Differential form approach for stationary axisymmetric Maxwell fields in general relativity

Abstract

A formulation for stationary axisymmetric electromagnetic fields in general relativity is derived by casting them into the form of an anisotropic fluid. Several simplifications of the formalism are carried out in order to analyze different features of the fields, such as the derivation of electromagnetic sources for the Maxwell field in the form of thin layers, construction of new solutions, and generation techniques.

Magnetic surfaces in stationary axisymmetric general relativity

“…When A is zero, that is in the static case, this formula is pretty similar to the classical one (12b), corrected by metric factors. Its first term also coincides in the static case with the formula introduced in [10], where the contribution of the moment density arising from the layer of monopoles (second term) was not taken into account.…”

“…so that the electromagnetic Faraday, F , and Maxwell, 4 * F , forms have a simple expression after introducing the Hodge duality ( * θ 2 = θ 3 ) in the space orthogonal to the Killing fields [10]. Taking into account Cartan's structure equations, expressing the torsionfree connection coefficients as one-forms, a, w, s, b and ν, (cfr.…”

“…Taking into account Cartan's structure equations, expressing the torsionfree connection coefficients as one-forms, a, w, s, b and ν, (cfr. [10], [12] for the physical meaning of these forms)…”

Non-regular Potentials and Sources for Static Axisymmetric Electrovacs

“…After the fashion of [3], [6], [7], an asymptotically cartesian function Z will be introduced and will be taken to satisfy the same differential equation as V :…”

“…The electric charge distribution is easily obtained from the integration of the Maxwell equations [2], but electric and magnetic moment distributions cannot be calculated in that way. In order to achieve that goal a generalization of the approach followed in [3], [4] will be attempted. In these references magnetic and electric sources for static electrovacs were constructed thanks to the introduction of an asymptotically cartesian coordinate Z, provided that the electric field was not monopolar.…”