General structure of the solutions of the Hamiltonian constraints of gravity

TL;DR

This paper develops a general framework to classify and construct solutions to the Hamiltonian constraints in pure gravity, offering a systematic approach to understanding Einstein gravity solutions.

Contribution

It introduces a mathematical classification scheme for solutions of gravity constraints and provides complete families of solutions in certain cases.

Findings

Classified solutions into four distinct categories.

Constructed complete families of solutions in specific cases.

Suggested a systematic approach for studying Einstein gravity solutions.

Abstract

A general framework for the solutions of the constraints of pure gravity is constructed. It provides with well defined mathematical criteria to classify their solutions in four classes. Complete families of solutions are obtained in some cases. A starting point for the systematic study of the solutions of Einstein gravity is suggested.