On the derivation of several second order partial differential equations from a generalization of the Einstein equation

TL;DR

This paper generalizes Einstein's equation for complex line elements, deriving various second-order PDEs and analyzing their nonrelativistic limits, spatial variance effects, and dissipative properties.

Contribution

It introduces a novel generalization of Einstein's equation for complex line elements and derives new second-order PDEs with analysis of their properties.

Findings

Derived several second-order semilinear PDEs from the generalized Einstein equation.

Analyzed nonrelativistic limits of the PDEs.

Studied spatial variance effects and dissipative behaviors.

Abstract

A generalization of the Einstein equation is considered for complex line elements. Several second order semilinear partial differential equations are derived from it as semilinear field equations in uniform and isotropic spaces. The nonrelativistic limits of the field equations are also considered. The roles of spatial variance are studied based on energy estimates,and several dissipative or antidissipative properties are remarked.