A new form of Einstein’s equations

Abstract

We derive the field equations of general relativity from a complex Lagrangian of particular simplicity. In this way we exhibit that Einstein’s vacuum equations are equivalent to the existence of certain covariantly constant vector- and bivector-valued two-forms. We discuss the implications this brings along for the exploration of new (hidden) symmetries as well as for the construction of new solutions. We even engage the equations’ simplicity for the construction of the most general (local) solution of the vacuum equations of general relativity and show that formally the general solution can be expressed in terms of Weyl’s conformal tensor and several arbitrary forms acting as ‘‘constants of integration.’’ In a 3+1 decomposition we demonstrate how the simple structure of the Lagrangian equations continues for the Hamilton equations, and compare the result with Ashtekar’s approach.