N-dimensional geometries and Einstein equations from systems of PDE's

TL;DR

This paper demonstrates how n-dimensional Riemannian and Lorentzian metrics can be derived from specific PDE systems related to the Hamilton-Jacobi equation and explores their connection to Einstein equations.

Contribution

It introduces a novel approach linking systems of second-order PDEs to the construction of Einstein metrics in arbitrary dimensions.

Findings

All n-dimensional metrics can be generated from particular PDE systems.

Imposing Einstein equations constrains these PDE systems.

The work bridges PDE systems with geometric structures in general relativity.

Abstract

The aim of the present work is twofold: first, we show how all the $n$-dimensional Riemannian and Lorentzian metrics can be constructed from a certain class of systems of second-order PDE's which are in duality to the Hamilton-Jacobi equation and second we impose the Einstein equations to these PDE's.