[Paper Review] Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter

Mihalis Dafermos, Alan D. Rendall|arXiv (Cornell University)|Jan 5, 2007

Cosmology and Gravitation Theories4 citations

TL;DR

This paper establishes strong cosmic censorship for surface-symmetric cosmological spacetimes with self-gravitating collisionless matter (described by the Einstein-Vlasov system), proving inextendibility of maximal Cauchy developments under generic conditions. It resolves the conjecture affirmatively for $k \leq 0$ or $\Lambda \geq 0$, and provides a geometric characterization of spacetime boundaries in the $k=1$, $\Lambda>0$ case, with the remaining obstruction tied to extremal black hole formation.

ABSTRACT

This paper addresses strong cosmic censorship for spacetimes with self-gravitating collisionless matter, evolving from surface-symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature $k$ of the symmetric surfaces and the cosmological constant $Λ$. With a suitable formulation, the question of strong cosmic censorship is settled in the affirmative if $Λ=0$ or $k\le0$, $Λ>0$. In the case $Λ>0$, $k=1$, we give a detailed geometric characterization of possible "boundary" components of spacetime; the remaining obstruction to showing strong cosmic censorship in this case has to do with the possible formation of extremal Schwarzschild-de Sitter-type black holes. In the special case that the initial symmetric surfaces are all expanding, strong cosmic censorship is shown in the past for all $k,Λ$. Finally, our results also lead to a geometric characterization of the future boundary of black hole interiors for the collapse of asymptotically flat data: in particular, in the case of small perturbations of Schwarzschild data, it is shown that these solutions do not exhibit Cauchy horizons emanating from $i^+$ with strictly positive limiting area radius.

Motivation & Objective

• To establish strong cosmic censorship in general relativity for spacetimes with self-gravitating collisionless matter under surface symmetry.
• To resolve the global dynamics of the Einstein-Vlasov system with compact initial data and surface symmetry.
• To determine whether maximal Cauchy developments are inextendible, ensuring deterministic evolution from generic initial data.
• To characterize the structure of spacetime boundaries, especially in the $k=1$, $\Lambda>0$ case, and identify obstructions to inextendibility.

Proposed method

• Analyzes the Einstein-Vlasov system in null coordinates using surface symmetry to reduce the problem to a 2+1-dimensional effective system.
• Employs the Hawking mass and conservation laws for particle current to control the evolution of matter and curvature.
• Applies spacetime integral estimates and Penrose diagram techniques to study the global structure and boundary behavior of solutions.
• Uses the extension theorem and inextendibility criteria to analyze radial null geodesics and boundary components.
• Applies local and higher-order estimates on Christoffel symbols and curvature to control regularity and singularities.
• Investigates the behavior of solutions near $r=0$, $r=\infty$, and horizon points to determine whether Cauchy horizons form.

Experimental results

Research questions

• RQ1 Under what conditions is the maximal Cauchy development of surface-symmetric initial data with collisionless matter inextendible?
• RQ2 What is the geometric structure of the future and past boundaries of spacetime in the $k=1$, $\Lambda>0$ case?
• RQ3 Can extremal Schwarzschild-de Sitter-type black holes form, and how do they obstruct strong cosmic censorship?
• RQ4 Do solutions with $k\leq0$, $\Lambda\geq0$ generically exhibit inextendible Cauchy developments?
• RQ5 Are there arbitrarily small perturbations of expanding homogeneous solutions that lead to horizon points or global horizons?

Key findings

• For $k\leq0$ or $\Lambda\geq0$, strong cosmic censorship holds: the maximal Cauchy development is inextendible, ensuring deterministic evolution.
• In the $k=1$, $\Lambda>0$ case, the spacetime boundary is geometrically characterized, with the only obstruction to censorship being the potential formation of extremal black holes.
• In the case of antitrapped initial data, strong cosmic censorship holds in the past for all $k$ and $\Lambda$, regardless of curvature or cosmological constant.
• For asymptotically flat collapse, black hole interiors do not develop Cauchy horizons emanating from $i^+$ with strictly positive limiting area radius under small perturbations of Schwarzschild data.
• In arbitrarily small neighborhoods of expanding homogeneous solutions, both horizon points and global horizons occur for solutions with non-empty interior, indicating that null boundary components are generic.
• The finiteness of the $\mathbb{S}^1$ factor in $\mathbb{S}^1\times\mathbb{S}^2$ topology limits perturbation techniques used in $\mathbb{R}\times\mathbb{S}^2$, but similar behavior persists for large initial lengths.