Static Gravitational Fields. I. Eight Theorems

Abstract

In a fresh look on static gravitational fields in general relativity, eight new theorems have resulted. Also in the process of deriving theorems a new solution has emerged. In the first of these theorems an invariant necessary integral condition for the existence of a solution has been derived. Physically this condition corresponds to the equilibrium of matter. In the second theorem a scalar condition has been found which implies the flatness of the static gravitational universe. In the third theorem, it has been proved that there cannot occur any group of motion along ``the lines of forces.'' In Theorems 5 and 6, the questions of whether the spatial part of a static gravitational universe can be Einstein, projectively flat, or Stäckel are investigated. In the seventh theorem, the static gravitational field equations have been reduced to the geometrized equations in a spatial universe. In the last theorem, all conformastat gravitational universes have been found. One of these is the universe due to ``an infinite plate,'' and this is a new solution.