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Volume 12: Pages 457-467, 1999

A Representation of the Electromagnetic Field in the Presence of Curvature

Raymond J. Beach

Lawrence Livermore National Laboratory, L‐482, PO Box 808, Livermore, California 94551 U.S.A.

Classical treatments of electromagnetism in the presence of a gravitational field generally consist of embedding Maxwell's equations into the metrical continuum by writing them in a covariant form. Except for the coupling through the energy‐momentum tensor, this treatment gives the electromagnetic field equations an existence that is distinct from and independent of that of the gravitational field equations of general relativity. Here an alternative formulation is presented in which electromagnetic fields along with their sources are described in terms of the metric tensor g_μν and a continuous four‐component vector field u^λ. In the presence of ponderable mass, the continuous four‐component vector field u^λ is interpreted as the ordinary 4‐velocity describing the mass's motion. Because the proposed equations describing the electromagnetic field differ from the classical Maxwell equations, their ability to describe classical physics is shown by direct calculation. This is done using Einstein's classical gravitational field equations along with an energy‐ momentum tensor constructed using the proposed electromagnetic field equations. As an example, the asymptotic behavior of the presented equations is investigated under the restriction of axial symmetry. In this case, the equations lead naturally to the expected asymptotic descriptions of the gravitational and electromagnetic fields of a point charge superimposed on a magnetic dipole. Finally, the description of an electromagnetic plane wave in terms of the dynamical variables (g_μν, u^λ) in the weak field limit is presented.

Keywords: electromagnetism, gravity

Received: January 6, 1999; Published online: December 15, 2008