New Approach to Einstein's Empty Space Field Equations

Abstract

Tetrad formalism is used to derive a set of 36 scalar field equations which correspond to the ordinary field equations Rμν=0. The scalar equations are obtained by beginning with a given Petrov type of empty-space Riemann tensor and applying the Ricci identity to each of the tetrad vectors. The unknowns or field variables become the 24 Ricci rotation coefficients, the number of which can always be reduced by the Bianchi identities and occasionally by tetrad transformations which leave the form of the Riemann tensor invariant. The use of these scalar field equations is illustrated by their application to a degenerate case of Petrov type I. It is believed that by this method all possible solutions of this particular case have been found.