Electromagnetism in a Nonsymmetric Theory of Gravitation

In a previous work [1] a theory of gravitation based on a nonsymmetric metric was formulated (pure gravitation with no association of the antisymmetric part of the metric to the electromagnetic eld strength). The sources of the metric eld are the matter energymomentum tensor T and the matter fermionic particle number current density S . This is a conserved current with particle number fermionic charge F = R p gSdx, a constant which measures the coupling of the current to the geeometry. The theory was shown to be free of non-physical radiative negative-energy modes even when it is expanded about a Riemannian background, being outside of the class of ill-behaved nonsymmetric theories analyzed by Damour, Deser and McCarthy [2]. Only the symmetric part of the connection is present in the eld equations, making the theory as close as possible to general relativity. A solution of the eld equations for a spherically neutral point particle has been obtained [3], together with its implications for the motion of light and test particles. The theory is shown [3] to be consistent with all four general relativity solar tests. In the following we shall study the eld equations when electromagnetic elds are present. The electromagnetic eld Lagrangian that we shall use is the one given by Mann, Palmer and Mo at [4], which rescues the validity of the weak equivalence principle, which Will [5] showed to be violated if only a minimal electromagnetic coupling (same form as in general relativity) were adopted. This is because with such a minimal coupling the gravitational acceleration a of an electrically neutral body of total mass m composed of charged particles turns out to depend on its internal electrostatic energy Ee. Spedi cally, the accelaration is related to the one of gravity g by [6] a =g (1+ Ee=mc ), where depends on the metric coe ccients. Will then shows that in the case of a nonsymmetric theory with a minimal electromagnetic coupling is not zero. With the coupling proposed by Mann et al, as written in Eq. (1) below, turns out to be null, ensuring then the equivalence principle. The eld equations contain one free parameter, Z. However, the solution of the eld equations for a static and spherically symmetric eld turns out to be independent of Z. In Sec. II we establish the expression of the gravitationally modi ed Maxwell inhomogeneous equation and of the electromagnetic energy-momentum-stress tensor. In Sec. III we display the eld equations of the theory when the electromagnetic eld is present and in Sec. IV we determine the solution of the coupled eld equations for the case of a static and spherically symmetric point particle. We draw our conclusions in Sec. V.