M'&10 3fo. In fact, the situation is worse than this:The mass of an eventual condensation is M(M'AB/1V, or 3f(10 Mo. We therefore come to the conclusion that the time available is too short for the formation of baryon inhomogeneities, and that if inhomogeneities are the explanation of large-scale aggregations of matter, then they are an integral part of the universe from the beginning of its expansion.The case for density Quctuations is certainly no better. For a galactic mass of %=10" Mo, we have from (45), p«10 ' g™ (since M'))M), and the instability hypothesis leads to the improbable conclusion that the foundations of galactic structure are laid down in the radiation era. If we accept the primordialstructure hypothesis and assume that density inhomogeneities exist from the earliest moments, we are still in difhculty because small amplitude fluctuations are not ampli6ed, 30 and the density inhomogeneity is of the order 10' times greater than the required compositional inhomogeneity."Nor are they dissipated. Any dissipation mechanism (such as Misner's (Ref. 4) g in the lepton or hadron eras causes only the very shortwavelengths of ) (ct to decay.The 6eld equations governing the gravitational Geld of a uniformly rotating axially symmetric source are reformulated in terms of a simple variational principle. The new formalism affords a concise unified derivation of the solutions discovered by Weyl and Papapetrou, and permits a simple derivation of the Kerr metric in terms of prolate spheroidal coordinates. More complex solutions are identi6ed by applying perturbation theory.

The charged C-metric is transformed into another exact solution of the Einstein–Maxwell field equations corresponding to a massive charged particle accelerated by an electric field. When the appropriate equations of motion are satisfied, the nodal singularity associated with the C-metric disappears.

The coupled Einstein-Maxwell 6eld equations are reformulated in terms of a pair of complex functions which have especially simple forms in the case of known axially symmetric stationary solutions. The formalism affords, in particular, a simple derivation of a solution previously guessed by Newman et al.

We present a general procedure for transforming asymptotically flat axially symmetric solutions of the Einstein–Maxwell equations into solutions resembling Melvin’s magnetic universe. Specific applications yield metrics associated with black holes in a magnetic universe. It is hoped that these solutions will be of interest to astrophysicists studying gravitational collapse in the presence of strong magnetic fields.

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