ON THE STATIONARY EINSTEIN-MAXWELL FIELD-EQUATIONS

The stationary gravitational equations in the presence of the electromagnetic fields, outside charged gravitating sources, are investigated, (i) The action integral of Kramer-Neugebauer-Stephani (K.N.S.) is derived from the Hilbert action integral by using new variational techniques. (ii) It is shown that the classification scheme for the system of partial differential equations of general relativity depends on the coordinate system used. In particular, if orthogonal coordinates are chosen for the associated space then the system of Einstein-Maxwell equations is a hyperbolic one. (iii) The eigenvalues of the Ricci tensor of associated space are expressed in terms of the invariants of stationary electro-gravitational fields. It is proved that if these eigenvalues are equal then the fields must belong to the class of Peres-Israel-Wilson (PIW) solutions. (iv) The global integrability of some of the stationary Einstein-Maxwell equations and the consequent equilibrium conditions of the bodies" are investigated. (v) Boundary value problems for some of the field equations are pursued. It is proved that (ω≡ln|g44| is neither subharmonic nor superharmonic and the boundary value problem for this function does not yield a unique solution in general. A nontrivial solution of the stationary equations with ω≡0 is given. A special boundary value problem is explicitly solved. (vi) The PIW solutions are generated from the charged Kerr-Tomimatsu-Sato-Yamazaki (KTSY) solutions. The complex axially symmetric harmonic functions of these PIW solutions can be obtained from the real axially symmetric harmonic functions of the static Weyl class of electrovac solutions by a complex scale transformation of the coordinates. © 1980 American Institute of Physics."