Developed Adomian method for quadratic Kaluza-Klein relativity

TL;DR

This paper extends the Adomian decomposition method to solve nonlinear matrix differential equations in quadratic Kaluza-Klein gravity, deriving general solutions for stationary cylindrically symmetric metrics.

Contribution

It introduces modifications to the Adomian method for matrix differential equations in GR and finds the most general solutions in closed form.

Findings

Derived the most general solutions for specific matrix differential equations

Modified the Adomian decomposition method for nonlinear matrix equations

Solutions terminate in closed forms due to scalar constraints

Abstract

We develop and modify the Adomian decomposition method (ADecM) to work for a new type of nonlinear matrix differential equations (MDE's) which arise in general relativity (GR) and possibly in other applications. The approach consists in modifying both the ADecM linear operator with highest order derivative and ADecM polynomials. We specialize in the case of a 4$\times$4 nonlinear MDE along with a scalar one describing stationary cylindrically symmetric metrics in quadratic 5-dimensional GR, derive some of their properties using ADecM and construct the \textit{most general unique power series solutions}. However, because of the constraint imposed on the MDE by the scalar one, the series solutions terminate in closed forms exhausting all possible solutions.