Selfgravitating nonlinear scalar fields

TL;DR

This paper studies the Einstein-scalar field equations in spherically symmetric, asymptotically flat spacetimes, proving local existence and conditions for global solutions, and providing an example of a naked singularity.

Contribution

It establishes local well-posedness for the Einstein-scalar field system with specific initial data spaces and explores conditions leading to global solutions or singularities.

Findings

Local existence and uniqueness of solutions

Global solutions exist if no central singularity forms

Explicit example of a naked singularity at the center

Abstract

We investigate the Cauchy problem for the Einstein - scalar field equations in asymptotically flat spherically symmetric spacetimes, in the standard 1+3 formulation. We prove the local existence and uniqueness of solutions for initial data given on a space-like hypersurface in the Sobolev $H_1\cap H_{1,4} $ space. Solutions exist globally if a central (integral) singularity does not form and/or outside an outgoing null hypersurface. An explicit example demonstrates that there exists a local evolution with a naked initial curvature singularity at the symmetry centre.