the Jacobian of Lorentz Transform is:
The Maxwell’s equations can be viewed like this for matching the order and sign:
Lets simplify (2). the cross product can be changed to Einstein notation.
where . Set a matrix F be :
If we let the index i be zero. ,
the equation (2) can be written like :
where and . each column represent 1 equation. notices that the minus sign before the rot in equation (2) is automatically included.
If we extend the j to be zero, define , . the minus sign perseveres the matrix F to be anti-symmetric. Then, the equation 1 will be absorbed. if we define a 4-current.
the equation (1) and (2) can be combined into:
Using similar method, by defining a matrix G, such that:
The transformation of field is simply use the Jacobian. Since the 4-current also need to transform, and the Zero 4-vector in equation G also. Thus