General covariance, Lorentz covariance, the Lorentz force, and the Maxwell equations

Abstract

We examine the problem of defining the electric and magnetic fields in the presence of a gravitational field and/or an accelerating coordinate system. The standard definitions given in an inertial coordinate system do not apply in a noninertial frame and further are not unique. We relate this result to the apparent freedom of choice of which components of the electromagnetic second-rank F tensor, (contravariant, covariant, mixed), one uses to express in terms of the electric and magnetic three-vector fields. This same ambiguity carries over in determining the three-vector form of the Maxwell equations in a noninertial frame. We discuss the various definitions given in standard textbooks and research literature and relate this ambiguity to the fact that the space-time coordinate xμ is not a four vector in general relativity. Using the example of the one-piece Faraday generator we show that through lowest order in 1/c the contravariant and covariant conventions yield the same integral forms of the Faraday flux law.