Classical electromagnetism and special relativity

It gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another.

It motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity.

In fact, half of Einstein's 1905 first paper on special relativity, "On the Electrodynamics of Moving Bodies", explains how to transform Maxwell's equations.

In these two frames moving at relative velocity v, the E-fields and B-fields are related by:[2] where is called the Lorentz factor and c is the speed of light in free space.

Component by component, for relative motion along the x-axis v = (v, 0, 0), this works out to be the following: If one of the fields is zero in one frame of reference, that doesn't necessarily mean it is zero in all other frames of reference.

The components can be collected together as: Analogously for the charge density ρ and current density J,[6] Collecting components together: For speeds v ≪ c, the relativistic factor γ ≈ 1, which yields: so that there is no need to distinguish between the spatial and temporal coordinates in Maxwell's equations.

Authors usually derive magnetism from electrostatics when special relativity and charge invariance are taken into account.

13–6) uses this method to derive the magnetic force on charge in parallel motion next to a current-carrying wire.

[9] If the charge instead moves perpendicular to a current-carrying wire, electrostatics cannot be used to derive the magnetic force.

In this case, it can instead be derived by considering the relativistic compression of the electric field due to the motion of the charges in the wire.

A famous example of the intermixing of electric and magnetic phenomena in different frames of reference is called the "moving magnet and conductor problem", cited by Einstein in his 1905 paper on special relativity.

If a conductor moves with a constant velocity through the field of a stationary magnet, eddy currents will be produced due to a magnetic force on the electrons in the conductor.

Classical electromagnetic theory predicts that precisely the same microscopic eddy currents will be produced, but they will be due to an electric force.

[12] The laws and mathematical objects in classical electromagnetism can be written in a form which is manifestly covariant.

Here, this is only done so for vacuum (or for the microscopic Maxwell equations, not using macroscopic descriptions of materials such as electric permittivity), and uses SI units.

The above relativistic transformations suggest the electric and magnetic fields are coupled together, in a mathematical object with 6 components: an antisymmetric second-rank tensor, or a bivector.

This is called the electromagnetic field tensor, usually written as Fμν.

In matrix form:[13] where c the speed of light; in natural units c = 1.

There is another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacing E/c → B and B → −E/c, to get its Hodge dual Gμν.

The charge and current density, the sources of the fields, also combine into the four-vector called the four-current.

This short form of Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written using tensors.

The EM field tensor can also be written[14] where is the four-potential and is the four-position.