It states that one of the following must hold for any nonzero bivector: the bivector has one "repeated" principal null direction; in this case, the bivector itself is said to be null, the bivector has two distinct principal null directions; in this case, the bivector is called non-null. Furthermore, for any non-null bivector, the two eigenvalues associated with the two distinct principal null directions have the same magnitude but opposite sign, , so we have three subclasses of non-null bivectors: spacelike: ν = 0 timelike : ν ≠ 0 and non-simple: ν ≠ 0 and , where the rank refers to the rank of the linear operator F. Physical interpretation The algebraic classification of bivectors given above has an important application in relativistic physics: the electromagnetic field is represented by a skew-symmetric second rank tensor field (the electromagnetic field tensor) so we immediately obtain an algebraic classification of electromagnetic fields. In a cartesian chart on Minkowski spacetime, the electromagnetic field tensor has components where and denote respectively the components of the electric and magnetic fields, as measured by an inertial observer (at rest in our coordinates). As usual in relativistic physics, we will find it convenient to work with geometrised units in which . In the "Index gymnastics" formalism of special relativity, the Minkowski metric is used to raise and lower indices. Invariants The fundamental invariants of the electromagnetic field are: . (Fundamental means that every other invariant can be expressed in terms of these two.) A null electromagnetic field is characterised by . In this case, the invariants reveal that the electric and magnetic fields are perpendicular and that they are of the same magnitude (in geometrised units). An example of a null field is a plane electromagnetic wave in Minkowski space. A non-null field is characterised by . If , there exists an inertial reference frame for which either the electric or magnetic field vanishes. (These correspond respectively to magnetostatic and electrostatic fields.) If , there exists an inertial frame in which electric and magnetic fields are proportional. Curved Lorentzian manifolds So far we have discussed only Minkowski spacetime. According to the (strong) equivalence principle, if we simply replace "inertial frame" above with a frame field, everything works out exactly the same way on curved manifolds. See also Electromagnetic peeling theorem Electrovacuum solution Lorentz group Petrov classification Notes References See section 25. Mathematical physics Electromagnetism Lorentzian manifolds Scientific