Note concerning gravitation and electromagnetism

Abstract

The Maxwell equations for gravitational fields previously assumed by Sciama are derived from elementary considerations. The Lagrangian for a gravitating mass in a non-inertial coordinate system yields equations of motion leading to force definitions for a gravitational field intensity and a gravitational induction field. The non-inertial velocity of the coordinate system plays the role of a vector potential contributing to the generalized momenta of bodies moving in the system. A Lagrangian density constructed from the force-defined fields then lead to the source definitions of gravitational fields. It is found that positive field energy densities require repulsive gravitational forces, whereas attractive forces imply the violation of the conservation of energy. This paradox is resolved by representing gravitational quantities as pure-imaginary entities. Thus characterized, the equations which define gravitational fields become identical to Maxwell's equations but are pure-imaginary. This suggests a combined representation for gravitational and electromagnetic fields which, in covariant form, indicates both the well known equivalence of mass and energy and a possible equivalence of charge and energy. From orthogonality considerations, it is conjectured that this latter energy is gravitational, and that, whereas gravitational fields interact with electromagnetic energy, electromagnetic fields interact with gravitational energy.