Covariant form

A covariant form is a mathematical expression of physical laws, which satisfy the principle of special relativity. It is expressed with a set of covariant operators, special relative scalars, vectors and tensors. Electromagnetic theory is founded on Maxwell's equations. They are not expressed with a set of the covariant operators, the special relative scalars, vectors and tensors. They are not expressed in the covariant form. It is not clear that they satisfy the principle of special relativity. They should satisfy the principle of special relativity, because special relativity comes from electromagnetic theory. The following is the covariant form of electromagnetic theory.

(1)
(2)

(3)

In the future, a textbook of the electromagnetic theory will be founded on equations (1) and (2). Because equations (1) and (2) are fewer than Maxwell's equations, they have a plain physical sense and it is easy to memorize them. Equation (1) is a wave equation, which means the potential Aⁱ propagates at light velocity,c in vacuum from jⁱ such as a source. Equation (2) is called the Lorenz condition. Partially differentiating equation (1) by xⁱ , from equation (2)

(4)

(5)

Surprisingly, Maxwell's equations are deduced from equations (1) and (2). Consequently, we may not memorize Maxwell's equations. Hence, we deduce Maxwell's equations from equations (1) and (2). The deduction is not necessary to understand the general relativity. However, it is meaningful. Because we can compare the electromagnetic theory and the general relativity. We understand the electromagnetic theory very well. However, we don't understand general relativity very well.