Transformation properties of electric and magnetic fields under three-dimensional Lorentz transformation equations

By extending the conventional one-dimensional Lorentz transformations, we obtain here the extended transformations between inertial frames moving with constant relative velocity in three dimensions of space. We assess the importance of the extended three-dimensional transformations in the calculations of physical quantities such as electromagnetic fields in the relativistic regime. The invariance of the D’Alembertian operator, the Lorentz invariance of the charge continuity equation, and the invariance of spacetime interval, which are crucial concepts in special relativity and the cornerstone of the Lorentz transformation, are proven under the extended transformations. In this highly intriguing topic, our specific goal is to present a brief but careful derivation of three-dimensional Lorentz transformations that maintain the property of Maxwell field equations under transformations and make the equation invariant.