Reciprocal Static Solutions of the Equations of the Gravitational Field

Abstract

This paper deals with reciprocal static line-elements, previously defined by the author, the condition that their Ricci tensors vanish being no longer imposed. If the indices i, k run from 1 to n with the exception of the fixed index a (the line-elements being static with respect to xa) a certain quantity $$ appears with the remarkable property that in a reciprocal transformation $$ is invariant, whilst $$ merely changes sign. $$ is closely related to the Hamiltonian derivative of the Gaussian curvature, so that the general results obtained may be applied to the field equations of General Relativity Theory, with n=a=4. $$ is then the total energy density; and formally every static distribution of matter has a " reciprocal distribution" associated with it. In particular, the equation of state of a distribution of fluid reciprocal to a distribution of fluid possessing a given equation of state may be obtained directly from the latter, i.e. without the solution of the field equations being known.