Classical Solutions of Higher Dimensional Einstein-Maxwell-Higgs System With Nontrivial Potential: Global Existence and Completeness

TL;DR

This paper proves the existence and completeness of global solutions to the higher dimensional Einstein-Maxwell-Higgs system with nontrivial potential, using contraction mapping and analysis of local mass and charge functions.

Contribution

It introduces a reduction to a single integro-differential equation and demonstrates global existence and spacetime completeness for small initial data in higher dimensions.

Findings

Existence of a unique fixed point for the reduced equation

Global classical solutions for small initial data

Spacetime completeness established through local mass and charge functions

Abstract

We study the Cauchy problem of higher dimensional Einstein-Maxwell-Higgs system in the framework of Bondi coordinates. As a first step, the problem is reduced to a single first-order integro-differential equation by defining a generalized ansatz function. Then, we employ contraction mapping to show that there exists the unique fixed point of the problem. For a given small initial data, we prove the existence of a global classical solution. Finally, by introducing local mass and local charge functions in higher dimensions, we also show the completeness property of the spacetimes.