The Einstein-Maxwell system in 3+1 form and initial data for multiple charged black holes

TL;DR

This paper formulates the Einstein-Maxwell system in a 3+1 framework, analyzes its hyperbolic properties, and develops a method for constructing initial data for multiple charged black holes using a puncture-like approach.

Contribution

It provides a detailed 3+1 formalism for Einstein-Maxwell equations, proves their symmetric hyperbolicity, and introduces a new method for initial data construction for charged black holes.

Findings

Maxwell's equations are shown to be symmetric hyperbolic in curved spacetime.

A puncture-like method for initial data of multiple charged black holes is developed.

Previous results are recovered as special cases within this framework.

Abstract

We consider the Einstein-Maxwell system as a Cauchy initial value problem taking the electric and magnetic fields as independent variables. Maxwell's equations in curved spacetimes are derived in detail using a 3+1 formalism and their hyperbolic properties are analyzed, showing that the resulting system is symmetric hyperbolic. We also focus on the problem of finding initial data for multiple charged black holes assuming time-symmetric initial data and using a puncture-like method to solve the Hamiltonian and the Gauss constraints. We study the behavior of the resulting initial data families, and show that previous results in this direction can be obtained as particular cases of our approach.