In classical electromagnetism, electric and magnetic fields are treated as distinct entities. But in the language of special relativity, they are deeply unified: the electric field E and magnetic field B are components of a single object — the electromagnetic field tensor, also known as the Faraday tensor. This page explores how electric and magnetic fields transform between inertial frames, and how these transformations emerge naturally from the tensor formalism.
Consider a long wire lying along the x-axis. Suppose:
Positive charges (ions) with linear density λ+ are moving rightward at speed v.
Negative charges (electrons) with equal linear density λ− are moving leftward at −v.
In the lab frameS, the net charge is zero: λ++λ−=0, so no electric field is produced. But both sets of charges are in motion, creating a total current:
I=λ+v+λ−(−v)=2λv.
This current produces a magnetic field around the wire. A test charge near the wire will experience a magnetic force, but no electric force in this frame.
Now switch to a new inertial frame S′ moving rightward with velocity v. In S′:
The positive charges (ions) are at rest.
The electrons now move leftward faster, at a speed (taking into account the relativistic velocity addition rule):
v′=1−c2v2−2v,
Due to the negative charges motion relative to us in this frame, we would also witness a length contraction between the charge spacing in λ−′ such that
λ−′=−γλ.
The result? Length contraction increases the magnitude of negative charge density so that now an electric field appears, in addition to a modified magnetic field.
Conclusion: Electric and magnetic fields mix under Lorentz transformations. What looks like a pure magnetic field in one frame becomes a mixture of electric and magnetic fields in another.
The mixing of E and B under Lorentz transformations becomes much more transparent in the language of tensors.
If Λνμ is the Lorentz transformation matrix from frame S to S′, then the electromagnetic tensor transforms covariantly as
F′μν=ΛαμΛβνFαβ.
This compactly and correctly mixes the electric and magnetic fields under any Lorentz boost or rotation. The transformation laws for E and B derived earlier are simply the component-wise manifestation of this tensor equation under a boost along x.
Although E and B themselves mix under boosts, there are two Lorentz invariant quantities formed from the Faraday tensor:
FμνFμν=2(B2−c2E2),
ϵμνρσFμνFρσ=−c8E⋅B.
These scalar quantities are the same in all inertial frames. If E2/c2>B2, there exists a frame where B=0; if B2>E2/c2, there is a frame where E=0. When E⋅B does not equal zero, there is no frame where E and B are orthogonal.
✅ Conclusion:
We have recovered the standard result for Ey′ under a Lorentz boost in the x-direction:
Ey′=γ(Ey−vBz)
This verifies that the abstract tensor transformation formula correctly reproduces the component transformation laws derived earlier from physical reasoning.
Electric and magnetic fields are not invariant under Lorentz transformations — they mix.
A purely magnetic field in one frame can look like a combination of electric and magnetic fields in another.
The proper mathematical object to describe this mixing is the Faraday tensor, which unifies E and B into a single relativistic entity.
Under Lorentz boosts, Fμν transforms like any rank-2 tensor: F′μν=ΛαμΛβνFαβ.
This unification sets the stage for the covariant formulation of Maxwell's equations.