Applying covariant versus contravariant electromagnetic tensors to rotating media

Abstract

When the covariant form of Maxwell’s equations are applied to a rotating reference frame, a choice must be made to work with either a covariant electromagnetic tensor Fαβ or a contravariant electromagnetic tensor Fαβ. We argue that which tensor one chooses is ultimately dictated by whether one chooses to express the electric and magnetic fields in terms of a vector basis or in terms of a one-form basis, dual to the vector basis. We explain that when fields are expressed as one-forms, the covariant electromagnetic tensor is used; and when fields are expressed as vectors, the contravariant tensor is used. Using this formalism, we derive general field equations expressed in terms of vector and one-form fields in the rotating and laboratory frames when matter is present. Fields in the presence of matter are then related to those in a vacuum by using a covariant form of Minkowski’s constitutive equations, generalized to noninertial frames. Both vector and one-form field equations are used to derive the fields observed in the reference frame of a polarizable, permeable cylinder that rotates within an axially directed magnetic field. We find that the vector and one-form field equations both lead to predictions consistent with experimental results. We conclude that the choice between working with a covariant or contravariant electromagnetic tensor depends upon whether one chooses to express fields as vectors or as one-forms.