Maxwell’s Equations
are
Lorentz Invariant
Maxwell’s Equations
 
 B  0
   
 E  B  0
t
 
 E  
    
 B  E  j
t
Charge current obeys continuity eqn.
 
 E  
 
 B  0
   
 E  B  0
t
    
 B  E  j
t

    
  
   E         B  E  j
t
t
 

  j
t
charge
density

current
density




    (  u )  0
t
continuity equation,
or charge conservation
Charge current 4-vector
 
 E  
 
 B  0
   
 E  B  0
t
 

  j 
t
    
 B  E  j
t





jx 
jy 
jz  0
t
x
y
z
Define charge current 4-vector

J  (  , j )  (  , jx , j y , jz )
Divergence operator
is invariant under
Lorentz transformations

x
J  0
E&M potentials
 
 B  0
   
 E  B  0
t
 
 E  
    
 B  E  j
t
Define scalar and vector potentials,
 and

A.
  
  
B    A this works because   (  A)  0

  
E   A   this works because

t   
 
 
  [ E   A   ]  B  0 and   ( )  0
t
t
The Lorenz condition
 
 E  
 
 B  0
   
 E  B  0
t

How are  and A
Lorenz condition:
 

  A 
t
    
 B  E  j
t
related?




  Ax  Ay  Az  0
t
x
y
z
Define potential 4-vector

  ( , A)  ( , Ax , Ay , Az )
Its divergence is Lorentz invariant

x
  0
Wave equation for 
 
 B  0
   
 E  B  0
t
 
 E  
    
 B  E  j
t
Use the first inhomogeneous eqn to show that  obeys
the wave equation, travelling at c.

 [

  
E   A  
t
 

 A   
t
] 
1  2
2




2
2
c t

Wave equation for A
 
 B  0
   
 E  B  0
t
 
 E  
    
 B  E  j
t

Use the second inhomogeneous eqn to show that A obeys
the wave equation, travelling at c.
    
 B  E  j
t
  
B   A
 

  A  0
t

 
1  A
2
 2  A  j
2
c t
2
Maxwell’s equations in
Lorentz invariant form
 
 E  
 
 B  0
   
 E  B  0
t

2
t
2
(
   

2
2
t
2
    
 B  E  j
t

 A
2
t 2
 2 )  
 
 A  j
2
J
Maxwell’s Equations
 
 E  
 
 B  0
   
 E  B  0
t
Used
Used
Used
    
 B  E  j
t

 and j to make a charge current 4-vector, J



E and B to make a scalar and vector potentials,  and A

 and A to make a 4-potential, 
(
2
t
2
 )  
2
J