Relativistic electromagnetism is a scientific idea that explains how electricity and magnetism behave when objects move very quickly. This idea is studied using electromagnetic field theory, which includes Coulomb's law and Lorentz transformations.
Electromechanics
After James Clerk Maxwell created a mathematical model to describe the electromagnetic field in 1873, scientists began to question how fields work. This discussion happened, for example, during a lecture by Lord Kelvin at Johns Hopkins University in 1884, which was remembered 100 years later.
The need for equations to stay the same when viewed by observers moving at different speeds led to the development of special relativity. This theory describes space and time as connected concepts in a four-dimensional structure, where light and radiation act as intermediaries. This geometric view of space and time helped explain electrical technologies, such as generators, motors, and lighting. The Coulomb force, which describes how electric charges interact, was expanded into the Lorentz force. This model enabled the creation of transmission lines, power grids, and the study of radio frequency communication.
In the early 20th century, Leigh Page worked to build a complete theory of electromechanics based on relativity. From a project plan in 1912 to his textbook Electrodynamics published in 1940, he explored how electric and magnetic fields interact as seen by observers moving relative to each other. In this framework, charge density, which in electrostatics describes how much charge is present in a region, becomes proper charge density and creates magnetic fields for moving observers.
Interest in teaching electromagnetism using relativity grew again in the 1960s, influenced by Richard Feynman’s textbook. Books like Classical Electromagnetism via Relativity by Rosser and Physics by Anthony French helped explain these ideas, including how proper charge density is visualized. One author noted, "Maxwell — Out of Newton, Coulomb, and Einstein."
The use of retarded potentials to describe electromagnetic fields created by moving charges is a key part of relativistic electromagnetism. This concept shows how changes in charge positions affect electromagnetic fields over time, following the rules of relativity.
Principle
Understanding how an electric field appears in different frames of reference is important for studying fields created by moving sources. In a specific situation, the sources that create the field are not moving relative to one of the frames. If we know the electric field in the frame where the sources are at rest, we can determine the electric field in another frame. To do this, we need to know the electric field at a specific point in space and time in the rest frame and the speed at which the two frames move relative to each other. This information is enough to calculate the electric field at the same point in the other frame. The electric field in the second frame depends only on the local value of the electric field in the first frame at that point, not on how the source charges are arranged. Therefore, the electric field fully describes the effect of distant charges.
Basic explanations of magnetism often use the Biot-Savart law, which describes the magnetic field produced by an electric current. An observer who is not moving relative to a group of static, free charges will not detect a magnetic field. However, an observer who is moving relative to the same group of charges will see a current and, as a result, notice a magnetic field. This means the magnetic field is the same as the electric field, but observed from a moving frame of reference.
Redundancy
The title of this article is unnecessary because all mathematical theories about electromagnetism already include ideas from relativity. As Einstein explained, the special theory of relativity was a way to organize the earlier work of Maxwell and Lorentz on electricity and magnetism. Maxwell's theory combines space and time into a four-dimensional structure. The finite speed of light and other constant motion lines were described using analytic geometry. The perpendicular relationship between electric and magnetic fields in space was expanded to include time using hyperbolic orthogonality.
In 1914, Ludwik Silberstein wrote a textbook called The Theory of Relativity, connecting the new geometry to electromagnetism. Faraday's law of induction influenced Einstein when he wrote in 1905 about the "reciprocal electrodynamic action of a magnet and a conductor."
However, the goal of this article is to describe an analytic geometry of spacetime and charges that provides a logical way to understand forces and currents. While such a clear path may not exist, differential geometry offers a method. The tangent space at an event in spacetime is a four-dimensional space that can be transformed using linear operations. Symmetries observed by electricians are expressed through linear algebra and differential geometry. Using a mathematical method called exterior algebra, a 2-form F can be created from electric and magnetic fields, along with its dual 2-form ★F. The equations dF = 0 and d★F = J (current) describe Maxwell's theory using differential forms.