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Interior Weyl-type solutions to the Einstein-Maxwell field equations. (English) Zbl 1081.83510
Summary: Static solutions of the electro-gravitationalfield equations exhibiting a functional relationship between the electric and gravitational potentials arestudied. General results for these metrics are presented which extend previous work of Majumdar. Inparticular it is shown that for any solution of the field equations exhibiting such a Weyl-typerelationship, there exists a relationship between the matter density, the electric field density and the charge density. It is also found that the Majumdar condition can hold for a bounded perfect fluid only if the matter pressure vanishes (that is, charged dust). By restricting to spherically symmetric distributions of charged matter a number of exact solutions are presented in closed form which generalise the Schwarzschild interior solution. Some of these solutions exhibit functional relations between the electric and gravitational potentials different to the quadratic one of Weyl. All the non-dust solutions are well-behaved and, by matching them to the Reissner-Nordstrom solution, all of the constants of integration are identified in terms of the total mass, total charge and radius of the source. This is done in detail for a number of specific examples. These are also shown to satisfy the weak and strong energy conditions and many other regularity and energy conditions that may be required of any physically reasonable matter distribution.

MSC:
83C22 Einstein-Maxwell equations
83C15 Exact solutions to problems in general relativity and gravitational theory
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