Lorentz invariant.

2. Covariant

A.A =!"AA

!

= A0

( )2

!

!

A( )2

! F V t


20/23!+

!

Four Vectors

1. Contravariant

" Let us define derivatives (gradients) on this manifold.

!=

!

!x

=

1

c

!

!t,"

!

#

$

%&

'

()

"

DAlembertian operator is defined as

2. Covariant

! =

!

!x

=

1

c

!

!t,!

"

#

$%

&

'(

= !2

This is a Lorentz scalar.

=!"!

!"=

1

c2

!2

!t2"

!

#2$

%&

'

()

! F V t


21/23!*

!

Four Vectors

1. Contravariant

" Similarly, energy and momentum are combined into a four vector.

p

=

E

c,

!

p

!

"#

$

%&

"

The length of this four vector

2. Covariant p =E

c

,! !

p"

#

$%

&

'

p2=!

"p

p

"=

E2

c2 !

!

p2

"

#$

%

&'

"

This is also a Lorentz scalar. p2=m

2c2

! E2=

!

p2c2+m

2c4

mis the rest mass of the particle.

! C i f M ll E ti


22/23!!

!

Covariance of Maxwells Equations

1. Contravariant:

" Charge density and current density combine into a four vector form

J= c

!,

!

J

( )

"

The equation of continuity can be written in the covariant form

2. Covariant: J = c!,!

!

J( )

!J

= !

0J

0+!

iJ

i

"

Continuity equation is nothing but the vanishing of the four

divergence of the four vector current density.

=

!!

!t+

!

".!

J = 0

"

Four divergence is a scalar quantity. This implies that the equation of

continuity is invariant under Lorentz transformation.

! C i f M ll E ti


23/23

!

Covariance of Maxwells Equations

1. Contravariant:

" Scalar potential and Vector potential combine into a four potential

A=

!,

!

A

( )

"

Given a vector potential, we can construct a second rank

anti-symmetric tensor

2. Covariant: A = !,!

!

A( )

F! = !

A

! +!

!A

= !F

!

" SinceF!is an anti-symmetric tensor, it has only six independent

components.

" Let us find out each components.

class23 lorentz 1

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