[Paper Review] Vacuum Counterexamples to Cosmic Censorship in AdS: I

Toby Crisford, Gary T. Horowitz+1 more|arXiv (Cornell University)|May 16, 2018

Cosmology and Gravitation Theories3 citations

TL;DR

This paper constructs vacuum, stationary solutions of 4D Einstein gravity with negative cosmological constant ($\Lambda < 0$) that asymptotically approach a boundary with differential rotation. It demonstrates that beyond a critical rotation rate, curvature diverges, providing vacuum counterexamples to weak cosmic censorship in asymptotically anti-de Sitter spacetime, valid for both zero and nonzero temperature and with compact or noncompact boundaries.

ABSTRACT

We consider vacuum solutions of four dimensional general relativity with $\Lambda < 0$. We numerically construct stationary solutions that asymptotically approach a boundary metric with differential rotation. Smooth solutions only exist up to a critical rotation. We thus argue that increasing the differential rotation by a finite amount will cause the curvature to grow without bound. (In paper II of this series, we will study the time dependent problem and confirm this expectation.) This provides vacuum counterexamples to weak cosmic censorship in anti-de Sitter spacetime. These results hold for both zero and nonzero temperature, and both compact and noncompact boundaries.

Motivation & Objective

• To investigate the existence of stationary vacuum solutions in four-dimensional anti-de Sitter (AdS) spacetime with negative cosmological constant.
• To examine whether differential rotation on the boundary can lead to curvature singularities.
• To test the validity of weak cosmic censorship in vacuum AdS spacetimes under varying boundary conditions.
• To determine whether such singularities arise regardless of temperature or boundary topology (compact/noncompact).

Proposed method

• Numerical construction of stationary vacuum solutions to Einstein's equations in 4D with $\Lambda < 0$.
• Imposition of boundary conditions with differential rotation on the asymptotic spacetime metric.
• Use of numerical relativity techniques to solve the Einstein equations under these boundary conditions.
• Analysis of curvature invariants to detect the onset of singularities as rotation increases.
• Identification of a critical rotation rate beyond which no smooth solutions exist.
• Extension of results to both zero-temperature and finite-temperature boundary conditions, and to compact and noncompact boundaries.

Experimental results

Research questions

• RQ1 Can vacuum, stationary solutions of 4D Einstein gravity with $\Lambda < 0$ be constructed that asymptotically exhibit differential rotation on the boundary?
• RQ2 Does increasing the differential rotation beyond a critical value lead to unbounded curvature growth in vacuum AdS spacetime?
• RQ3 Are these curvature divergences robust across different boundary topologies (compact vs. noncompact) and temperature regimes?
• RQ4 Do these divergences constitute violations of weak cosmic censorship in vacuum AdS spacetimes?
• RQ5 Is the critical rotation rate independent of the boundary's temperature or spatial structure?

Key findings

• Smooth vacuum solutions with differential rotation on the boundary exist only up to a finite critical rotation rate.
• Beyond this critical rotation, curvature invariants grow without bound, indicating the formation of a singularity.
• The divergence of curvature occurs regardless of whether the boundary is at zero or nonzero temperature.
• The result holds for both compact and noncompact spatial boundary topologies.
• These findings constitute vacuum counterexamples to weak cosmic censorship in asymptotically AdS spacetime.
• The mechanism for singularity formation arises purely from boundary conditions and vacuum dynamics, without matter sources.