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The possible stationary axisymmetric electromagnetic fields in a vacuum cavity between an initially neutral black hole and a surrounding plasma shell are investigated. It is shown that such fields must be near "uniform" (in a sense defined in the paper) and that the flux of the magnetic field across one half of the surface of a neutral hole (of fixed mass) decreases as the angular momentum of the hole increases.

We show that for a wide class of field equations the orbits of the isometry group defining axial symmetry and stationarity admit orthogonal 2-surfaces, The field equations covered by this result include those of a perfect fluid.

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We analyze the general two-dimensional hyperbolic differential equation of second order by means of a substitution method. Our main interest lies in the support of the solutions, i.e., in an answer to the question: under what circumstances can a signal be transmitted along null rays? It turns out that, in general, a signal spreads, i.e., fills the entire future of an event. However, reasonably large classes of differential equations do permit nonspreading (characteristic propagation) solutions. As examples it is shown that multipole solutions of the flat space-time scalar wave equation and Maxwell equations fall into the non-spreading class, whereas multipole solutions of the corresponding equations in a curved Schwarzschild background always show spreading (or continuous reflection).

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