Differential Geometry from Differential Equations

TL;DR

This paper demonstrates how third-order differential equations and pairs of PDEs can be used to systematically construct Lorentzian conformal metrics, including all solutions to Einstein's equations, revealing deep geometric structures from differential equations.

Contribution

It introduces a novel method linking differential equations to Lorentzian conformal geometry, generalizing previous work to include all four-dimensional Lorentzian metrics and Einstein solutions.

Findings

Constructed Lorentzian metrics from third-order ODEs and PDE pairs.

Identified conditions under which these metrics admit conformal Killing fields.

Showed that all four-dimensional Lorentzian metrics can be obtained through this framework.

Abstract

We first show how, from the general 3rd order ODE of the form z'''=F(z,z',z'',s), one can construct a natural Lorentzian conformal metric on the four-dimensional space (z,z',z'',s). When the function F(z,z',z'',s) satisfies a special differential condition of the form, U[F]=0, the conformal metric possesses a conformal Killing field, xi = partial with respect to s, which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space (z,z',z'') or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z_ss = S(z,z_s,z_t,z_st,s,t) and z_tt = T(z,z_s,z_t,z_st,s,t), with z_s and z_t, the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z_s,z_t,z_st,s,t). When the S and T satisfy equations analogous to U[F]=0, namely equations of the form M[S,T]=0, the 6-space then possesses a pair of conformal Killing fields, xi =partial with respect to s and eta =partial with respect to t which allows, via the mapping to the four-space of z, z_s, z_t, z_st and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations.