Electromagnetic Lorentz invariants

Retain signature $(+,-,-,-)$, $x^0=ct$, $F^{0i}=-E_i/c$, and $F^{ij}=-\epsilon^{ijk}B_k$. Contracting both tensor indices gives an invariant. Every independent antisymmetric entry appears twice because both index orders are summed. The time-space entries contribute negatively after index lowering, while the purely spatial entries contribute positively. The result compares magnetic field squared with electric field squared divided by c squared. Both terms have units of tesla squared, making the subtraction physically meaningful in SI.

There is a second independent invariant, the ordinary spatial dot product $\mathbf E\cdot\mathbf B$. It is unchanged by proper Lorentz transformations, including the boosts used here. Under a spatial reflection it changes sign because the electric field is a polar vector and the magnetic field an axial vector; it is therefore called a pseudoscalar. For our classification only whether it vanishes is needed. Do not replace either invariant by the sum of field magnitudes: the sum contributes to energy density and is observer dependent.

If a frame has no electric field, its second invariant is zero and its first invariant is positive unless the entire tensor vanishes. Therefore nonzero $I_2$ immediately rules out both a purely electric and a purely magnetic frame. When the fields are perpendicular and $I_1>0$, the magnetic field is strong enough that a subluminal boost can remove the electric field. The necessary boost points along the original Poynting direction. Substituting into the field transformation proves sufficiency as well as necessity for this crossed-field case.

For perpendicular fields with $I_1<0$, a different subluminal boost removes the magnetic field. The remaining electric magnitude follows from the invariant without repeating the transformation. These statements are local: a boost chosen from the fields at one event may not remove a component everywhere in a nonuniform field. A single global inertial frame requires the appropriate velocity to be the same throughout the region. Fields that are parallel with a nonzero dot product cannot be reduced to a single kind by any inertial boost.

When both invariants vanish, either the tensor is zero or the field is nonzero and null. A plane electromagnetic wave in vacuum is the familiar nonzero example: its electric and magnetic vectors are perpendicular and their magnitudes satisfy $E=cB$. Trying either elimination velocity now requires speed c. No inertial observer reaches that speed. The zero invariants do not indicate zero energy density, zero force on a charge, or zero radiation. They indicate a special algebraic class whose electromagnetic energy cannot be brought to rest by an inertial boost.

A useful workflow is to compute both invariants before and after any proposed component transformation. A mismatch diagnoses an error, often a missing gamma, an incorrect boost sign, or a lost factor of c. Matching invariants is necessary but not sufficient to validate every component: many different tensors share the same two numbers. The invariants classify fields coarsely and constrain possible frames; they do not determine spatial direction, the complete tensor, or its derivatives. Maxwell's equations still supply information that these local scalars cannot contain.