Vacuum Einstein Equations in Terms of Curvature Forms

TL;DR

This paper presents a closed-form explicit representation of vacuum Einstein equations using curvature 2-forms, focusing on cases with non-zero conformal curvature invariants, and analyzes their algebraic and differential identities.

Contribution

It introduces a novel explicit formulation of vacuum Einstein equations in terms of curvature forms, expanding understanding of their algebraic and differential structure.

Findings

Derived a closed explicit form of Einstein equations using curvature 2-forms

Analyzed algebraic and differential identities among the equations

Discussed conditions with non-vanishing conformal curvature invariant

Abstract

A closed explicit representation of the vacuum Einstein equations in terms of components of curvature 2-forms is given. The discussion is restricted to the case of non-vanishing cubic invariant of conformal curvature spinor. The complete set of algebraic and differential identities connecting particular equations is presented and their consistency conditions are analyzed.