A class of exact solutions of Einstein's field equations in higher dimensional spacetimes, d${\bm\geq 4}$: Majumdar-Papapetrou solutions

摘要

The Newtonian theory of gravitation and electrostatics admit equilibriumconfigurations of charged fluids where the charge density can be equal to themass density, in appropriate units. The general relativistic analog for chargeddust stars was discovered by Majumdar and by Papapetrou. In the present work weconsider Einstein-Maxwell solutions in d-dimensional spacetimes and show thatthere are Majumdar-Papapetrou type solutions for all ${\rm d} \geq 4$. It isverified that the equilibrium is independent of the shape of the distributionof the charged matter. It is also showed that for perfect fluid solutionssatisfying the Majumdar-Papapetrou condition with a boundary where the pressureis zero, the pressure vanishes everywhere, and that the $({\rmd}-1)$-dimensional spatial section of the spacetime is conformal to aRicci-flat space. The Weyl d-dimensional axisymmetric solutions are generalizedto include electric field and charged matter.