What is the Difference Between Lorentz Gauge and Coulomb Gauge?

The Lorentz gauge and Coulomb gauge are two different mathematical methods used to describe the electromagnetic field. The main difference between the two lies in the choice of gauge condition, which is a mathematical constraint used to simplify the equations of motion for the electromagnetic field. Here are the key differences between the two:

1. Relation to Space: The Lorentz gauge is related to the Minkowski space, which is a 4D (four-dimensional) real vector space, while the Coulomb gauge is related to the Euclidean space, which is a 3D (three-dimensional) space.
2. Lorentz Covariance: The Coulomb gauge is not Lorentz covariant, meaning that if a Lorentz transformation to a new inertial frame is carried out, a further gauge transformation has to be made to retain the Coulomb gauge condition. The Lorentz gauge, on the other hand, is Lorentz covariant, which makes it suitable for use in relativistic quantum field theories.
3. Far Field Radiation Problems: The Lorenz gauge is commonly used for far field radiation problems due to its simple form when no sources are present and its usefulness for problems with radiation. The Coulomb gauge is not as commonly used for these problems.

Despite these differences, it has been shown that the Lorenz gauge leads to exactly the same fields for radiation as the Coulomb gauge. The choice of gauge is often a matter of convenience, as it simplifies certain problems while still maintaining the physical consistentency.

Comparative Table: Lorentz Gauge vs Coulomb Gauge

The main differences between Lorentz Gauge and Coulomb Gauge are related to the spaces they are associated with and their applications in electromagnetism. Here is a summary of their differences in tabular form:

Both Lorentz Gauge and Coulomb Gauge are used to simplify Maxwell's equations, express potentials in terms of source terms, and are vital for solving boundary value problems in electrostatics and magnetostatics. However, the Lorentz Gauge maintains Lorentz invariance, which the Coulomb Gauge does not. The choice between the two gauges depends on the specific problem being solved and the desired simplification of the equations.