Space of initial data for self-similar Schwarzschild solutions

TL;DR

This paper constructs and analyzes the space of initial data for self-similar Schwarzschild solutions in general relativity, linking event horizons to attractors of a scalar parabolic equation on the sphere.

Contribution

It introduces a framework for understanding initial data of black holes via scalar parabolic equations and characterizes the event horizon as a global attractor within this setting.

Findings

Event horizon corresponds to a global attractor of the parabolic equation.

Constructed the space of initial data for self-similar Schwarzschild solutions.

Explored properties of attractors and their solutions.

Abstract

The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic equation on the sphere with various singularities, and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar Schwarzschild solutions. Those describe, for example, the initial data of black holes. We construct the space of initial data for such solutions, and show that the event horizon is related with global attractors of such parabolic equations. Lastly, some properties of those attractors and its solutions are explored.