Lorentz invariants of the EM field
The two scalar quantities — E·B and |E|² − c²|B|² — that every observer agrees on, regardless of their motion. Constructed as tensor traces from F^{μν}; classify the field as 'electric-like', 'magnetic-like', or 'null'.
Definition
The two Lorentz invariants of the electromagnetic field are the scalar quantities E·B and |E|² − c²|B|², both of which retain the same numerical value in every inertial frame. In tensor language they correspond to two independent scalar contractions of the field tensor F^{μν} with itself: ½ F^{μν} F_{μν} = c²|B|² − |E|²/c² (the kinetic part of the EM Lagrangian density up to a sign convention), and ¼ F^{μν} F_{μν} ∝ −E·B*/c (the dual-tensor contraction that vanishes on parity-respecting field configurations). Because they are scalars built from a tensor by index contraction, they are Lorentz-invariant by construction.
Operationally these two numbers classify the electromagnetic field into three types. (1) If E·B = 0 and |E|² > c²|B|², the field is "electric-like" and a frame exists where B = 0 entirely (a pure electrostatic field viewed from the right inertial observer). (2) If E·B = 0 and |E|² < c²|B|², the field is "magnetic-like" and a frame exists where E = 0 (a pure magnetostatic field). (3) If E·B ≠ 0, no boost can eliminate either field — the situation of a generic electromagnetic wave or a generic configuration with both static and dynamic components. The invariants therefore answer the natural question "is this configuration secretly just a Coulomb field?" with a frame-independent yes/no test, and they recover the §11.4 result (a magnetic-only force in the lab frame is generically electric-like in the right boosted frame) as a special case of the second classification.